Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy: Proceedings of the Second International Conference, MMCITRE 2021 9811659516, 9789811659515

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Advances in Intelligent Systems and Computing 1405

Manoj Sahni José M. Merigó Ritu Sahni Rajkumar Verma   Editors

Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy Proceedings of the Second International Conference, MMCITRE 2021

Advances in Intelligent Systems and Computing Volume 1405

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland Advisory Editors Nikhil R. Pal, Indian Statistical Institute, Kolkata, India Rafael Bello Perez, Faculty of Mathematics, Physics and Computing, Universidad Central de Las Villas, Santa Clara, Cuba Emilio S. Corchado, University of Salamanca, Salamanca, Spain Hani Hagras, School of Computer Science and Electronic Engineering, University of Essex, Colchester, UK László T. Kóczy, Department of Automation, Széchenyi István University, Gyor, Hungary Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX, USA Chin-Teng Lin, Department of Electrical Engineering, National Chiao Tung University, Hsinchu, Taiwan Jie Lu, Faculty of Engineering and Information Technology, University of Technology Sydney, Sydney, NSW, Australia Patricia Melin, Graduate Program of Computer Science, Tijuana Institute of Technology, Tijuana, Mexico Nadia Nedjah, Department of Electronics Engineering, University of Rio de Janeiro, Rio de Janeiro, Brazil Ngoc Thanh Nguyen , Faculty of Computer Science and Management, Wrocław University of Technology, Wrocław, Poland Jun Wang, Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, Hong Kong

The series “Advances in Intelligent Systems and Computing” contains publications on theory, applications, and design methods of Intelligent Systems and Intelligent Computing. Virtually all disciplines such as engineering, natural sciences, computer and information science, ICT, economics, business, e-commerce, environment, healthcare, life science are covered. The list of topics spans all the areas of modern intelligent systems and computing such as: computational intelligence, soft computing including neural networks, fuzzy systems, evolutionary computing and the fusion of these paradigms, social intelligence, ambient intelligence, computational neuroscience, artificial life, virtual worlds and society, cognitive science and systems, Perception and Vision, DNA and immune based systems, self-organizing and adaptive systems, e-Learning and teaching, human-centered and human-centric computing, recommender systems, intelligent control, robotics and mechatronics including human-machine teaming, knowledge-based paradigms, learning paradigms, machine ethics, intelligent data analysis, knowledge management, intelligent agents, intelligent decision making and support, intelligent network security, trust management, interactive entertainment, Web intelligence and multimedia. The publications within “Advances in Intelligent Systems and Computing” are primarily proceedings of important conferences, symposia and congresses. They cover significant recent developments in the field, both of a foundational and applicable character. An important characteristic feature of the series is the short publication time and world-wide distribution. This permits a rapid and broad dissemination of research results. Indexed by DBLP, INSPEC, WTI Frankfurt eG, zbMATH, Japanese Science and Technology Agency (JST). All books published in the series are submitted for consideration in Web of Science.

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Manoj Sahni · José M. Merigó · Ritu Sahni · Rajkumar Verma Editors

Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy Proceedings of the Second International Conference, MMCITRE 2021

Editors Manoj Sahni Department of Mathematics School of Technology Pandit Deendayal Energy University Gandhinagar, Gujarat, India Ritu Sahni Department of Mathematics Pandit Deendayal Energy University Gandhinagar, Gujarat, India

José M. Merigó School of Information Systems and Modelling University of Technology Sydney Broadway, NSW, Australia Rajkumar Verma Department of Management Control and Info System University of Chile Santiago, Chile

ISSN 2194-5357 ISSN 2194-5365 (electronic) Advances in Intelligent Systems and Computing ISBN 978-981-16-5951-5 ISBN 978-981-16-5952-2 (eBook) https://doi.org/10.1007/978-981-16-5952-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

We are delighted to provide the conference proceedings of the 2nd International Conference on Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy (MMCITRE2021), which took place in Pandit Deendayal Energy University (formerly known as Pandit Deendayal Petroleum University), Gandhinagar, Gujarat, on February 6–8, 2021. Due to the COVID-19 pandemic, the conference was streamed live using the Zoom application and the sessions were held in hybrid mode using Microsoft Teams. The primary goal of the conference is to bring together academics, researchers, professionals and educators to interact and share experiences and research results on topics related to science, engineering, computers and mathematics. This conference is attended by many researchers, scholars and industry persons from nearly all over India as well as many other countries including the USA, UK, Australia, Jordan, Spain, Japan, Chile, Oman, Nepal, etc. Papers in the form of a parallel oral presentation were delivered. This book comprises research papers on numerous topics, primarily focused on the mathematical modeling of many fields, situations based on uncertainty, the modeling of energy systems, statistical analysis, optimization approaches, etc. Since this conference was organized in a hybrid mode, many researchers from Ahmedabad and Gandhinagar only participated in person and visited Pandit Deendayal Energy University (PDEU), which is the Gujarat’s top private university. The 100-acre campus of PDEU is located in Gandhinagar, Gujarat. It provides numerous courses ranging from engineering, arts and management to its students through varied national and international exchange programs with the best universities all around the world. It was set up as a private university by GERMI under the State Act on April 4, 2007. The university has broadened the scope of its programs since its foundation in 2007, delivering a wide variety of courses in technology, management, petroleum, solar and nuclear energy and liberal education through various SOT, SPT, SPM and SLS schools in a relatively short span of time. It aims to extend students and professionals’ possibilities to gain key subject knowledge which is appropriately supported by leadership training activities and helps students to create a worldwide imprint. A

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Preface

variety of well-planned courses such as undergraduates, postgraduates and doctorate programs and intensive research initiatives pursue this aim further. This was the second international conference on the same topic, “Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy (MMCITRE2021)” hosted by the Department of Mathematics, PDEU. The conference’s keynote and prominent presenters were requested to participate in the conference as well as in the review process. A significant number of research publications from all around the world were submitted to this international conference. All of these papers are rigorously reviewed by experts in the relevant fields in a doubleblind peer review procedure, and only the highest quality research papers are chosen for oral presentation at the conference. Finally, papers that passed the review process were chosen, and a total of 42 research papers are included in this proceedings in the form of chapters based on the quality of work assessed by experts from various fields. These chapters were considered to be relevant not only for researchers but also for postgraduate and undergraduate students in the fields of physics, mathematics and engineering as well as for industrial persons working in many sectors, such as medical, energy and stock market. Both new basic mathematical discoveries and mathematical and computational approaches utilized in interdisciplinary applications are presented in these articles. We must learn how to deal with new and different challenges at a moment of instability and change on every societal sphere. The growth of future academicians, researchers, programmers, educators and industries is strongly dependent on their capacity to use mathematical tools in diverse applications in real life. In addition, this conference has been organized to appreciate how young researcher and educationalist and prominent scientist should disseminate new information and progress in all aspects of computational and mathematical advancements and their applications. This knowledge must be turned into transformational leadership that motivates and assists practitioners in making meaningful changes in their communities. This conference is a very significant interconnection in the network of change that many believe will lead to a future, more conscious and capable general population. Basically, the conference is intended to give a place to promote and exchange knowledge of current research and achievements in the field of mathematics and mathematical education to scientific, research and educational staff. It might lead to fresh insights that allow us to form meaningful connections with others and have a good impact on society as a whole, however little. We believe that the papers in this proceedings will aid in broadening scientific understanding and enriching our mathematical abilities for new standard education and will benefit all the academics, researchers and industrialists looking for new mathematical tools. Gandhinagar, India Broadway, Australia Gandhinagar, India Santiago, Chile

Manoj Sahni José M. Merigó Ritu Sahni Rajkumar Verma

Contents

Advanced Mathematical Concepts Strongly Prime Radicals and S-Primary Ideals in Posets . . . . . . . . . . . . . . . J. Catherine Grace John

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Association Schemes Over Some Finite Group Rings . . . . . . . . . . . . . . . . . . Anuradha Sabharwal and Pooja Yadav

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Rings Whose Nonunits Are Multiple of Unit and Strongly Nilpotent Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dinesh Udar

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Improved Lower Bounds on Second Order Non-linearities of Cubic Boolean Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ruchi Telang Gode, Shahab Faruqi, and Ashutosh Mishra

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Cesàro-Riesz Product Summability φ − |C1 R; δ|q Factor for an Infinite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Smita Sonker

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Ostrowski-Type Inequalities with Exponentially Convex Functions and Its Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anulika Sharma and Ram Naresh Saraswat

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Energy of 2-Corona of Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Veninstine Vivik and P. Xavier

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Generalized KKM Mapping Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bhagwati Prasad Chamola, Ritu Sahni, and Manoj Sahni

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Optimal Mild Solutions of Time-Fractional Stochastic Navier-Stokes Equation with Rosenblatt Process in Hilbert Space . . . . . . K. Anukiruthika and P. Muthukumar

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Shape Preserving Hermite Interpolation Reproducing Ellipse . . . . . . . . . . 107 Shubhashree Bebarta and Mahendra Kumar Jena vii

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Mellin Transform of Bose–Einstein Integral Functions . . . . . . . . . . . . . . . . 121 Akbari Jahan A Note on Solution of Linear Partial Differential Equations with Variable Coefficients Formed by an Algebraic Function Using Sumudu Transform with Sm Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Sandip M. Sonawane and S. B. Kiwne Mathematical Modelling and Simulation in Various Disciplines A Computational Approach for Disease Diagnosis Using Information Embedded in the Relationships Between MicroRNA and Their Target Messenger RNA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Srirupa Dasgupta, Medhashree Ghosh, Abhinandan Khan, Goutam Saha, and Rajat Kumar Pal Computational Reconstruction of Gene Regulatory Networks Using Half-Systems Incorporating False Positive Reduction Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Prianka Dey, Abhinandan Khan, Goutam Saha, and Rajat Kumar Pal A Study with Magnetic Field on Stenosed Artery of Blood Flow . . . . . . . . 169 Sarfraz Ahmed and Biju Kumar Dutta Estimation of Reproduction Number of COVID-19 for the Northeastern States of India Using SIR Model . . . . . . . . . . . . . . . . . 181 Prabhdeep Singh, Arindam Sharma, Sandeep Sharma, and Pankaj Narula Novel Generalized Divergence Measure for Intuitionistic Fuzzy Sets and Its Applications in Medical Diagnosis and Pattern Recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Adeeba Umar and Ram Naresh Saraswat Development of 2D Axisymmetric Acoustic Transient and CFD Based Erosion Model for Vibro Cleaner Using COMSOL Multiphysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Vipulkumar Rokad and Divyang H. Pandya Performance Analysis of Nonlinear Isothermal CSTR for Its Different Operating Conditions by Sliding Mode Controller Design . . . . 215 N. S. Patil and B. J. Parvat Production Inventory Model for Deteriorating Items with Hybrid-Type Demand and Partially Backlogged Shortages . . . . . . . 229 Sushil Bhawaria and Himanshu Rathore Influence of Electric and Magnetic Fields on Rayleigh–Taylor Instability in a Power-Law Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 Krishna B. Chavaraddi, Praveen I. Chandaragi, Priya M. Gouder, and G. B. Marali

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An Eco-Epidemic Dynamics with Incubation Delay of CDV on Amur Tiger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Jyoti Gupta, Joydip Dhar, and Poonam Sinha Optimization and Statistical Analysis Comparison of Optimization Results of RSM Approaches for Transesterification of Waste Cooking Oil Using Microwave-Assisted Method Catalyzed by CaO . . . . . . . . . . . . . . . . . . . . . . 271 Nirav Prajapati, Pravin Kodgire, and Surendra Singh Kachhwaha Developing a Meta-Heuristic Method for Solving Multi-objective COTS Selection Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 Surbhi Tilva and Jayesh Dhodiya SNOV-DNOV Mesh Sorting for Multi-objective Optimization . . . . . . . . . . 299 Rupande Desai and Narendra Patel Multivariate Analysis and Human Health Risk Assessment of Groundwater in Amreli District of Gujarat State, India . . . . . . . . . . . . . 313 S. D. Dhiman Tuning P max in RED Gateways for QoS Enhancement in Wireless Packet Switching Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 N. G. Goudru Development of a Mathematical Framework to Evaluate and Validate the Performance of Smart Grid Communication Technologies: An Indian Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 Jignesh Bhatt, Omkar Jani, and V.S.K.V. Harish Early Prediction of Cardiovascular Disease Using One-vs-All Model . . . 351 Sarita Mishra, Manjusha Pandey, Siddharth Swarup Rautaray, and Mahendra Kumar Gourisaria Signature Analysis of Series–Parallel System . . . . . . . . . . . . . . . . . . . . . . . . . 361 Akshay Kumar, Mangey Ram, Soni Bisht, Nupur Goyal, and Vijay Kumar Modified Goel-Okumoto Software Reliability Model Considering Uncertainty Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 Md. Asraful Haque and Nesar Ahmad A Neoteric Technique Using ARIMA-LSTM for Time Series Analysis on Stock Market Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 Hetvi Shah, Vishva Bhatt, and Jigarkumar Shah Collaborative Deployment Strategy for Efficient Connectivity in the Internet of Things . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 Ritik Bansal, Utkarsh Khandelwal, and Saurabh Kumar

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Theoretical Development of Fuzzy and Energy System with Applications Fuzzy Beta C-Open Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 P. Xavier and J. Veninstine Vivik On Intuitionistic Fuzzy Measures of Generalized Bounded Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 Purvak K. Patel and Rajendra G. Vyas Fuzzy Model of Transmission Dynamics of COVID-19 in Nepal . . . . . . . . 423 Gauri Bhuju, Ganga Ram Phaijoo, and Dil Bahadur Gurung Cryptanalysis of Fuzzy-Based Mobile Lightweight Protocol Scheme . . . . 439 Nishant Doshi Energetic and Exergetic Analyses of Hybrid Wind-Solar Energy Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 Hardik K. Jani, Surendra Singh Kachhwaha, and Garlapati Nagababu Analysis of State of Health Estimation for Lithium-Ion Cell Using Unscented and Extended Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 Chaitali Mehta, Amit V. Sant, and Paawan Sharma Control of 7-Level Simplified Generalized Multilevel Inverter Topology for Grid Integration of Photovoltaic System . . . . . . . . . . . . . . . . . 473 Nirav R. Joshi and Amit V. Sant Implementation of Constraint Handling Techniques for Photovoltaic Parameter Extraction Using Soft Computing Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 P. Ashwini Kumari and P. Geethanjali Single Zonal Building Energy Modelling and Simulation . . . . . . . . . . . . . . 503 Nayan Kumar Singh and V. S. K. V. Harish Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515

About the Editors

Manoj Sahni is working as Associate Professor and Head at the Department of Mathematics, School of Technology, Pandit Deendayal Energy University, Gandhinagar, Gujarat, India. He has more than 17 years of teaching and research experience. He holds a M.Sc. degree in Mathematics from Dayalbagh Educational Institute (Deemed University), Agra, M.Phil. from IIT Roorkee, and a Ph.D. degree in Mathematics from Jaypee Institute of Information Technology (Deemed University), Noida, India. He has published more than 60 research papers in peer-reviewed journals, conference proceedings, and chapters with reputed publishers like Springer, Elsevier, etc. He also serves as Reviewer for many international journals of repute. He has conducted the 1st International Conference on Mathematical Modeling, Computational Techniques, and Renewable Energy on February 21–23, 2020. He has contributed to the scientific committee of several conferences and associations. He has delivered many expert talks at the national and international level. He has organized many seminars, workshops, and short-term training programs at PDEU and various other universities. He has also organized special Symposia in an International Conference (AMACS2018) on Fuzzy Set Theory: New Developments and Applications to Real-Life Problems held at London in 2018. He is Member of many international professional societies, including the American Mathematical Society, Forum for Interdisciplinary Mathematics, Indian Mathematical Society, IAENG, and many more. José M. Merigó is Professor at the School of Information, Systems and Modeling at the Faculty of Engineering and Information Technology at the University of Technology Sydney (Australia) and Part-Time Full Professor at the Department of Management Control and Information Systems at the School of Economics and Business at the University of Chile. Previously, he was Senior Research Fellow at the Manchester Business School, University of Manchester (UK), and Assistant Professor at the Department of Business Administration at the University of Barcelona (Spain). He holds a Master and a Ph.D. degree in Business Administration

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from the University of Barcelona. He also holds a B.Sc. and M.Sc. degree from Lund University (Sweden). He has published more than 500 articles in journals, books, and conference proceedings, including journals such as Information Sciences, IEEE Computational Intelligence Magazine, IEEE Transactions on Fuzzy Systems, European Journal of Operational Research, Expert Systems with Applications, International Journal of Intelligent Systems, Applied Soft Computing, Computers & Industrial Engineering, and Knowledge-Based Systems. He has also published several books with Springer and with World Scientific. He is on the editorial board of several journals, including Computers & Industrial Engineering, Applied Soft Computing, Technological and Economic Development of Economy, Journal of Intelligent & Fuzzy Systems, International Journal of Fuzzy Systems, Kybernetes, and Economic Computation and Economic Cybernetics Studies and Research. He has also been Guest Editor for several international journals, Member of the scientific committee of several conferences, and Reviewer in a wide range of international journals. Recently, Thomson & Reuters (Clarivate Analytics) has distinguished him as Highly Cited Researcher in Computer Science (2015–present). He is currently interested in decision making, aggregation operators, computational intelligence, bibliometrics, and applications in business and economics. Ritu Sahni is visiting faculty at Pandit Deendayal Energy University, Gandhinagar, Gujarat, India and former Assistant Professor in the Institute of Advanced Research, Gandhinagar, Gujarat, India. She has more than 15 years of teaching and research experience. She has published more than 30 research papers in peer-reviewed International Journals and Conferences. She is the reviewer of many International Journals and is member of Indian Science Congress and many other well-renowned societies. She has delivered many talks in National and International Conference across the globe. She has also organized many National level Seminars, Workshops in her Institute. She has published many papers in SCI, Scopus, ESCI and many other journals. She is presently working in the area of Fixed Point Theorems, Numerical Methods, Fuzzy Decision Making Systems and other allied areas. Rajkumar Verma is working as Postdoctoral Research Fellow at the Department of Management Control and Information Systems at the University of Chile. Before joining the University of Chile, he was Assistant Professor at the Department of Mathematics, Delhi Technical Campus, India. He holds a M.Sc. degree in mathematics from Chaudhary Charan Singh University, Meerut, and a Ph.D. degree in Mathematics from the Jaypee Institute of Information Technology, Noida, India. He has published more than 45 research papers in journals and conference proceedings, including the International Journal of Intelligent Systems, Journal of Intelligent & Fuzzy Systems, Kybernetika, Informatica, International Journal of Machine Learning and Cybernetics, International Journal of Uncertainty, Fuzziness, and Knowledge-Based Systems, Soft Computing, Applied Mathematical Modelling, and Neural Computing and Applications. He has contributed to the scientific committee

About the Editors

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of several conferences and associations. He also serves as Reviewer for many international journals. He is member of many international professional societies, including the American Mathematical Society, IEEE Computational Intelligence Society, Forum for Interdisciplinary Mathematics, Indian Mathematical Society, and Indian Society of Information Theory and Applications.

Advanced Mathematical Concepts

Strongly Prime Radicals and S-Primary Ideals in Posets J. Catherine Grace John

Abstract The notion of the strongly prime radical of an ideal in posets is defined in this study. Also, we studied the concepts of S-primary ideals in posets. Characterizations of S-primary ideals with respect to strongly prime radical are discussed. Further, the S-primary decomposition of an ideal is obtained. Keywords Poset · Ideals · Strongly prime ideal · Strongly m-system · Strongly prime radical · S-primary

1 Introduction Various radicals play an important role in algebraic structures. All maximal ideals intersection in a commutative ring with unity is named as Jacobson radical, and all prime ideals intersection is called prime radical of ring. The primary ideal that was a development of prime ideal principles was launched using the radical notion [1]. In mathematics, the theory of primary ideals is crucial, particularly in abstract algebra, since a classic pillar of ideal theory is the deconstruction of an ideal into primary ideals. It offers the algebraic basis for decomposing an algebraic variety into its irreducible components. In another sense, primary decomposition is just an extension of the unique-prime-factorization theorem, which states that in number theory, each integer higher than 1 is either a prime number or can be represented as the product of prime integers and that this representation is unique. Theory of primary ideals played a major position of significance in commutative ring theory, and then, it was taken to commutative semi-groups [2]. Anjaneyulu [3] developed the theory of primary ideals in the arbitrary semigroup. Satyanarayana [11] developed commutative primary semi-groups, in which each ideal in the semi-group is primary. He distinguishes its structure from that of J. C. G. John (B) Karunya Institute of Technology and Sciences, Karunya Nagar, Coimbatore, Tamil Nadu 641114, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_1

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J. C. G. John

commutative primary rings. In addition, he established the necessary and sufficient condition for semi-group to primary semi-group. Badawi [4] established the notion of a 2-absorbing primary ideal for a commutative ring R with 1 = 0 and established certain 2-absorbing primary ideal features. He also presented a few examples of primary ideals that are 2-absorbing. Murata [10] applied the idea of a primary ideal into a compactly constructed multiplicative lattice, made use of the m-system. Further, he developed a primary decomposition theorem for ideals in lattice. Joshi [8] later expanded the primary ideal notion to all posets. Following [8], we have studied the notion of S-primary ideals in posets in this paper.

2 Preliminaries Throughout the whole paper, a poset with the smallest element 0 is represented by (X, ≤). We refer [1, 3] to the terminology for fundamental definitions and notations for posets. For W ⊆ X, W  = {r ∈ X : r ≤ a ∀a ∈ W }(W u = {r ∈ X : a ≤ r ∀a ∈ W }) indicate a lower (upper) cone of W in X. For H, W ⊆ X, it may express (H, W ) instead (H ∪ W ) and (H, W )u instead of (H ∪ W )u . If P = {r1 , r2 , . . . , rn } is a finite set of X, then we are using the notation for the (r1 , r2 , . . . , rn ) instead of the ({r1 , r2 , . . . , rn }) and dually for the notion of upper cone. This is undeniable that for any subset D of X, we have D ⊆ D u and D ⊆ D u . If D ⊆ E, then we have E  ⊆ D  and E u ⊆ D u . Also, D u = D  and D uu = D u . Following [4], a subset D(= φ) of X is referred as semi-ideal of X if s ∈ D and r ≤ s, then r ∈ D. Let D ⊆ X. Then, D is referred as ideal if q, w ∈ D implies (q, w)u ⊆ D [9]. I d(X) denotes set of all ideals in X. Let D be a proper semi-ideal (ideal) of X. Then D is said to be prime whenever (s, q) ⊆ D implies either s ∈ D or q ∈ D for all s, q ∈ X [3]. An ideal D of X is termed as semi-prime whenever (s, v) ⊆ D and (s, w) ⊆ D together imply (s, (v, w)u ) ⊆ D for all s, v, w ∈ X [9]. Given w ∈ X, a principal ideal of X generated by an element w is (w] = (w) = {k ∈ X : k ≤ w}, and a principle filter of X constructed by an element w is [w) = (w)u = {k ∈ X : k ≥ w}. Following [6], an ideal D of X is referred as strongly prime if whenever (I ∗ , W ∗ ) ⊆ D implies either I ⊆ D or W ⊆ D for all different proper ideals I, W of X, where I ∗ = I \{0}. An ideal D of X has been said that strongly semi-prime if (I ∗ , J ∗ ) ⊆ D and (I ∗ , S ∗ ) ⊆ D together imply (I ∗ , (J ∗ , S ∗ )u ) ⊆ D for different proper ideals I, J and S of X. Following [5], a subset S(= φ) of X is referred as m-system if ∀w1 , w2 ∈ S, there exists r ∈ (w1 , w2 ) such that r ∈ S. Strongly m-system is defined as an extension of m-system as seen below: A subset S = φ of X is termed strongly m-system if I ∩ S = φ , J ∩ S = φ imply that (I ∗ , J ∗ ) ∩ S = φ for any proper different ideals I, J of X.

Strongly Prime Radicals and S-Primary Ideals in Posets

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It needs to be noted that for an ideal D of X, D is strongly prime ⇔ X\D is a strongly m-system in X. Each strongly m-system is also a m-system. However, in general, reverse part does not need to be true.

3 Main Results Definition 1 Let D be an ideal of X. We have defined strongly prime radical sr (D) of D to be the set of all c ∈ X such that every strongly m-system of X which contains c has a non-empty intersection with D. Theorem 1 Let D1 and D2 be ideals of a poset X. Then (i) (ii) (iii) (iv) (v) (vi)

D1 ⊆ sr (D1 ). sr (sr (D1 )) = sr (D1 ). If D2 ⊆ D1 , then sr (D2 ) ⊆ sr (D1 ). sr ((D1∗ , D2∗ ) ) = sr (D1 ∩ D2 ) = sr (D1 ) ∩ sr (D2 ). In case, D1 is a strongly prime ideal of X which implies that sr (D1 ) = D1 . If D1 is strongly prime ideal and D2 ⊆ D1 , then sr (D2 ) ⊆ D1 .

Proof (i) Let c ∈ D1 . Then, obviously every strongly m-system containing c has a non-empty intersection with D1 . Therefore, c ∈ sr (D1 ). / sr (D1 ). Then, a strongly m-system Mc (ii) Let c ∈ sr (sr (D1 )) and assume that c ∈ exists such that c ∈ Mc and Mc ∩ D1 = φ. Since c ∈ sr (sr (D1 )), we have got Mc ∩ sr (D1 ) = φ. Let p ∈ Mc ∩ sr (D1 ). As p ∈ sr (D1 ), then every strongly m-system containing p must intersect D1 . So, in particular Mc ∩ D1 = φ, a contradiction. (iii) It is trivial. (iv) For any ideals D1 and D2 of X, we have (D1∗ , D2∗ ) ⊆ D1 ∩ D2 ⊆ D1 . Then, by (iii), we have sr ((D1∗ , D2∗ ) ) ⊆ sr (D1 ∩ D2 ) ⊆ sr (D1 ) which implies sr ((D1∗ , D2∗ ) ) ⊆ sr (D1 ∩ D2 ) ⊆ sr (D1 ) ∩ sr (D2 ). Let x ∈ sr (D1 ) ∩ sr (D2 ) and K be a strongly m-system of X containing x. Then, we have K ∩ D1 = φ and K ∩ D2 = φ which implies (D1∗ , D2∗ ) ∩ K = φ which implies x ∈ sr ((D1∗ , D2∗ ) ). (v) Let D1 be strongly prime ideal of X. Suppose sr (D1 )  D1 . Then, there exists / D1 . As D1 is strongly prime, we got X\D1 is a strongly c ∈ sr (D1 ) such that c ∈ m-system of X containing c and (X\D1 ) ∩ D1 = φ, a contradiction to the fact that c ∈ sr (D1 ). Hence, D1 = sr (D1 ). (vi) Let D1 and D2 be ideals of X and D1 be strongly prime such that D2 ⊆ D1 .  Then, by (iii) and (v), we have sr (D2 ) ⊆ sr (D1 ) = D1 . Corollary 1 Let X be a poset and D be a maximal ideal of X. If D is strongly semi-prime, then sr (D) = D. Theorem 2 ([6], Theorem 2.1) Let T be a non-void strongly m-system of X and K be an ideal of X with K ∩ T = φ. Then, K is contained in a strongly prime ideal D of X with D ∩ T = φ.

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 Theorem 3 For an ideal B of X, we have { Q i : Q i is strongly prime ideal of i X containing B} = {c ∈ X : every strongly m-system of S which contains c has a non-empty intersection with B}. Proof Let H = {c ∈ X: Every strongly m-system of X which contains c has a nonempty intersection with B} and c ∈ / H . Then, a strongly m-system S of X which contains c and S ∩ B is an empty set. With the support of Theorem 2, ∃ a strongly prime ideal Q of X B ⊆ Q together with Q ∩ S = φ implies that c ∈ / ∩Q i . So, / ∩Q i . Then, there will be a strongly prime ideal Q i ∩Q i ⊆ H . Conversely, let c ∈ of X with c ∈ / Q i which implies c ∈ X\Q i and X\Q i is a strongly m-system in X. / H . Hence H ⊆ ∩Q i .  Since (X\Q i ) ∩ B = φ, we have c ∈ Definition 2 For an ideal D of X and a strongly prime ideal B of X with D ⊆ B, B is referred as minimal strongly prime ideal of D if there is not any strongly prime ideal K of X with D ⊂ K ⊂ B. Sspec(X) represents all strongly prime ideal collections of X, and Smin(X) indicates all ideal collections of X. For any ideal D of X, S P(D) =  minimal strongly prime Q i and S P(X) = Q i , where Q i s are strongly prime ideal of X. We also Q i ⊇D have S P(D) = S P(X) if D = 0. Remark 1 For an ideal  D of X, we have S P(D) = sr (D). Moreover, sr (D) is an D remains an ideal in X. From [7], for an ideal D of ideal of X because D∈I d(X)  X, Pi = D, where Pi s are prime ideals in X Pi ⊇D

But here is an illustration for in X.

 Q i ⊇D

Q i = D, where Q i s are strongly prime ideals

Example 1 Let X = {0, q, r, h, g} be a poset with the relation ≤ on X as follows:

g

h

r

q

0 Here, I1 = {0, q, r }; I2 = {0, q, h, g} are strongly prime ideals of X, and the ideal D1 = {0, q} gives S P(D1 ) = I1 ∩ I2 = {0, q} = D1 . But for the ideal D2 = {0, q, h}, we have S P(D2 ) = I2 = {0, q, h, g} = D2 . 

Strongly Prime Radicals and S-Primary Ideals in Posets

7

Definition 3 An ideal B( X ) of X is referred as S-primary, if (I ∗ , D ∗ ) ⊆ B imply that I ⊆ B or D ⊆ sr (B) for any different proper ideals I, D of X and I ∗ = I \{0}. It is a prompt findings that in a poset X, each strongly prime ideal is also a Sprimary ideal. However, reverse does not need to be valid in general using the below example. Example 2 Let X = {0, h, s, t, u, v, w} be a poset with the relation ≤ on X as follows:

w v u s t h

0 Here, I = {0, h, t} is a S-primary ideal of X, but not a strongly prime ideal  (((u) )∗ , ((s) )∗ ) ⊆ I and (u) )  I and (s)  I . Definition 4 Let D be an ideal of X. Then, D is referred as S Q -primary if D is S-primary with sr (D) = Q, for some strongly prime ideal Q of X. The S-primary decomposition of D is an expression of the type D = K 1 ∩ K 2 ∩ · · · ∩ K n , where each K i is a S Q i -primary. Definition 5 Let X be a poset and D be an ideal of X with a S-primary decomposition D = K 1 ∩ K 2 ∩ · · · ∩ K m . A S-primary decomposition of D is called minimal if Ki  K j for every i = 1, 2, . . . , m and all these K i s are distinct. j=i

m K i with sr (K i ) = Q i , Definition 6 For a poset X and an ideal D of X, D = i=1 i = 1, 2, . . . , m be a minimal S-primary decomposition of D in X. Then, the strongly prime ideals Q i , i = 1, 2, . . . , m, are said to be collection of associated strongly prime ideals of the decomposition. Moreover, D is called decomposable if the Sprimary decomposition exists. Example 3 Let X = {0, k, u, v, w} be a poset with the relation ≤ on X as follows:

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J. C. G. John

v

w

k

u

0 Here, D = {0, k} is a S Q -primary ideal of X where sr (D) = (v] = Q.



Remark 2 For a semi-ideal D of X and V ⊆ X, we have described V, D = {q ∈ X : L(v, q) ⊆ D ∀v ∈ V } = v, D [5]. If V = {z},, then we are writing v∈V

{z}, D = z, D. We say that D satisfies (∗) condition if (S, Q) ⊆ D implies S ⊆ Q, D ∀S, Q ⊆ X [6]. Theorem 4 Let D be a S Q -primary ideal of a poset X for some strongly prime Q of X and v ∈ X. Then, (i) v ∈ / D and v, D has (∗) condition that implies that v, D is a S Q -primary ideal; (ii) v ∈ / Q gives that v, D = D. Proof (i) Let h ∈ v, D, then (((v) )∗ , ((h) )∗ ) ⊆ (v, h) ⊆ D. As D is S Q primary and (v)  D, we have got h ∈ (h) ⊆ sr (D) = Q for some strongly primary ideal Q of X. Therefore, D ⊆ v, D ⊆ Q. By Theorem 1, Q = sr (D) ⊆ sr (v, D) ⊆ Q. Thus, Q = sr (v, D). Now, we have established that v, D is S-primary. Consider (((h) )∗ , ((w) )∗ ) ⊆ v, D. If (h)  v, D, then L(v, h)  D. So, there is a t ∈ L(v, h) and t ∈ / D. Consequently, we get sr (t, D) = Q. Since L(t, w) ⊆ L(h, w) ⊆ v, D, we have L(v, t, w) ⊆ D. As t ≤ v, we have L(t, w) ⊆ D. Accordingly, w ∈ t, D ⊆ sr (t, D) = Q = sr (v, D. Thus, w ∈ sr (v, D) and (w) ⊆ sr (v, D). On the other side, if (h)  sr (v, D) = Q, then we are showing that (w) ⊆ v, D. Let t ∈ / Q= (v, w) . So, (t, h) ⊆ (v, h, w) ⊆ D. Since D is S-primary and h ∈ sr (D), We have got t ∈ D. (ii) Assume that v ∈ / Q. Suppose that D ⊂ v, D. Then there remains that there is / Q = sr (D), we have t ∈ D, t ∈ v, D and t ∈ / D. As D is S Q -primary and v ∈ which is a contradiction. Hence, v, D = D.  m Di is a minimal Theorem 5 Consider a decomposable ideal W of X, if W = i=1 S-primary decomposition of W , where Q i = sr (Di ), i = 1, 2, 3 . . . m be associated strongly prime ideals of the decomposition, then each strongly prime ideal of the form sr (h, W ) for some h ∈ X is one of the associated strongly prime ideals Q i for some i, and moreover, for each associated strongly prime ideal Q i , ∃h i ∈ X sr (h i , W ) = Q i .

Strongly Prime Radicals and S-Primary Ideals in Posets

9

m Proof Consider W = D1 ∩ D2 ∩ · · · ∩ Dm . For h ∈ / W , h, W  = h, Di  = i=1  m h, Di . Hence h, W  = (h, D j ), 1 ≤ j ≤ m. By Theorem 1, we get i=1

h ∈D / j



sr (h, W ) = sr (

h ∈D / j

h, D j ) =

 h ∈D / j

sr h, D j , 1 ≤ j ≤ m. If sr (h, W ) is

strongly prime, thereafter it needs to be noted that sr (h, W ) = Q j , for some j, 1 ≤ j ≤ m. Therefore, each strongly prime ideal of the structure sr (h, W ) is one of the Q j s for some j, 1 ≤ j ≤ m. Take the associated strongly prime ideal Q j , 1 ≤ j ≤ m. We must look out h ∈ X sr (h, W ) = Q j . As the decomposition of W is minimal, so D j  Di for each j ∈ {1, 2, 3, . . . , m}. It gives that ∃ h j ∈ i= j  n n Di and h j ∈ / D j . Now, h j , W  = h j , Dj = h j , Di . Since i=1 i=1 i= j hj ∈ Di we get that h j , W  = h j , D j . This implies that sr (h j , W ) = i= j

sr (h j , D j ) = Q j , by Theorem 4.



Example 4 In Example 1, consider the ideal W = (q]. Observe that (q] = (r ] ∩ (h] ∩ (g] is a S-primary decomposition of W = (q] and a minimal S-primary decomposition of (q] is (q] = (r ] ∩ (h]. Further, Sr ((r ]) = (r ] and sr ((h]) = (g]. Thus, (r ] and (g] are associated strongly prime ideals of the minimal S-primary decomposition of W = (q]. For the associated strongly prime ideal (r ], there exists h such that sr (h, W ) = (r ], and for that associated strongly prime ideal (g], there exists r such that sr (r, W ) = (g]. Following [5], let X be a poset and B be an ideal of X. For a strongly prime  ideal  Q of X, we have stated B Q = {w ∈ X : (w, s) ⊆ B for some s ∈ X\Q} = s, B. s∈X\Q

n Theorem 6 Let B be an ideal of X with B = Di , a minimal S-primary i=1 decomposition of B, where sr (Di ) = Q i be associated strongly prime ideals of the decomposition. Then, the below proclamation holds. If Q is strongly prime and B ⊆ Q which also contains Q 1 , Q 2 , . . . , Q k (1 ≤ k ≤ n) but does not having Q k+1 , Q k+2 , . . . , Q n , then B Q = D1 ∩ D2 ∩ · · · ∩ Dk and if Q contains none of the Q i ’s then B Q = X. Proof Suppose that Q contains Q 1 , Q 2 , . . . , Q k , but does not contain Q k+1 , / Q, (v, r ) ⊆ B. This implies that Q k+2 , . . . , Q n . Let v ∈ B Q , that is, for some r ∈  (v, r ) ⊆ Di ⊆ sr (Di ), for each i = 1, 2, . . . , n. It can be quickly found that r is precisely not in Q 1 , Q 2 , . . . , Q k . For otherwise, if r ∈ Q i , for some i ∈ {1, 2, . . . , k}, then by assumption, r ∈ Q, an incoherence. So, r ∈ / sr (D1 ), sr (D2 ), . . . , sr (Dk ) which gives (r )  sr (D1 ), sr (D2 ), . . . , sr (Dk ). As D1 , D2 , . . . , Dk are S-primary, we have got (v) ∈ D1 , D2 , . . . , Dk . Hence, (v) ∈ D1 ∩ D2 ∩ · · · ∩ Dk . Conversely, let x ∈ D1 ∩ D2 ∩ ∩ ∩ Dk . As Q k+1 , Q k+2 , . . . , Q n  Q, there exist vk+1 ∈Q k+1 \Q, vk+2 ∈ Q k+2 \Q, . . . , vn ∈Q n \Q. Since v j ∈ Q j =sr (D j ), j = k+1, . . . , n, hence every strongly m-system including v j intersects with D j . In particular, X\Q is a strongly m-system containing v j which intersects D j for every j = k + 1, . . . , n.

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Choose h j ∈ D j ∩ (X\Q), j = k + 1, . . . , n. As Q is strongly prime, we have ({h k+1 , h k+2 , . . . , h n })  Q. Therefore, there exists h ∈ ({h k+1 , h k+2 , . . . , h n }) n and h ∈ / Q. Thus, (x, h) ⊆ Di = B with h ∈ / Q. Hence, x ∈ B Q . Suppose i=1 Q i  Q for every i = 1, 2, . . . , n. Then, ∃ vi ∈ Q i \Q for each i = 1, 2, . . . , n. Since X\Q is a strongly m-system of X, applying similar technical in procedure as above, we get h i ∈ Di ∩ (X\Q), i = 1, . . . , n. As Q is strongly prime ideal, we get ({h 1 , h 2 , . . . , h n })  Q. Therefore, there exists h ∈ ({h 1 , h 2 , . . . , h n }) n h∈ / Q. It is obvious that h ∈ Di = B and therefore for any x ∈ X, we have i=1

(x, h) ⊆ B, which gives x ∈ B Q . Hence, B Q = X.



Theorem 7 ([6], Theorem 2.4) For an ideal B of X, we have if B has the property below that for n > 2, if pairwise distinct ideals K 1 , K 2 , . . . , K n of X with (K 1∗ , K 2∗ , . . . , K n∗ ) ⊆ B, then at least (n − 1) of n subsets (K 2∗ , K 3∗ , . . . , K n∗ ) , ∗ ) are subsets of B. (K 1∗ , K 3∗ , . . . , K n∗ ) , . . . , (K 1∗ , K 2∗ , . . . , K n−1 Theorem 8 Let X be a poset and B be an ideal of X. If B has two minimal primary decomposition I1 ∩ I2 ∩ · · · ∩ Ik = D1 ∩ D2 ∩ · · · ∩ Ds , where Ii is S Ai -primary and D j is S B j -primary and each Ai and B j are isolated strongly prime, then k = s. Proof Let I1 ∩ I2 ∩ · · · ∩ Ik = D1 ∩ D2 ∩ · · · ∩ Ds where Ii is S Ai -primary and D j is S B j -primary. Then, A1 ∩ A2 ∩ · · · ∩ Ak = sr (I1 ) ∩ sr (I2 ) ∩ · · · ∩ sr (Ik ) = sr (I1 ∩ I2 ∩ · · · ∩ Ik ) = sr (D1 ∩ D2 ∩ · · · ∩ Ds ) = sr (D1 ) ∩ sr (D2 ) ∩ · · · ∩ sr (Ds ) = D1 ∩ D2 ∩ · · · ∩ Ds . Now, L(A∗1 , A∗2 , . . . , A∗k ) ⊆ A1 ∩ A2 ∩ · · · ∩ Ak ⊆ D j for all j. Since B j is strongly prime ideal and Theorem 7, we have got Ai ⊆ B j for some i. Also, L(B1∗ , B2∗ , . . . , Bs∗ ) ⊆ B1 ∩ B2 ∩ · · · ∩ Bs ⊆ Ai for all i. Since Ai is strongly prime ideal and Theorem 7, we have got Br ⊆ Ai for some r . So, Br ⊆ Ai ⊆ B j . Since B j is an isolated strongly prime, we have Br = B j which implies B j = Ai , so k = s. 

4 Conclusion The definition and its generalization of the prime ideal have a distinguished place in algebraic geometry and commutative algebra. These are useful tools to determine the properties of algebraic structure. In this article, more generalization of primary ideals in posets is given, and some properties of these S-primary ideals are obtained. Also, the S-primary decomposition of an ideal in posets is discussed.

References 1. Anderson, D.D., Bataineh, M.: Generalization of prime ideals. Comm. Algebra 36, 686–696 (2008). https://doi.org/10.1080/00927870701724177

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2. Anderson, D.D., Mahaney, L.A.: Commutative rings in which every ideal is a product of primary ideals. J. Algebra 106, pp. 528–535 (1987). https://doi.org/10.1016/0021-8693(87)90014-7 3. Anjaneyulu, A.: Primary ideals in semi groups. Semigroup Forum. 20, 129–144(1980). https:// doi.org/10.1007/BF02572675 4. Badawi, A.,Tekir, U., Yetkin, E.: On 2-absorbing primary ideals in commutative rings. Bull. Korean Math. Soc. 51(4), 1163–1173 (2014). https://doi.org/10.4134/BKMS.2014.51.4.1163 5. Catherine Grace John, J., Elavarasan, B.: z J -ideals and strongly prime ideals in posets. Kyungpook Math. J. 57, 385–391 (2017). https://doi.org/10.5666/KMJ.2017.57.3.385 6. Catherine Grace John, J., Elavarasan, B.: Strongly prime ideals and primal ideals in posets. Kyungpook Math. J. 56, 727–735 (2016). https://doi.org/10.5666/KMJ.2016.56.3.727 7. Halas, R..: On extension of ideals in posets. Discr. Math. 308, 4972–4977 (2008). https://doi. org/10.1016/j.disc.2007.09.022 8. Joshi, V., Mundlik, N.: On primary ideals in posets, Math. Slovaca. 65(6), 1237–1250(2015). https://doi.org/10.1515/ms-2015-0085 9. Kharat, V.S., Mokbel, K.A.: Primeness and semiprimeness in posets. Math. Bohem. 134,(1), 19–30 (2009). https://doi.org/10.21136/mb.2009.140636 10. Murata, K.: Primary decomposition of elements in compactly generated integral multiplicative lattices. Osaka J. Math. 7, 97–115 (1970). https://projecteuclid.org/euclid.ojm/1200692690 11. Satyanarayana, M.: Commutative primary semigroups. Czech. Math. J. 22(4), 509–516 (1972). https://doi.org/10.21136/CMJ.1972.101121 12. Venkatanarasimhan, P.V.: Semi ideals in posets. Math. Ann. 185(4), 338–348 (1970).https:// doi.org/10.1007/BF01349957

Association Schemes Over Some Finite Group Rings Anuradha Sabharwal and Pooja Yadav

Abstract In this paper, we study non-symmetric commutative association schemes for cyclic groups Z p × Z p × · · · × Z p ( p is prime), Z p1 × Z p2 × · · · × Z pr ( pi s are    r times

distinct primes), dihedral group and symmetric group without using conjugacy classes. We also construct commutative association schemes for finite group rings over Zn , the ring of integers mod n. Moreover, we construct association scheme for n × n circulant matrices over Z p , for p prime. Keywords Group ring · Association scheme · Symmetric group · Dihedral group · Circulant matrices

1 Introduction In the theory of algebraic combinatorics, association scheme plays a vital role. Association schemes were introduced by Bose and Shimamoto [1]. They are used in coding theory, graph theory, design theory and group theory. Association schemes may also be seen as colorings of the edges of complete graphs which satisfies nice regularity conditions. Jørgensen [5] has listed non-symmetric association schemes with classes less than 96 vertices which stimulate us to study non-symmetric association scheme for various finite groups and group rings. We start with a brief introduction of association scheme. For more basic results on association schemes, we refer to [8]. In this paper, for a finite set X , we denote by G the partition of X × X .

A. Sabharwal (B) Government College for Girls, Gurugram, Haryana, India P. Yadav Kamala Nehru College, University of Delhi, New Delhi, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_2

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A. Sabharwal and P. Yadav

2 Preliminaries Definition 1 Association Scheme (AS): Let X be a finite set, and G be a partition of X × X with R0 , R1 , . . . , Rn binary relations on G. Then, χ = (X, G) is an association scheme of n-class if the following conditions hold: 1. Existence of identity relation R0 = {(x, x) : x ∈ X } in G. 2. For any relation R ∈ G, there exists a relation R ∗ ∈ G such that for every (x, y) ∈ R, (y, x) ∈ R ∗ . 3. For each i, j, k, if (x, y) ∈ Rk , the cardinality |x Ri ∩ y R ∗j | is a constant pikj which does not depend on choice of x and y. The order of G is the number of elements in X . The non-negative integers { pikj }0≤i, j,k≤n are the intersection numbers or parameters of G. The association scheme G is commutative if pikj = p kji ∀ 0 ≤ i, j, k ≤ n, and it is symmetric if each relation Ri is a symmetric relation, that is, Ri = Ri∗ ∀ i ∈ {0, 1, . . . , n}. If (x, y) ∈ Ri with x = y, then x and y are called ith associates. For x ∈ X and R ∈ G, let the set x R be the set of all elements y ∈ X when (x, y) ∈ R. Example 1 Let X = Z2 × Z2 = {(0, 0), (0, 1), (1, 0), (1, 1)}. Let us define the following relations in G: R0 = {((0, 0), (0, 0)), ((0, 1), (0, 1)), ((1, 0), (1, 0)), ((1, 1), (1, 1))} R1 = {((0, 0), (1, 1)), ((0, 1), (1, 0)), ((1, 0), (0, 1)), ((1, 1), (0, 0))} R2 = {((0, 0), (1, 0)), ((0, 1), (1, 1)), ((1, 0), (0, 0)), ((1, 1), (0, 1))} R3 = {((0, 0), (0, 1)), ((0, 1), (0, 0)), ((1, 0), (1, 1)), ((1, 1), (1, 0))} Then, there exist an identity relation R0 , and if (x, y) ∈ Rk , the cardinality |x Ri ∩ y R ∗j | is a constant 0 and 1 depending on i, j, k . Also since, Rk = Rk∗ for all 0 ≤ k ≤ 3, (X, G) is a symmetric AS. Note: Every symmetric AS is commutative. Definition 2 Group Association Scheme(GAS): A finite group G having conjugacy classes C0 , C1 , . . . , Cd yields a commutative association scheme with a class of relations Rk on G defined by Rk = {(x, y)|yx −1 ∈ Ck } ∀0 ≤ k ≤ d. This scheme is called group association scheme of G. Any finite group (X, ∗) yields an association scheme with the following class of relations: Rk = {(x, y)|x = k ∗ y|x, y ∈ X } for all k ∈ X. This association scheme is commutative iff X is an abelian group. In this paper, we will study association schemes using these kinds of relations for some finite groups, and further, we will compute its parameters also. We define relations for association schemes in such a way that their intersection numbers are either 0 or 1.

Association Schemes Over Some Finite Group Rings

15

Theorem 1 [4, Theorem 3.3] An association scheme of a group is commutative, if its order is a prime number. Therefore, for X = Z p where p is prime, association scheme (X, G) defined in Lemma 1 is commutative.

3 Main Results We now study association schemes for Z p1 × Z p2 × · · · × Z pr , where p1 , p2 , . . . , pr are distinct primes and for Z p × Z p × · · · × Z p , where p is odd prime.    r times

Lemma 1 Let X = Z p . Let us define relations Rk in G by Rk = {(i, j)|i = k + j|i, j ∈ Z p } ∀ k ∈ Z p . Then, (X, G) is a non-symmetric commutative AS, and intersection numbers of this AS are as follows: pikj

 1 if k = i + j, = 0 if k = i + j

Proof As Z p is an abelian group, (Z p , G) with the given relations forms a commutative association scheme. Let Ri , R j , Rk be arbitrary relations in G. To find cardinality pikj such that for all (x, y) ∈ Rk , we have |x Ri ∩ y R ∗j | = pikj , let (x, y) ∈ Rk and let x Ri = x  and y R ∗j = y  . That is, x = x  + i, y  = y + j and x = y + k. Since every pair of points (x, y) are ith associates for exactly one i, pikj can be either 0 or 1. Therefore, pikj = 1 if x  = y  that is, if k = i + j, and pikj = 0 if x  = y  that is, if k = i + j. Lemma 2 Let X = Z p × Zq , where p and q are distinct primes. Then, the relations Rk in G defined by Rk = {(x, y)|x1 ≡ (k + y1 ) mod p, x2 ≡ (k + y2 ) mod q |x = (x1 , x2 ), y = (y1 , y2 ) ∈ Z p × Zq } ∀ 0 ≤ k ≤ pq − 1 is a non-symmetric commutative AS with intersection numbers  pikj

=

1 if k ≡ (i + j) mod pq, 0 otherwise

Proof Since Z p × Zq is an abelian group, it can be easily verified that (X, G) is a commutative AS with non-symmetric relations as Ri∗ = R j if i + j = pq. Let Ri , R j , Rk ∈ G. Now, we will find cardinality pikj such that for all (x, y) ∈ Rk we have |x Ri ∩ y R ∗j | = pikj .

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Let (x, y) ∈ Rk , where x = (x1 , x2 ), y = (y1 , y2 ) ∈ Z p × Zq , and let x Ri = x  = and y R ∗j = y  = (y1 , y2 ). That is, x1 ≡ (y1 + k) mod p, x2 ≡ (y2 + k) mod q; x1 ≡ (x1 + i) mod p, x2 ≡ (x2 + i) mod q; y1 ≡ (y1 + j) mod p, y2 ≡ (y2 + j) mod q. Using these equations, we have pikj = 1 iff x  = y  that is, if k ≡ (i + j) mod pq.

(x1 , x2 )

Theorem 2 Let X = Z p1 × Z p2 × · · · × Z pr , where p1 , p2 , . . . , pr are distinct primes and r ≥ 3. For each t ∈ Z pr −1 pr and ds ∈ Z ps ∀ 1 ≤ s ≤ r − 2, the relations Rk in G defined by Rk = {(x, y)| yr ≡ (k + xr ) mod pr , yr −1 ≡ (k + xr −1 ) mod pr −1 , ys ≡ (ds + xs ) mod ps ∀ 1 ≤ s ≤ r − 2| x = (x1 , x2 , . . . , xr ), y = (y1 , y2 , . . . , yr ) ∈ Z p1 × Z p2 × · · · × Z pr } ∀ k = p2 p3 · · · pr d1 + p3 p4 · · · pr d2 + · · · + pr −1 pr dr −2 + t is a non-symmetric commutative AS with intersection numbers ⎧ ⎪ ⎨1 if k ≡ (i + j) mod pr −1 pr , and k pi j = ds ≡ (ds(1) + ds(2) ) mod ps ∀ 1 ≤ s ≤ r − 2 ⎪ ⎩ 0 otherwise where k = p2 p3 · · · pr d1 + p3 p4 · · · pr d2 + · · · + pr −1 pr dr −2 + t; i = p2 p3 · · · (2) (1) pr d1(1) + p3 p4 · · · pr d2(1) + · · · + pr −1 pr dr(1) −2 + t ; j = p2 p3 · · · pr d1 + p3 p4 · · · (2) (2) pr d2 + · · · + pr −1 pr dr −2 + t (2) for some t, t (1) , t (2) ∈ Z pr −1 pr and ds , ds(1) , ds(2) ∈ Z ps ∀ 1 ≤ s ≤ r − 2. Proof |X | = |Z p1 × Z p2 × · · · × Z pr | = p1 p2 · · · pr = |Rk | for all 0 ≤ k ≤ p1 p2 · · · pr − 1. All the relations Rk are disjoint, and they form partition of G. Let Ri , R j , Rk be arbitrary relations in G. We will show that for each pair x, y with (x, y) ∈ Rk , the cardinality |{z ∈ X |(x, z) ∈ Ri , (z, y) ∈ R j }| is a constant. Let (x, y) ∈ Rk , where x = (x1 , x2 , . . . , xr ), y = (y1 , y2 , . . . , yr ) ∈ X , and let x Ri = x  = (x1 , x2 , . . . , xr ), y R ∗j = y  = (y1 , y2 , . . . , yr ). Now, (x, y) ∈ Rk implies yr ≡ (k + xr ) mod pr , yr −1 ≡ (k + xr −1 ) mod pr −1 , ys ≡ (ds + xs ) mod ps ∀ 1 ≤ s ≤ r − 2 where k = p2 p3 · · · pr d1 + p3 p4 · · · pr d2 + · · · + pr −1 pr dr −2 + t for some t ∈ Z pr −1 pr and ds ∈ Z ps ∀ 1 ≤ s ≤ r − 2. Similarly, as (x, x  ) ∈ Ri and (y  , y) ∈ R j , we have xr ≡ (i + xr ) mod pr , xr −1 ≡ (i + xr −1 ) mod pr −1 , xs ≡ (ds(1) + xs ) mod ps ∀ 1 ≤ s ≤ r − 2 where i = p2 p3 · · · (1) pr d1(1) + p3 p4 · · · pr d2(1) + · · · + pr −1 pr dr(1) for some t (1) ∈ Z pr −1 pr , ds(1) ∈ −2 + t Z ps ∀ 1 ≤ s ≤ r − 2, and yr ≡ ( j + yr ) mod pr , yr −1 ≡ ( j + yr −1 ) mod pr −1 , ys ≡ (ds(2) + ys ) mod ps ∀ 1 ≤ s ≤ r − 2 where j = p2 p3 · · · pr d1(2) + p3 p4 · · · pr d2(2) + (2) · · · + pr −1 pr dr(2) for some t (2) ∈ Z pr −1 pr , ds(2) ∈ Z ps ∀ 1 ≤ s ≤ r − 2. −2 + t Using above equations, we have pikj = 1 if and only if x  = y  that is, if k ≡ (i + j) mod pr −1 pr and ds ≡ (ds(1) + ds(2) ) mod ps ∀ 1 ≤ s ≤ r − 2. Hence, (X, G) is a non-symmetric commutative AS.

Association Schemes Over Some Finite Group Rings

17

Lemma 3 Let X = Z p × Z p , where p is odd prime. For each d, t ∈ Z p , the relations Rk in G defined by Rk = {(x, y)|x1 ≡ (k + d + y1 ) mod p, x2 = (k + t + y2 ) mod p |x = (x1 , x2 ), y = (y1 , y2 ) ∈ Z p × Z p } ∀ k = pd + t yields a non-symmetric commutative association scheme, and its intersection numbers are  1 if 2k + d + t ≡ (2i + 2 j + d1 + d2 + t1 + t2 ) mod p, k pi j = 0 otherwise where k = pd + t; i = pd1 + t1 ; j = pd2 + t2 for some d, d1 , d2 , t, t1 , t2 ∈ Z p . Proof |X | = |Z p × Z p | = p 2 and |Rk | = p 2 ∀0 ≤ k ≤ p 2 − 1. All the relations Rk are disjoint and ∪{Rk : 0 ≤ k ≤ p 2 − 1} = G. Let Ri , R j , Rk be arbitrary relations in G. We will show that for each (x, y) ∈ Rk , the cardinality |{z ∈ X |(x, z) ∈ Ri , (z, y) ∈ R j }| is a constant. Let (x, y) ∈ Rk where x = (x1 , x2 ), y = (y1 , y2 ) ∈ Z p × Z p and let x Ri = x  = (x1 , x2 ), y R ∗j = y  = (y1 , y2 ). Now, (x, y) ∈ Rk implies x1 ≡ (y1 + k + d) mod p, x2 ≡ (y2 + k + t) mod p, where k = pd + t for some d, t ∈ Z p . Similarly, as (x, x  ) ∈ Ri and (y  , y) ∈ R j , x1 ≡ (x1 + i + d1 ) mod p, x2 ≡ (x2 + i + t1 ) mod p where i = pd1 + t1 for some d1 , t1 ∈ Z p and y1 ≡ (y1 + j + d2 ) mod p, y2 ≡ (y2 + j + t2 ) mod p where j = pd2 + t2 for some d2 , t2 ∈ Z p . Using above equations, we have pikj = 1 if and only if x  = y  that is, if 2k + d + t ≡ (2i + 2 j + d1 + d2 + t1 + t2 ) mod p. Theorem 3 Let X = Z p × Z p × · · · × Z p , where p is odd prime and r ≥ 3. For    r times

each t, ds ∈ Z p ∀ 1 ≤ s ≤ r − 1, the relations Rk in G defined by Rk = {(x, y)| yr ≡ (k + t + xr ) mod p, ys ≡ (ds + xs ) mod p ∀ 1 ≤ s ≤ r − 1| x = (x1 , x2 , . . . , xr ), y = (y1 , y2 , . . . , yr ) ∈ X } ∀ k = pr −1 d1 + pr −2 d2 + · · · + pdr −1 + t is a non-symmetric commutative AS with intersection numbers ⎧ 



( j) r −1 (k) r −1 (i) (k) (i) ⎪ ⎪ +k ≡ + t ( j) + i + j s=1 ds + t s=1 (ds + ds ) + t ⎨1 if pikj = mod p ⎪ ⎪ ⎩0 otherwise

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(k) r −1 (i) where k = pr −1 d1(k) + pr −2 d2(k) + · · · + pdr(k) d1 + pr −2 d2(i) + · · · −1 + t ; i = p ( j) ( j) ( j) (i) r −1 + pdr(i) d1 + pr −2 d2 + · · · + pdr −1 + t ( j) for some t (k) , t (i) , t ( j) , −1 + t ; j = p ( j) (k) (i) ds , ds , ds ∈ Z p ∀ 1 ≤ s ≤ r − 1.

Proof |X | = pr and |Rk | = pr for all 0 ≤ k ≤ pr − 1. All the relations Rk are disjoint, and they form partition of G. Let Ri , R j , Rk be arbitrary relations in G. We will show that for each (x, y) ∈ Rk , the cardinality |{z ∈ X |(x, z) ∈ Ri , (z, y) ∈ R j }| is a constant. Let (x, y) ∈ Rk where x = (x1 , x2 , . . . , xr ), y = (y1 , y2 , . . . , yr ) ∈ X and let x Ri = x  = (x1 , x2 , . . . , xr ), y R ∗j = y  = (y1 , y2 , . . . , yr ). Now, (x, y) ∈ Rk implies yr ≡ (k + t (k) + xr ) mod p, ys ≡ (ds(k) + xs ) mod p (k) ∀ 1 ≤ s ≤ r − 1 where k = pr −1 d1(k) + pr −2 d2(k) + · · · + pdr(k) for some t (k) , −1 + t (k) ds ∈ Z p ∀ 1 ≤ s ≤ r − 1. Similarly, as (x, x  ) ∈ Ri and (y  , y) ∈ R j , we have xr ≡ (i + t (i) + xr ) mod p, xs ≡ (ds(i) + xs ) mod p ∀ 1 ≤ s ≤ r − 1 where i = pr −1 d1(i) + pr −2 d2(i) + · · · + (i) pdr(i) for some t (i) , ds(i) ∈ Z p ∀ 1 ≤ s ≤ r − 1, and yr ≡ ( j + t ( j) + yr ) mod −1 + t ( j) ( j) ( j) p, ys ≡ (ds + ys ) mod p ∀ 1 ≤ s ≤ r − 1 where j = pr −1 d1 + pr −2 d2 + · · · + ( j) ( j) pdr −1 + t ( j) for some t ( j) , ds ∈ Z p ∀ 1 ≤ s ≤ r − 1. k We have pi j = 1 if and only if x  = y  that is, if k + t (k) ≡ (i + j + t (i) + ( j)

t ( j) ) mod p, and ds(k) ≡ (ds(i) + ds ) mod p ∀ 1 ≤ s ≤ r − 1. Adding these equations, we have the desired result, and hence, (X, G) is a non-symmetric commutative association scheme. Yue Meng-tian, Li Zeng-ti (see [7]) constructed symmetric association schemes on the dihedral group. In next theorem, we provide a new family of commutative association schemes with non-symmetric relations on the dihedral group. Theorem 4 Let X = D2n = a, b|a 2 , bn , ab = b−1 a . That is, the canonical form of any element of D2n is a l bm where 0 ≤ l ≤ 1 and 0 ≤ m ≤ n − 1. Define relations Rk in G by Rk = {(a l bm , a k+l bk+m )|0 ≤ l ≤ 1, 0 ≤ m ≤ n − 1} for all 0 ≤ k ≤ 2n − 1. Then, (X, G) is a non-symmetric AS, and intersection numbers of this scheme are as follows:  1 if k = i + j, pikj = 0 if k = i + j Proof R0 = {(x, x)|x ∈ D2n } is an identity relation. It can be verified that (X, G) is an AS and the relations are non-symmetric. Let Ri , R j , Rk ∈ G. To find cardinality pikj such that for each (x, y) ∈ Rk , |x Ri ∩ y R ∗j | = pikj . Let (x, y) ∈ Rk and let x Ri = x  and y R ∗j = y  . That is, x = a l bm , y = a k+l bk+m ; x = a l1 bm 1 , x  = a i+l1 bi+m 1 ; y  = a l2 bm 2 , y = a j+l2 b j+m 2 where 0 ≤ l, l1 ,

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19

l2 ≤ 1 and 0 ≤ m, m 1 , m 2 ≤ n − 1. Since every pair of points (x, y) are ith associates for exactly one i, pikj can be either 0 or 1. Therefore, using above equations, we have pikj = 1 if x  = y  that is, if k = i + j and pikj = 0 if x  = y  that is, if k = i + j. Let us denote Sn by the symmetric group of degree n. Tomiyama and Yamazaki [6] have characterised group association schemes for symmetric group. In next theorem, we have determined non-symmetric commutative association scheme for symmetric groups S3 and S4 without using conjugacy classes. Theorem 5 Let X = S3 = {τ i σ j : 0 ≤ i ≤ 1, 0 ≤ j ≤ 2} where τ 2 = 1 and σ 3 = 1. Then, the relations Rk in G defined by Rk = {(τ l σ m , τ k+l σ k+m )|0 ≤ l ≤ 1, 0 ≤ m ≤ 2} for all 0 ≤ k ≤ 6 is a non-symmetric commutative AS, and intersection numbers of this association scheme are as follows:  1 if k = i + j, pikj = 0 if k = i + j Proof Since the canonical form of S3 and D6 is same, we can conclude this theorem. Lemma 4 The symmetric group of degree 4, S4 = {τ a σ b τ σ l : 0 ≤ a ≤ 1, 0 ≤ b ≤ 2, 0 ≤ l ≤ 3} where τ 2 = 1 and σ 4 = 1. Proof Let A = {τ a σ b τ σ l : 0 ≤ a ≤ 1, 0 ≤ b ≤ 2, 0 ≤ l ≤ 3}, where τ 2 = 1 and σ 4 = 1. Claim: Elements of the form {σ 3 τ σ j : 0 ≤ j ≤ 3} and {τ σ 3 τ σ j : 0 ≤ j ≤ 3} are also of the form given in A. A shorter presentation for Sn is given in [2] which is Sn = τ, σ |τ 2 = σ n = (τ σ )n−1 = 1, (τ σ −1 τ σ )3 = 1, (τ σ − j τ σ j )2 = 1 for 2 ≤ j ≤ n/2 . Therefore, the presentation for S4 is τ, σ |τ 2 = σ 4 = (τ σ )3 = 1, (τ σ −1 τ σ )3 = 1, (τ σ −2 τ σ 2 )2 = 1 . As (τ σ )3 = 1, σ 3 τ = τ σ τ σ . Therefore, σ 3 τ σ j = τ σ τ σ j+1 for 0 ≤ j ≤ 3. Again (τ σ )3 = 1 implies τ σ 3 τ = σ τ σ . Therefore, τ σ 3 τ σ j = τ 0 σ τ σ j+1 for 0 ≤ j ≤ 3. Hence, every element of S4 can be written of the form τ a σ b τ σ l where 0 ≤ a ≤ 1, 0 ≤ b ≤ 2, 0 ≤ l ≤ 3. Theorem 6 Let X = S4 and G be a partition of X × X . Then, the following relations Rk = {(τ a σ b τ σ l , τ a+k σ b τ σ l+k )|0 ≤ a ≤ 1, 0 ≤ b ≤ 2, 0 ≤ l ≤ 3} ∀ 0 ≤ k ≤ 24 form a non-symmetric association scheme for S4 , and its intersection numbers are  1 if k = i + j, pikj = 0 if k = i + j

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Proof By previous lemma, any element of S4 can be written of the form {τ a σ b τ σ l : 0 ≤ a ≤ 1, 0 ≤ b ≤ 2, 0 ≤ l ≤ 3}. R0 = {(τ a σ b τ σ l , τ a σ b τ σ l |0 ≤ a ≤ 1, 0 ≤ b ≤ 2, 0 ≤ l ≤ 3} is an identity relation. For each Ri ∈ G, R ∗ ∈ G. Also, |Rk | = |X | = 24 and {∪Rk : 0 ≤ k ≤ 23} = G. Let Ri , R j , Rk be arbitrary relations in G. To find cardinality pikj such that for each (x, y) ∈ Rk , we have |x Ri ∩ y R ∗j | = pikj . Let (x, y) ∈ Rk and let x Ri = x  and y R ∗j = y  . That is, x = τ a σ b τ σ l , y = τ a+k σ b τ σ l+k ; x = τ a1 σ b1 τ σ l1 , x  = τ a1 +i σ b1 τ σ l1 +i ; y  = τ a2 σ b2 τ σ l2 , y = τ a2 + j σ b2 τ σ l2 + j where 0 ≤ a, a1 , a2 ≤ 1, 0 ≤ b, b1 , b2 ≤ 2, 0 ≤ l, l1 , l2 ≤ 3. Using these equations, we have pikj = 1 if x  = y  that is, if k = i + j and pikj = 0 if x  = y  that is, if k = i + j. The study of association schemes for finite groups stimulates us to do the computations on finite group rings. Theorem 7 Let X be a finite group ring Zn [G] over Zn , where G is any finite group, and let G be a partition of X × X . Then, relations Rω on G defined as ⎧⎛ ⎫ ⎞ ⎨  ⎬    Rω = ⎝ αg g, βg g ⎠ : αg , βg ∈ Zn , αg g + ω = βg g ⎩ ⎭ g∈G

g∈G

g∈G

g∈G

for all ω ∈ Zn [G], is a non-symmetric commutative AS for X with intersection numbers  1 if k = i + j, k pi j = 0 otherwise Proof When ω = 0, R0 = {(x, x)|x ∈ Zn [G]} is an identity relation. |X | = n |G| and |Rω | = n |G| ∀ω ∈ Zn [G]. All the relations Rω are disjoints and {∪Rω : ω ∈ Zn [G]} = G. Let Ri , R j , Rk be arbitrary relations in G. To find cardinality pikj such that for each (x, y) ∈ Rk , we have |x Ri ∩ y R ∗j | = pikj . Let (x, y) ∈ Rk and let x Ri = x  and y R ∗j = y  . That is, x + k = y; x + i = x  ; y  + j = y. Using these equations, we have pikj = 1 if x  = y  that is, if k = i + j and pikj = 0 if x  = y  that is, if k = i + j. Hence, (X, G) is a commutative AS with non-symmetric relations as Rω∗ = Rω if ω + ω = 0. Let Crn (R) denote the algebra of n × n circulant matrices over a ring R [3]. We will now construct association scheme for circulant matrices over the ring of integers modulo p, where p is odd prime.    ab : a, b ∈ Z p , where p is odd prime. Let Theorem 8 Let X = Cr2 (Z p ) = ba d, t ∈ Z p . Then, the relations Rk in G defined by

Association Schemes Over Some Finite Group Rings

Rk =

21

    b b a1 a2 , 1 2 : b1 ≡ (a1 + k + d) mod p, a2 a1 b2 b1  b2 ≡ (a2 + k + t) mod p : ai , bi ∈ Z p ∀ k = pd + t.

is a non-symmetric commutative AS for X , and its intersection numbers are  pikj

=

1 if 2k + d + t ≡ (2i + 2 j + d1 + d2 + t1 + t2 ) mod p, 0 otherwise

where k = pd + t; i = pd1 + t1 ; j = pd2 + t2 for some d, d1 , d2 , t, t1 , t2 ∈ Z p .      a1 a2 a a Proof For k = 0, R0 = , 1 2 ; ai ∈ Z p is an identity relation. a2 a1 a2 a1 Also, Ri∗ ∈ G for all Ri ∈ G. Let Ri , R j , Rk ∈ G. To find intersection numbers pikj , let (A, B) ∈ Rk where A =           a1 a2 b b a a b b and B = 1 2 and let A Ri = A = 1 2 , B R ∗j = B  = 1 2 . Now, a2 a1 b2 b1 a2 a1 b2 b1 (A, B) ∈ Rk implies b1 ≡ (a1 + k + d) mod p, b2 ≡ (a2 + k + t) mod p where k = pd + t for some d, t ∈ Z p . Similarly, as (A, A ) ∈ Ri and (B  , B) ∈ R j , a1 ≡ (a1 + i + d1 ) mod p, a2 ≡ (a2 + i + t1 ) mod p where i = pd1 + t1 for some d1 , t1 ∈ Z p ; and b1 ≡ (b1 + j + d2 ) mod p, b2 ≡ (b2 + j + t2 ) mod p where j = pd2 + t2 for some d2 , t2 ∈ Z p ; As all the relations Rk in G are disjoint and {∪Rk : k ∈ {0, 1, . . . , p 2 − 1}} = G, there is a unique value for each x  and y  , that is, the intersection numbers can be either 0 or 1. Therefore, we have pikj = 1 if and only if x  = y  that is, if 2k + d + t ≡ (2i + 2 j + d1 + d2 + t1 + t2 ) mod p. ⎡ ⎤ a1 a2 . . . an ⎢an a1 . . . an−1 ⎥ ⎢ ⎥ Let us denote n × n circulant matrix ⎢ . . . . ⎥ by [a1 , a2 , . . . , an ]. ⎣ .. .. . . .. ⎦ a2 a3 . . . a1 ⎫ ⎧⎡ ⎤ a1 a2 . . . an ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎢ ⎬ ⎨ ⎥ a a . . . a n 1 n−1 ⎢ ⎥  : ai s ∈ Z p , where p is odd Theorem 9 Let X = Crn (Z p ) = ⎢ . . . ⎥ . ⎪ ⎪ ⎣ .. .. . . .. ⎦ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ a2 a3 . . . a1 prime and n ≥ 3. Let t, ds ∈ Z p ∀ 1 ≤ s ≤ n − 1. Then, the relations Rk in G defined by Rk = {([a1 , a2 , . . . , an ], [b1 , b2 , . . . , bn ]) : bn ≡ (an + k + t) mod p, bs ≡ (as + ds ) mod p ∀ 1 ≤ s ≤ n − 1 : ai s, bi s ∈ Z p } ∀ k = p n−1 d1 + p n−2 d2 + · · · + pdn−1 + t is a non-symmetric commutative AS for X , and its intersection numbers are

22

pikj =

A. Sabharwal and P. Yadav

⎧ 



( j) n−1 (k) n−1 (i) (k) (i) ⎪ ⎪ +k ≡ + t ( j) + i + j s=1 ds + t s=1 (ds + ds ) + t ⎨1 if mod p ⎪ ⎪ ⎩0 otherwise

(k) where k = p n−1 d1(k) + p n−2 d2(k) + · · · + pdn−1 + t (k) ; i = p n−1 d1(i) + p n−2 d2(i) ( j) ( j) ( j) (i) + · · · + pdn−1 + t (i) ; j = p n−1 d1 + p n−2 d2 + · · · + pdn−1 + t ( j) for some t (k) , ( j) t (i) , t ( j) , ds(k) , ds(i) , ds ∈ Z p ∀ 1 ≤ s ≤ n − 1.

Proof |X | = p n and |Rk | = p n for all 0 ≤ k ≤ p n − 1. All the relations Rk are disjoint, and they form partition of G. Let Ri , R j , Rk be arbitrary relations in G. Now, as in previous results, we will show that for each pair A, B with (A, B) ∈ Rk , the cardinality |{C ∈ X |(A, C) ∈ Ri , (C, B) ∈ R j }| is a constant. Let (A, B) ∈ Rk where A = [a1 , a2 , . . . , an ], B = [b1 , b2 , . . . , bn ] ∈ X . Also, let A Ri = A = [a1 , a2 , . . . , an ] and B R ∗j = B  = [b1 , b2 , . . . , bn ]. Now, (A, B) ∈ Rk implies bn ≡ (k + t (k) + an ) mod p, bs ≡ (ds(k) + as ) mod p (k) ∀ 1 ≤ s ≤ n − 1 where k = p n−1 d1(k) + p n−2 d2(k) + · · · + pdn−1 + t (k) ; for some (k) (k) t , ds ∈ Z p ∀ 1 ≤ s ≤ n − 1. Similarly, as (A, A ) ∈ Ri and (B  , B) ∈ R j , we have an ≡ (i + t (i) + an ) mod p,  as ≡ (ds(i) + as ) mod p ∀ 1 ≤ s ≤ n − 1 where i = p n−1 d1(i) + p n−2 d2(i) + · · · + (i) pdn−1 + t (i) ; for some t (i) , ds(i) ∈ Z p ∀ 1 ≤ s ≤ n − 1, and bn ≡ ( j + t ( j) + bn ) mod ( j) ( j) ( j) p, bs ≡ (ds + bs ) mod p ∀ 1 ≤ s ≤ n − 1 where j = p n−1 d1 + p n−2 d2 + · · · + ( j) ( j) pdn−1 + t ( j) ; for some t ( j) , ds ∈ Z p ∀ 1 ≤ s ≤ n − 1. Since every pair (A, B) are ith associates for exactly one i, pikj can be either 0 or 1. We have pikj = 1 if and only if the two circulant matrices A = B  that is, ( j)

if k + t (k) ≡ (i + j + t (i) + t ( j) ) mod p, and ds(k) ≡ (ds(i) + ds ) mod p ∀ 1 ≤ s ≤ n − 1. Adding these equations, we have the desired result, and hence, (X, G) is a commutative AS with non-symmetric relations.

4 Conclusions In this article, a methodology for constructing association schemes is presented. We have computed non-symmetric commutative association schemes for some cyclic groups, dihedral groups and symmetric groups without using conjugacy classes. We have also constructed commutative non-symmetric association schemes for circulant matrices and finite group rings over Zn . One can work on the construction of symmetric association schemes of these groups and group rings. Also, one can construct zeta functions of these association schemes. Acknowledgements I would like to thank International Centre for Theoretical Sciences, where we participated in the program-Group Algebras, Representations and Computation (Code: ICTS/Prog-

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garc 2019/10) and got objective for this research. I would like to express my sincere gratitude and deep appreciation to Professor R. K. Sharma for his advice, suggestions and encouragement in our research work.

References 1. Bose, R.C., Shimamoto, T.: Classification and analysis of partially balanced incomplete block designs with two associate classes. J. Am. Stat. Assoc. 47(258), 151–184 (1952). https://doi. org/10.1080/01621459.1952.10501161 2. Bray, J., Conder, M., Leedham-Green, C., O’Brien, E.: Short presentations for alternating and symmetric groups. Trans. Am. Math. Soc. 363(6), 3277–3285 (2011). https://doi.org/10.1090/ S0002-9947-2011-05231-1 3. Davis, P.J.: Circulant Matrices. American Mathematical Society (2013) 4. Hanaki, A., Uno, K.: Algebraic structure of association schemes of prime order. J. Alg. Combin. 23(2), 189–195 (2006). https://doi.org/10.1007/s10801-006-6923-7 5. Jørgensen, L.K.: Schur rings and non-symmetric association schemes on 64 vertices. Discr. Math. 310(22), 3259–3266 (2010). https://doi.org/10.1016/j.disc.2010.03.002 6. Tomiyama, M., Yamazaki, N.: Characterization of the group association scheme of the symmetric group. Eur. J. Combin. 19(2):237–255 (1998). https://doi.org/10.1006/eujc.1997.0168 7. Yue, M., Li, Z.: Construction of an association scheme over dihedral group. J. Math. 35(1):103– 109 (2015). http://sxzz.whu.edu.cn/html/2015/1/20150112.htm 8. Zieschang, P.-H.: An Algebraic Approach to Association Schemes. Springer (2006)

Rings Whose Nonunits Are Multiple of Unit and Strongly Nilpotent Element Dinesh Udar

Abstract A new class of rings in which each nonunit element of a ring R is a product of a unit and a strongly nilpotent element is introduced in this paper. We obtain various properties and a complete characterization of these rings. We also investigate the subclass of these rings in which this multiplicative decomposition of nonunit elements is unique. In last section, we study the group ring of these rings and obtain a complete characterization for the same. Keywords Units · Nilpotents · Strongly nilpotents · UN rings

1 Introduction In the last two decades, there has been tremendous research in the direction of finding the structure of a ring in terms of some specific elements like idempotents, units and nilpotents. Although it was started in 1977 by Nicholson [7], where he introduced clean rings, but much of the momentum in this direction is gained in last two decades and various generalizations and subclasses of clean rings have been introduced and investigated. If each element of a ring R is sum of an idempotent and a unit, then it is called a clean ring. Researchers have either tried to express the ring elements as either sum or as product of idempotents, nilpotents and units. In [4], Diesl introduced nil clean rings, i.e., a ring with each element as sum of a nilpotent element and an idempotent. In [2], Chen et al. introduced and studied a subclass of nil clean rings which they named it as strongly P-clean rings. In this sub class of nil clean rings, nilpotent elements are replaced with strongly nilpotent elements and which commute with the corresponding idempotent. As far as product decomposition of elements is concerned C˘alug˘areanu in [1] introduced UN-rings. A ring R is UN-ring if each nonunit of it be expressed as product of a unit and a nilpotent element. These rings were further investigated by Vámos in [10]. Motivated by strongly P-clean rings and D. Udar (B) Department of Applied Mathematics, DTU, New Delhi, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_3

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UN-rings, we investigate that what properties a ring R exhibit if each nonunit of it is product of a unit and a strongly nilpotent element. By following the usual norm, we call these rings as US-rings. In Sect. 2, various properties of US-rings are obtained, and this section contains Theorem 1 as the chief result of this section is which states that R is US-ring iff R/P(R) is a skew field (division ring). We also study uniquely US-rings. In Sect. 3, US group rings are studied and the chief result obtained in this section is that a group ring RG is a US-ring iff R is a US-ring, G is locally finite p-group and p ∈ P(R). Through out the paper, we take R to be an associative ring with 1 = 0. The Jacobson radical is denoted by J (R), the prime radical by P(R) and the unit group by U (R). For ring theoretic results and notations, we refer to Lam [5] and for group rings we refer to Connell [3] and Passman [9].

2 US-Rings If each sequence a = a0 , a1 , a2 , . . . for ai+1 ∈ ai Rai is eventually terminating to zero, then a ∈ R is said to be strongly nilpotent element. The class strongly nilpotent elements are a subclass of nilpotent elements in a ring. Example 1 Let  usconsider the 2 × 2 full matrix ring M2 (R) over the field of reals. 01 The element is nilpotent in M2 (R) but not strongly nilpotent. 00 If we take R to be a commutative ring, then nilpotent and strongly nilpotent elements coincide. Now, we begin with certain basic properties of US-rings. Lemma 1 The foctor ring of a US-ring is US. Proof et f : R → S be an epimorphism, where R be a US-ring. Since, image of a unit is a unit and a strongly nilpotent element is strongly nilpotent, we get that ring S is a US-ring. The converse of lemma is not true. Example 2 The ring Z /2Z is US-ring but Z is not a US-ring. A reduced ring is defined to be a ring without nonzero nilpotent elements (except 0). It can be easily seen that a reduced US-ring is a skew field. We are listing it below in form of a Lemma, so as to use it at a later stage. Lemma 2 If R is a reduced (in particular with no nonzero strongly nilpotent element) US-ring, then R is a skew field. Let Z (R) denote the center of R.

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Lemma 3 If R is a US-ring, then is Z (R) also a US-ring. Proof Let us take an element x ∈ Z (R) which is a nonunit in Z (R). As Z (R) is a direct summand of R, so by [5, Exercise 5.0], we get that x is a nonunit of R. So, we have x = ub where u ∈ U (R) and b is strongly nilpotent in R. Thus, b = u −1 x and as b is nilpotent (say bn = 0, for some n ∈ N) and x ∈ Z (R), we get 0 = bn = (u −1 )n x n . Which gives us that x n = 0, i.e., x is nilpotent in Z (R). It is evident that a nilpotent element is same as strongly nilpotent in a commutative ring. Thus, x is strongly nilpotent in Z (R). What we have got is that Z (R) comprises of only units or strongly nilpotent elements, which concludes that Z (R) is a US-ring. Lemma 4 Let R be a US-ring, then R has no central idempotents except identity. Proof As R is a US-ring, so this implies that Z (R) consists of only units and strongly nilpotent elements. An idempotent can neither be a unit nor a strongly nilpotent. Thus, Z (R) can not contain an idempotent and hence the result. A ring is termed as 2-good if it satisfies the property each of its element is sum of two units. Lemma 5 Let R be a US-ring. If identity is 2-good, then R is a 2-good ring. Proof First of all we show that all the nonunits of a US-ring R are 2-good. Let x ∈ R be a nonunit element. So we can have x = ub, where u is a unit and b is strongly nilpotent. We can write x = u(1 + b) − u. It is evident that if b is nilpotent, then 1 + b is a unit. So, x = u(1 + b) − u is a decomposition of x as sum of two units, and hence x is 2-good. Now we show that units of a US-ring are also 2-good. Let v ∈ U (R), we can write v as v = 1.v. As 1 is 2-good (say 1 = μ1 + μ2 , where μ1 , μ2 ∈ U (R)), so v = (μ1 + μ2 ).v = μ1 .v + μ2 .v ∈ U (R) + U (R). Hence, v is 2-good. So, we get that R is a 2-good ring. Let e ∈ R be such that e2 = e. In many classes of rings, it is seen that if e Re as well as (1 − e)R(1 − e) are satisfying a certain condition, then the ring R also satisfy that condition. For example if for an idempotent e, the e Re as well as (1 − e)R(1 − e) are clean, then it follows that R is also clean ([8], Lemma). But this property may not hold in case of US-ring. Example 3 Let us take the ring Z /6Z . Here, 3¯ is an idempotent in Z /6Z . We can verify that e Re ∼ = Z /2Z and (1 − e)R(1 − e) ∼ = Z /3Z which are both US-ring, but Z /6Z is not. Now, we present the chief theorem of this section. Theorem 1 A ring R is US-ring iff R/P(R) is a skew field.

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Proof Let R be a US-ring. First of all, we will prove that R is equipped with only one maximal ideal and then show that this maximal ideal is P(R). From Lemma 4, it is clear that no central idempotents except identity are there in R. So it implies that R is an indecomposable ring. Now, let R has more than ideal (say two)  one maximal such that M = N . Then, it can be seen that R/(M N ) ∼ R/N , which = R/M implies that R is decomposable and hence a contradiction. Thus, R is equipped with only one maximal ideal. Let us denote this maximal ideal by M. Now, let x ∈ M, so x is a nonunit and hence can be written as x = ub for some u ∈ U (R) and b a strongly nilpotent element. From [6, Proposition 1, p. 56], b ∈ P(R), and so x ∈ P(R). Thus, M ⊆ P(R) and so P(R) is the only maximal ideal of R. It is evident that P(R) ⊆ J (R), so we get that P(R)=J (R). The two cases which arise are: Case 1: (P(R) = 0) By Lemma 2 and [6, Proposition 1, p. 56] we get that R is a skew field, and hence R/P(R) is a skew field. Case 2: (P(R) = 0) From Lemma 1, R/P(R) is a US-ring and R/P(R) do not contain any non zero strongly nilpotent elements. So from Lemma 2, it follows that R/P(R) is a skew field. Conversely, Let R/P(R) be a skew field. This implies that R consists of only units and strongly nilpotent elements, and hence, R is a US-ring. If each finitely generated subring of an ideal I is nilpotent, then I is said to be locally nilpotent ideal. Corollary 1 For R to be a US-ring it is necessary and sufficient that R is local and J (R) is locally nilpotent. Corollary 2 A commutative ring R is a US-ring iff R is local and J (R) is nilpotent. Corollary 3 A US-ring R is a clean ring. Let us now consider a subclass of US-ring in which the multiplicative decomposition is unique, i.e., each element of R is expressed uniquely as the product of a unit and a strongly nilpotent element. We call these rings as uniquely US-rings. Theorem 2 A ring R is uniquely US-ring iff R is a skew field. Proof Let us consider any strongly nilpotent element ϑ ∈ R. It is evident that 1 + ϑ ∈ U (R). We first show that (1 + ϑ)−1 ϑ = ϑ(1 + ϑ)−1 . We have ϑ(1 + ϑ) = (1 + ϑ)ϑ ⇒ (1 + ϑ)−1 ϑ(1 + ϑ) = ϑ ⇒ (1 + ϑ)−1 ϑ = ϑ(1 + ϑ)−1 As ϑ is nilpotent and (1 + ϑ)−1 ϑ = ϑ(1 + ϑ)−1 , so we get that (1 + ϑ)−1 ϑ is nilpotent. Since a uniquely US-ring is a US-ring, we get from Theorem 1 that (1 + ϑ)−1 ϑ is strongly nilpotent. We see that s has two representations as product of a unit and a strongly nilpotent element, namely

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ϑ = 1.ϑ and ϑ = (1 + ϑ)[(1 + ϑ)−1 ϑ]. Thus 1 = 1 + ϑ =⇒ s = 0, i.e., R is a reduced ring. By using Lemma 2, we get the R to be a skew field.

3 US Group Rings Let RG denote the group ring. Let ω(G) be the augmentation ideal of RG, which is generated by {1 − g|g ∈ G}. It is well established that RG/ω(G) ∼ = R. For I  R, one can verify that I G  RG and also we have RG/I G ∼ = (R/I )G. We first prove a result by taking base ring as skew field. Lemma 6 If D is a skew field with char D = p and G be a p-group which is locally finite, then DG is a US-ring. Proof By using [3, Corollary, p. 682], we obtain that ω(G) is locally nilpotent and hence ω(G) ⊆ P(R). As we have DG/ω(G) ∼ = D, so ω(G) is maximal in DG and hence ω(G) = P(R). Thus, DG/P(RG) ∼ = D. By using theorem 1, we get that DG is a US-ring. We now present the chief result of this section. Theorem 3 For a ring R and a nontrivial group G; the RG is a US-ring iff R is a US-ring, G is a locally finite p-group and p ∈ P(R). Proof Let RG be a US-ring, then the augmentation map f : RG → R given by: ⎛ ⎞   f⎝ ag g ⎠ = ag g∈G

g∈G

is an epimorphism. So, by Lemma 1 we get that R is a US-ring. Now taking the epimorphism φ : RG → (R/P(R))G, we get that (R/P(R))G is a US-ring. Since R is a US-ring, we obtain by Theorem 1 that R/P(R) is a skew field (say D). So, DG is a US-ring. Let ω(G) be the augmentation ideal of DG, then we have DG/ω(G) ∼ = D. Thus, ω(G) is maximal ideal of DG, which implies that ω(G) = P(DG), and hence, ω(G) is locally nilpotent. Now from [3, Corollary, p. 682] we get that G is a p-group which is locally finite and 0 = p ∈ D = R/P(R), i.e., p ∈ P(R). Conversely, let R be a US-ring, G a locally finite p-group and p ∈ P(R). By [3, Proposition 9], we obtain that P(R)G ⊆ P(RG). Also, we have P(RG)/P(R)G = P(RG/P(R)G). So, from these two results we get RG/P(R)G ∼ RG RG ∼ RG/P(R)G = = = P(RG) P(RG)/P(R)G P(RG/P(R)G) P(RG)

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where R = R/P(R) is a skew field (by Theorem 1). By using Lemma 6, we get RG RG RG is a US-ring and then applying Theorem 1 we get P(RG) is a skew field. Thus, P(RG) is a skew field. Again using theorem 1, we conclude that RG is a US-ring.

References 1. C˘alug˘areanu, G.: UN-rings. J. Alg. Appl. 15(9), 9 (2016) 2. Chen, H., Köse, H., Kurtulmaz, Y.: Strongly P-clean rings and matrices. Int. Electron. J. Alg. 15, 116–131 (2014) 3. Connell, I.G.: On the group ring. Can. J. Math. 15, 650–685 (1963) 4. Diesl, A.J.: Nil clean rings. J. Alg. 383, 197–211 (2013) 5. Lam, T.Y.: A First Course in Noncommutative Rings, 2nd edn. Springer, New York (2001) 6. Lambek J.: Lectures on Rings and Modules. AMS Chelsea Publishing (1966) 7. Nicholson, W.K.: Lifting idempotent and exchange rings. Trans. Amer. Math. Soc. 229, 269– 278 (1977) 8. Nicholson, W.K.: Extensions of clean rings. Commun. Alg. 29(6), 2589–2595 (2001) 9. Passman, D.S.: The Algebraic Structure of Group Rings. Wiley, New York (1977) 10. Vámos, P.: On rings whose nonunits are a unit multiple of a nilpotent. J. Alg. Appl. 16(6), 13 (2017)

Improved Lower Bounds on Second Order Non-linearities of Cubic Boolean Functions Ruchi Telang Gode, Shahab Faruqi, and Ashutosh Mishra

Abstract Nonlinearity of 2nd order is important in the study of Boolean functions from a cryptographic viewpoint. This property is useful in choosing efficient Boolean functions resisting quadratic approximation attacks. We obtain improved lower bounds with respect to the nonlinearities of 2nd order of some classes of i j degree 3 monomial Boolean functions of type g(x) = T r (x 2 +2 +1 ) (here i, j ∈ N) and i ≤  n2 , j ≤ i. We were able to obtain these bounds in an efficacious way for Boolean functions upto 13 variables. Obtained results refine the bounds obtained by Gode and Gangopadhyay [10] and Mesnager et al. [21]. This shows the efficiency of these classes of functions with respect to nonlinearity of 2nd order. Keywords Symmetric key cryptography · Boolean functions · Derivative of Boolean function · Nonlinearity of 2nd order · Linearized polynomial

1 Introduction Boolean functions are important units for symmetric key cipher systems. Substitution box (S-box) used in block ciphers is a vectorial Boolean function. Several stream ciphers centered on LFSR use Boolean functions either as “Nonlinear Combiner Functions” or as “Nonlinear Filter Functions.” Block type ciphers and stream ciphers are vulnerable to several attacks, and one may refer [4] for details on these attacks. So, there arises a requirement to design Boolean functions having cryptographically useful attributes. Some of the relevant cryptography-related attributes of Boolean functions are Balancedness, Nonlinearity, Algebraic Immunity, Correlation Immunity, and Autocorrelation [4].

R. T. Gode (B) · S. Faruqi National Defence Academy, Pune, India A. Mishra NIT Manipur, Imphal, Manipur, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_4

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1.1 Representation of Boolean Functions Definition 1 Suppose F2n refer the finite field having order 2n over F2 . A function g : F2n → F2 is called a Boolean function on n variables. Bn denotes the collection of all n- variable Boolean functions. For a basis S = {s1 , s2 , . . . , sn } ∈ F2n , an element α ∈ F2n can be expressed as α = α1 s1 + α2 s2 + · · · + αn sn where αi ∈ F2 , and i ∈ {1, 2, . . . , n}. The tuple of n elements (α1 , α2 , . . . , αn ) is called the coordinate vector of α ∈ F2n for a fixed basis S. A function g ∈ Bn is expressed by a polynomial in α1 , α2 , . . . , αn over F2 , also called as the “Algebraic Normal Form” of g given as 

g(α1 , α2 , . . . , αn ) =

λa

a=(a1 ,a2 ,...,an )∈{0,1}n

 n 

 αiai

, where λa ∈ F2 .

i=1

The function T r : F2n → F2 is called as Trace function defined as 2

T r (y) = y + y 2 + y 2 + · · · + y 2

n−1

, ∀ y ∈ F2n .

T r (x y) refers an “inner product” of y and x for x, y ∈ F2n . g ∈ Bn can also be given as g(x) = T r (μx r ) where x ∈ F2n , and the whole number r is called a monomial Boolean function for μ ∈ F2n . Here r is the exponent of g and deg(g) = wt (r ).

1.2 Some Definitions Definition 2 The Walsh Hadamard transform of g ∈ Bn at μ ∈ F2n is characterized as  (−1)g(x)+T r (μx) . Wg (μ) = x∈F2n

The multi-set given by [Wg (μ) : μ ∈ F2n ] is called as the Walsh Hadamard Spectrum of Boolean function g. Definition 3 The Hamming distance of function g ∈ Bn from h ∈ Bn is denoted by d(g, h) and described as d(g, h) = |{y : g(y) = h(y), y ∈ Fn2 }|.

Improved Lower Bounds on Second Order …

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1.3 Nonlinearity of Boolean Functions Rothaus [24] introduced the notion of nonlinearity of first order (also called as nonlinearity) for a Boolean function. Nonlinearity of g ∈ Bn is symbolized as nl(g). Matsui [20] discovered that symmetric ciphers that uses Boolean functions with low nonlinearity are vulnerable to “Best Affine Approximation Attacks.” Extensive theoretical and computational research has been done on the nonlinearity property of Boolean function since nonlinearity is associated with the Walsh Hadamard transform, and there is an algorithm to compute fast Walsh transform [19]. Walsh Hadamard Spectrum and nonlinearity of g ∈ Bn are related as: nl(g) = 2n−1 −

1 max |Wg (μ)|. 2 μ∈F2n

Parseval’s identity proves that 

Wg (μ)2 = 22n .

μ∈F2n

it is deduced that max{|Wg (μ)| : μ ∈ F2n } ≥ 2n/2 , which shows that nl(g) ≤ 2n−1 − n n 2 2 −1 . Thus 2n−1 − 2 2 −1 is the upper bound of nl(g). Definition 4 A function g ∈ Bn is called a bent function if and only if its Walsh n n n Hadamard transform Wg (μ) ∈ {2 2 , −2 2 } ∀μ ∈ F2n or nl(g) = 2n−1 − 2 2 −1 . From definition, it is understood that bent function exists only for even values of n. Bent functions are best possible immune to “Best Affine Approximation Attacks.” Refer [1, 4, 5, 16, 23] for more results related to non-linearities of Boolean functions.

1.4 Nonlinearities of Higher Order Definition 5 Suppose g ∈ Bn . The r th-order nonlinearity nlr (g) of the Boolean function g is defined as min{d(g, h) : h ∈ Bn , deg h ≤ r }. The succession of values nlr (g), for 1 < r < n is referred as the nonlinearity profile of g. With respect to Boolean functions the notion of nonlinearities of higher order is associated to “Higher Order Correlation Attacks” and “Low Order Approximations Attacks” [7, 14]. Carlet [3, 6] did an organized study of nonlinearity profile and higher order nonlinearity with respect to Boolean functions, Carlet developed some recursive techniques for obtaining lower bounds of nonlinearities of higher order. Using these recursive technique Carlet computed, the lower bounds of the nonlinearities of 2nd order of various classes of monomial functions including the inverse function n t f inv (y) = T r (y 2 −2 ) and the Welch function f welch (y) = T r (y 2 +3 ), for t = n − 1

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where n is odd, or for t = n + 1 where n is odd. Further, Sun and Wu [25] and Gangopadhyay et al. [11], Gode et al. [13], Li et al. [18] have obtained the bounds on nonlinearity of 2nd order for some classes of monomial functions of degree 3. Further, Mesnager et al. [21] have computed lower bounds of the nonlinearities of 2r r second order of degree 3 Boolean function of the form gμ (x) = T r (μx 2 +2 +1 ), ∗ μ ∈ F2n , r is a whole number. Millan [22] has presented an algorithm allowing the identification of good low order approximations and discussed required trade-offs in resisting both linear and quadratic approximations. For details on algorithms to obtain nonlinearities of 2nd order of Boolean functions one may refer [8, 9, 15]. These algorithms can work efficiently for n ≤ 11 as well as n ≤ 13 in some particular cases. In our work, we consider a class of degree 3 monomial functions defined i j by g(x) = T r (x 2 +2 +1 ) where 0 < i ≤  n2 , 0 < j ≤ i. An algorithm to compute lower bounds on the nonlinearities of 2nd order of the above classes of monomial functions is designed using properties of linearized polynomials. It is worth mentioning that the same results for the functions of these forms were derived earlier by Mesnager et al. [21], Gode and Gangopadhyay [10], and Li et al. [18]. The bounds obtained in this paper are improved upon those obtained in [10, 18, 21]. In Sect. 2 some preliminary results and recursive formulae on nonlinearities of higher order of functions are given.

2 Preliminary Results 2.1 Recursive Lower Bounds on Nonlinearities of Higher Order of Boolean Functions Carlet [6] proposed recursive formulae for estimating the lower bounds on the nonlinearities of r th-order of a Boolean function g, these bounds are determined by (r − 1)th order of non-linearities of the derivative of g. Definition 6 Derivative of g ∈ Bn reference to c ∈ F2n ,is symbolized by Dc g and defined as Dc g(y) = g(y) + g(y + c) ∀y ∈ F2n . Proposition 1 ([6], Corollary 2, Remark) Let g ∈ Bn , r < n, r ∈ N ∪ {0}. We have nlr (g) ≥ 2n−1 −

  1 2n 2 −2 nlr −1 (Dc (g)). 2 n c∈F2

In particular for r = 2, nl2 (g) ≥ 2n−1 −

  1 2n 2 −2 nl(Dc (g)). 2 n c∈F2

Improved Lower Bounds on Second Order …

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Proposition 1 is helpful for obtaining the lower bounds of nl2 (g) for any cubic Boolean function g. Because algebraic degree of a first-order derivative of a function of degree 3 is at most two, and the Walsh Hadamard spectrum of a degree two function can be evaluated as described in Sect. 2.2.

2.2 Quadratic Boolean Functions [2, 23] Suppose g ∈ Bn is degree two Boolean function. The bilinear form B(x, y) corresponding to g is defined as B(x, y) = g(0) + g(x) + g(y) + g(x + y). Let Eg denote the kernel of B(x, y), which is Eg = {x ∈ F2n : B(x, y) = 0 for all y ∈ F2n } .

2.3 Walsh Hadamard Spectrum of a Quadratic Boolean Function Lemma 1 ([2, p. 224], [19, p. 441, Theorem 5] If g : F2n → F2 is a Boolean function of degree 2, α ∈ F2n and B(x, y) is its associated bilinear form, here the Walsh Hadamard Spectrum of g , Wg (μ), is dependent only on the dimension d, of Eg (the kernel) of B(x, y). The weight dispersion of Wg (μ) is (Table 1) [2, p. 224].

2.4 Linearized Polynomial Let q = p n for a prime p. The definition of linearized polynomial is as follows. Definition 7 ([17, p. 108]) A polynomial Q(x) =

n 

αi x q

i

i=0

where αi ∈ Fq r (an extension field of Fq ) is called a linearized polynomial over Fq r . Table 1 Walsh Hadamard Spectrum of a degree two Boolean function

Wg (μ)

Count of μ

0

2n − 2n−d

2

(n+d) 2

−2

(n+d) 2

2n−d−1 + (−1)g(0) 2 2n−d−1 − (−1)g(0) 2

(n−d−2) 2 (n−d−2) 2

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Lemma 2 ([17, p. 108]) Let Q(x) be a nonzero linearized polynomial over Fq m and let the extension field Fq r of Fq m contains all the roots of Q(x). Then 1. The multiplicity of every zero of Q(x) is identical which is either some exponent of q or 1. 2. The zeros of Q(x) make a subspace of Fq r , where Fq r is considered as a vector space over Fq .

3 Algorithms An algorithm for finding the count of zeros of linearized polynomial over F2n is given below. Algorithm 1: Count of zeros of Q(x) INPUT: An integer n and the linearized polynomial Q(x) over F2n n COMPUTE: gcd(Q(x), x 2 + x) k OBTAIN: gcd is of the form x 2 + x. OUTPUT: Integer k Here k denotes the count of zeros of corresponding Q(x) over F2n . Our second algorithm computes the requisite lower bound. Algorithm 2: Lower Bounds on nl2 (g) INPUT: A Boolean function of form g = T r (x 2 +2 +1 ) over F2n . COMPUTE: The first order derivative Dc (g) of g. COMPUTE: Bilinear form B(x, y) associated with Dc (g) and corresponding Linearized Polynomial Qa (x). OBTAIN: Using Algorithm 1 obtain k for different values of c. CALCULATE: Walsh Spectrum and Nonlinearity of Dc (g) CALCULATE: Lower bound on nl2 (g) using Proposition 1. i

j

4 Previous Results Some known results on the bounds of nl2 (g) where g(x) = T r (x 2 +2 variable function are as follows: i

j

+1

) is an n-

Theorem 1 ([10, Theorem 3]) Let g ∈ Bn characterized by the class g(x) = i j T r (x 2 +2 +1 ) then for n > 2i  nl2 (g) ≥

2n−1 − 23n+2i−4/4 , n even, 2n−1 − 23n+2i−5/4 , n odd.

Improved Lower Bounds on Second Order …

37

Theorem 2 ([21, Example 18, Corollary 19]) Lower bounds on the nonlinearities 2r r of 2nd order of gλ = T r (λx 2 +2 +1 ) over F2n for n = sr with s = 3, 4, 5, 6, 7 are 1. For n = 3r nl2 (gλ ) ≥ 23r −1 −

1  3r 2 + (23r − 1)22r . 2

nl2 (gλ ) ≥ 24r −1 −

1  4r 2 + (24r − 1)23r . 2

nl2 (gλ ) ≥ 25r −1 −

1  5r 2 + (25r − 1)24r . 2

nl2 (gλ ) ≥ 26r −1 −

1  6r 2 + (26r − 1)25r . 2

nl2 (gλ ) ≥ 27r −1 −

1  7r 2 + (27r − 1)25r . 2

2. For n = 4r

3. For n = 5r

4. For n = 6r

5. For n = 7r

Theorem 3 ([21, Theorem 21]) Let gμ = T r (μx 2 is coprime to r , then

2r

+2r +1

) where μ ∈ F2n and n ≥ 4

1. For n ≡ 2 mod 6, 4 mod 6 nl2 (gμ ) ≥ 2

n−1

1 n 7 3n 10 n − 2 + 2 2 − 22 . 2 3 3

2. For n ≡ 0 mod 6 n−1

1 n 8 3n 8 n − 2 + 2 2 − 22 . 2 3 3

n−1

1 n 3n+1 n+3 − 2 + 32 2 − 2 2 . 2

nl2 (gμ ) ≥ 2 3. If n ≥ 5 but n ≡ ±1 mod 6 nl2 (gμ ) ≥ 2 Corollary 1 ([21, Corollary 24]) 1. For

2n 2. For

2n

n = sr here s > 3 is an odd integer, we have nl2 (gμ ) ≥ 2n−1 −

1 2

n+3r

+ (2n − 1)2 2 . n = sr here s > 2 is an even integer, we have nl2 (gμ ) ≥ 2n−1 − n

+ (2n − 1)2 2

22r +1 . 2r +1

1 2

38

R. T. Gode et al.

5 Main Results 5.1 Improved Lower Bounds Mesnager et al. [21] and Gode and Gangopadhyay [10] have derived bounds for non2r r linearities of 2nd order for some functions of degree 3 of type g(x) = T r (x 2 +2 +1 ), i j for a positive integer r and g(x) = T r (x 2 +2 +1 ) for positive integers i, j respectively. Using Algorithms 1 and 2, we got improved lower bounds for general class i j of degree 3 Boolean functions given as g(x) = T r (x 2 +2 +1 ), integers i, j > 0 and i ≥  n2 , j < i for 7 ≤ n ≤ 13 as follows. The result is given below. Theorem 4 Let g ∈ Bn be given as g(x) = T r (x 2 +2 +1 ) for 7 ≤ n ≤ 13 and for i ≤ n2 and j < i then lower bounds on nl2 (g) for 7 ≤ n ≤ 13 are given in Table 2. i

j

Proof First derivative, Dc g, of g having reference to c ∈ F∗2n given as Dc g(x) = g(x + c) + g(x) = T r ((x + c)2 +2 i

2 +2 i

2 +1 2

j

i

j

+1

2 +1 2

j

j

i

) + T r1n (x 2 +2 i

i

= T r (x c+x c +x c +x c 2 j 2i +1 2i +2 j 2i +2 j +1 +x c + xc +c ). 2

j

+1

)

2 +1 j

If Dc g is quadratic then the Walsh Hadamard Spectrum of Dc g is equivalent to the Walsh Hadamard Spectrum of h(x), where h(x) is the quadratic part of Dc g. h(x) = T r (x 2 +2 c + x 2 +1 c2 + x 2 i

j

i

j

j

+1 2i

c ).

Let B(x, y) be the associated bilinear form corresponding to h(x), the kernel Eh of B(x, y) is given as Eh = {x ∈ F2n : B(x, y) = 0 for all y ∈ F2n } , and k the dimension of Eh . Now, B(x, y) = h(0) + h(y) + h(x) + h(x + y) = T r (x 2 +2 c + x 2 +1 c2 + x 2 i

j

i

2i +2 j

+T r ((x + y)

j

j

+1 2i

c ) + T r (y 2 +2 c + y 2 +1 c2 + y 2 i

j

2i +1 2 j

c + (x + y)

2j

2j

2i

j

j

i

j

j

i

i

c + (x + y)

2i

2i

2j

j

j

+1 2i

j

i

c )

2 j +1 2i

c )

2i 2 j

j

i

= T r (((xc + x c)y + (xc + x c)y + (x c + x 2 c2 )y)) i

i

j

i

j

= T r ((xc2 + x 2 c)y 2 ) + T r ((xc2 + x 2 c)y 2 ) + T r ((x 2 c2 + x 2 c2 )y) = T r ((xc2 + x 2 c)y 2 )2 i

j

j

n−i

i

i

j

+ T r ((xc2 + x 2 c)y 2 )2

n− j

i

+T r ((x 2 c2 + x 2 c2 )y) n−i

= T r ((x 2 c2

n−i+ j

+ x2

n−i+ j

n−i

n

c2 )y 2 ) + T r ((x 2

n− j

c2

n+i− j

+ x2

n+i− j

c2

n− j

n

)y 2 )

Improved Lower Bounds on Second Order …

39

Table 2 Lower bounds on the nonlinearities of 2nd order for functions of form g(x) = i j T r (x 2 +2 +1 ) for 7 ≤ n ≤ 13, i ≥ n2 and j < i Lower bounds on nl2 (g)

n

i

j

g(x)

7

2

1

T r (x 7 )

36

7

3

2

T r (x 13 )

35

7

3

1

T r (x 11 )

35

8

2

1

T r (x 7 )

81

8

3

2

T r (x 13 )

79

8

3

1

T r (x 11 )

79

8

4

3

T r (x 25 )

80

8

4

2

T r (x 21 )

63

8

4

1

T r (x 19 )

80

9

2

1

T r (x 7 )

174

9

3

2

T r (x 13 )

174

9

3

1

T r (x 11 )

174

9

4

3

T r (x 25 )

174

9

4

2

T r (x 21 )

174

9

4

1

T r (x 19 )

174

10

2

1

T r (x 7 )

378

10

3

2

T r (x 13 )

378

10

3

1

T r (x 11 )

352

10

4

3

T r (x 25 )

375

10

4

2

T r (x 21 )

343

10

4

1

T r (x 19 )

375

10

5

1

T r (x 35 )

378

10

5

2

T r (x 37 )

378

10

5

3

T r (x 41 )

378

10

5

4

T r (x 49 )

378

11

2

1

T r (x 7 )

802

11

3

2

T r (x 13 )

793

11

3

1

T r (x 11 )

793

11

4

3

T r (x 25 )

793

11

4

2

T r (x 21 )

802

11

4

1

T r (x 19 )

793

11

5

4

T r (x 49 )

793

11

5

3

T r (x 41 )

793

11

5

2

T r (x 37 )

793

11

5

1

T r (x 35 )

793

12

2

1

T r (x 7 )

1671

12

3

2

T r (x 13 )

1633

12

3

1

T r (x 11 )

1656

12

4

3

T r (x 25 )

1659

12

4

2

T r (x 21 )

1659

12

4

1

T r (x 19 )

1659

12

5

1

T r (x 35 )

1549

13

2

1

T r (x 7 )

3468

40

R. T. Gode et al. i

j

j

i

+T r ((x 2 c2 + x 2 c2 )y) i

j

j

i

−j

i− j

−j

i− j

= T r (y(x 2 c2 + x 2 c2 + x 2 c2 + x 2 c2

−i

+ x 2 c2

j−i

j−i

−i

+ x 2 c2 ))

= T r (y Pc (x)). Therefore, Eh = {x ∈ F2n : Pc (x) = 0}. The size of the kernel Eh is same as the count of roots of Pc (x), or identically to the count of roots of (Pc (x))2i . Put (Pc (x))2i = Qc (x). Thus, i

j

j

i

i− j

−j

−j

i− j

j−i

−i

−i

j−i

i

Qc (x) = (x 2 c2 + x 2 c2 + x 2 c2 + x 2 c2 + x 2 c2 + x 2 c2 )2 (1) 2i i+ j i+ j 2i 2i− j i− j i− j 2i− j j j = x 2 c2 + x 2 c2 + x 2 c2 + x 2 c2 + x 2 c + xc2 . (2) Thus deg(Qc (x)) ≤ 22i . Now substituting different values of n, i for 2 ≤ i ≤  n2  and j < i in Eq. (2), we may obtain different classes of functions on n variables with their corresponding linearized polynomial Qc (x). The values of k can be evaluated using Algorithm 1. Further we can calculate the nonlinearity of derivative of g by substituting the values of n and k in 1 n+k nl(Dc g) ≥ 2n−1 − 2 2 2 for all c ∈ F∗2n . Finally putting the corresponding values of nl(Dc g) in below formula (Proposition 1)   1 2n nl2 (g) ≥ 2n−1 − 2 −2 nl(Dc (g)) 2 n c∈F2

we can get bounds for the nonlinearities of 2nd order for g(x). The above procedure was implemented by us using SAGE, and the results are as given in Table 2. Hence the proof completed.  For example, if n = 11, i = 2, j = 1 we obtained the following results (Table 3). We can obtain different forms of Qc (x) by varying c over F211 . For example if c = 1 1 we have Q 1 (x) = x 16 + x 8 + x 8 + x 2 + x 2 + x and gcd(Q 1 (x), x 2 1 + x) = x 2 + x, which gives k = 1. Substituting n = 11, i = 2, j = 1 and c = 1 in nl(Dc g) ≥ n+k 11+1 2n−1 − 21 2 2 we obtain nl(D1 g) ≥ 211−1 − 21 2 2 = 992. Applying Proposition 1, we get   1 22 nl2 (g) ≥ 211−1 − 2 −2 nl(Dc (g)) = 802. 2 c∈F 211

Table 3 Results for n = 11, i = 2, j = 1 n i j g(x) Qc (x) 11

2

1

T r (x 7 )

c8 x 16 + c16 x 8 + c2 x 8 + c8 x 2 + cx 2 + c2 ∗ x

n

gcd(Q1 (x), x 2 + x)

k

x2 + x

l

Improved Lower Bounds on Second Order …

41

Table 4 Comparison of different lower bounds on the non-linearities of 2nd order of g(x) = i j T r (x 2 +2 +1 ) n

i

j

g(x)

Our

Bounds of [21]

Bounds

6

2

1

T r (x 7 )

–

bounds

15

10

7

2

1

T r (x 7 )

36

25

32

7

3

2

T r (x 13 )

35

25

19

7

3

1

T r (x 11 )

35

25

19

8

2

1

T r (x 7 )

81

79

64

8

3

2

T r (x 13 )

79

–

38

8

3

1

T r (x 11 )

79

–

38

8

4

3

T r (x 25 )

80

–

–

8

4

2

T r (x 21 )

63

–

64

8

4

1

T r (x 19 )

80

–

–

9

2

1

T r (x 7 )

174

–

166

9

3

2

T r (x 13 )

174

–

128

9

3

1

T r (x 11 )

174

–

128

9

4

3

T r (x 25 )

174

–

75

9

4

2

T r (x 21 )

174

–

75

9

4

1

T r (x 19 )

174

–

75

10

2

1

T r (x 7 )

378

–

331

10

3

2

T r (x 13 )

378

–

256

10

3

1

T r (x 11 )

352

–

256

10

4

3

T r (x 25 )

375

–

150

10

4

2

T r (x 21 )

343

256

150

10

4

1

T r (x 19 )

375

–

150

10

5

1

T r (x 35 )

378

–

–

10

5

2

T r (x 37 )

378

–

–

10

5

3

T r (x 41 )

378

–

–

10

5

4

T r (x 49 )

378

–

–

11

2

1

T r (x 7 )

802

710

768

11

3

2

T r (x 13 )

793

–

662

11

3

1

T r (x 11 )

793

–

662

11

4

3

T r (x 25 )

793

–

512

11

4

2

T r (x 21 )

802

710

512

11

4

1

T r (x 19 )

793

–

512

11

5

4

T r (x 49 )

793

–

300

11

5

3

T r (x 41 )

793

–

300

11

5

2

T r (x 37 )

793

–

300

11

5

1

T r (x 35 )

793

–

300

12

2

1

T r (x 7 )

1671

1629

1536

12

3

2

T r (x 13 )

1633

–

1324

12

3

1

T r (x 11 )

1656

–

1324

12

4

3

T r (x 25 )

1659

–

1024

12

4

2

T r (x 21 )

1659

1024

1024

12

4

1

T r (x 19 )

1659

–

1024

12

5

1

T r (x 35 )

1549

–

600

13

2

1

T r (x 7 )

3468

3208

3372

of [10]

42

R. T. Gode et al.

6 Comparisons Comparisons of our results with previous bounds are tabulated in Table 4. Results depict the improvement of our bounds than previous ones, and our algorithm can be i j applied for general class of cubic functions of form T r (x 2 +2 +1 ). r

Remark 1 T r (x k ) = T r ((x k )2 ) for all k, r ∈ N. Remark 2 When gcd(2n−1 , k) = 1 then T r (x k ) is affinely equivalent to T r (μx k ) where μ ∈ F2n .

7 Conclusion We computed lower bounds for nonlinearities of 2nd order of few classes of degree 3 monomial functions. Our bounds improved upon previously known bounds [10, 21] when 7 ≤ n ≤ 13 . Our results also provide the idea related to the choice of i i j and j for which the class of functions having the form T r (x 2 +2 +1 ) possess high nonlinearities of second order. Results given in Sect. 6 indicate that there are classes of cubic functions which have a large Hamming distance from quadratic functions and therefore these functions are optimally resistive to “Quadratic Function Approximation Attacks." We hope that the results obtained will play a significant role in choosing cryptographically useful Boolean functions.

References 1. Berlekamp, E.R., Welch, L.R.: Weight distributions of the cosets of the (32; 6) Reed-Muller code. IEEE Trans. Inform. Theory 18(1), 203–207 (1972) 2. Canteaut, A., Charpin, P., Kyureghyan, G.M.: A new class of monomial bent functions. Finite Fields Appl. 14, 221–241 (2008) 3. Carlet, C.: The complexity of Boolean functions from cryptographic viewpoint. In: Dagstuhl Seminar Proceedings, Schloss Dagstuhl-Leibniz-Zentrum für Informatik (2006) 4. Carlet, C.: Boolean functions for cryptography and error correcting codes. In: Crama, Y., Hammer, P. (eds.) Boolean Methods and Models in Mathematics, Computer Science, and Engineering, pp. 257–397. Cambridge University Press (2010) 5. Carlet, C.: Vectorial Boolean functions for cryptography. In: Crama, Y., Hammer, P. (eds.) Boolean Methods and Models in Mathematics, Computer Science, and Engineering, pp. 398469. Cambridge University Press (2010) 6. Carlet, C.: Recursive lower bounds on the nonlinearity profile of Boolean functions and their applications. IEEE Trans. Inform. Theory 54(3), 1262–1272 (2008) 7. Courtois N.: Higher order correlation attacks, XL algorithm and cryptanalysis of Toyocrypt. In: Proceedings of the ICISC’02, Lecture Notes in Computer Science, vol. 2587, pp. 182–199. Springer (2002) 8. Dumer, I., Kabatiansky, G., Tavernier, C.: List decoding of second-order Reed-Muller codes up to the Johnson bound with almost linear complexity. In: Proceedings of the IEEE International Symposium on Information Theory, pp. 138–142. Seattle, WA (2006)

Improved Lower Bounds on Second Order …

43

9. Fourquet, R., Tavernier, C.: An improved list decoding algorithm for the second order Reed Muller codes and its applications. Des. Codes Crypt. 49, 323–340 (2008) 10. Gode, R., Gangopadhyay, S.: On second order nonlinearities of cubic monomial Boolean functions. In: IACR Cryptology, ePrint Archive, pp. 502 (2009). Available at: https://eprint. iacr.org/2009/502.pdf 11. Gangopadhyay, S., Sarkar, S., Gode, R.: On the lower bounds of the second-order nonlinearities of some Boolean functions. Inform. Sci. 180, 266–273 (2010) 12. Gode, R., Gangopadhyay, S.: Third-order nonlinearities of a subclass of Kasami functions. In: Cryptography and Communications: Discrete Structures, Boolean Functions and Sequences, vol. 2, pp. 69–83 (2010) 13. Gode, R., Gangopadhyay, S.: On lower bounds of second-order nonlinearities of cubic bent functions constructed by concatenating Gold functions. Int. J. Comput. Math. 88(15), 3125– 3135 (2011) 14. Golic, J.: Fast low order approximation of cryptographic functions. In: Proceedings of the EUROCRYPT’96, Lecture Notes in Computer Science, pp. 268–282. Springer (1996) 15. Kabatiansky, G., Tavernier, C.: List decoding of second-order Reed-Muller codes. In: Proceedings of the Eighth International Symposium of Communication Theory and Applications. Ambleside, UK (2005) 16. Kavut, S., Yucel, M.D.: 9-variable Boolean functions with nonlinearity 242 in the generalized rotation symmetric class. Inform. Comput. 208(4), 341–350 (2010) 17. Lidl, R., Niederreiter, H.: Introduction to finite fields and their applications. Cambridge University Press (1983) 18. Li, X., Hu, Y., Gao, J.: The lower bounds on the second order nonlinearity of cubic Boolean functions. Int. J. Found. Comput. Sci. 22(6), 1331–1349 (2011) 19. MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error Correcting Codes. North-Holland, Amsterdam (1977) 20. Matsui, M.: Linear cryptanalysis method for DES cipher. In: Proceedings of the EUROCRYPT93, Lecture Notes in Computer Science, vol. 765. Springer, pp. 386–397 (1994) 21. Mesnager, S., Kim, K.H., Jo, M.S.: On the number of the rational zeros of linearized polynomials and the second-order nonlinearity of cubic Boolean functions. Cryptogr. Commun., 1–16 (2019) 22. Millan, W.: Low order approximation of cipher functions. In: International Conference on Cryptography: Policy and Algorithms, Lecture Notes in Computer Science, vol. 1029. Springer, pp. 144–155 (1996) 23. Patterson, N.J., Wiedemann, D.H.: The covering radius of the (215 , 16) Reed-Muller code is at least 16276. IEEE Trans. Inform. Theory 29(3), 354–356 (1983) 24. Rothaus, O.S.: On bent functions. J. Comb. Theory A 20, 300–305 (1976) 25. Sun, G., Wu, C.: The lower bounds on the second-order nonlinearity of three classes of Boolean functions with high nonlinearity. Inform. Sci. 179(3), 267–278 (2009)

Cesàro-Riesz Product Summability φ − |C1 R; δ|q Factor for an Infinite series Smita Sonker

Abstract In this article, absolute Cesàro-Riesz summability φ − |C1 R|q for infinite series has been introduced, and a  generalized theorem has been determined for φ − |C1 R; δ|q summability factor for an λn . Further, a set of corollaries has been developed from the main result by using appropriate conditions. Summability techniques are used to reduce inaccuracy. By using the appropriate conditions, previous results can be easily obtained. Like this, the Bounded Input Bounded Output (BIBO) stoutness of drive is enhanced by absolute summability because it is necessary and sufficient condition for BIBO stability. Keywords Minkowski’s inequality · Cesàro-Riesz mean · Hölder’s inequality · Abel’s transform · Absolute summability

1 Introduction In 1890, Cesàro [9] defined a new class of methods, known as Cesàro summability methods to deal with non-converging series. After his pioneer work, summability became a topic of interest for many mathematicians such as Mittal [14], Rhoades and Savas [19, 20] who established many interesting results on absolute summability. Balci [1] defined absolute φ-summability factors and obtained some new results. Bor [2, 3] worked out many theorems on absolute summability and used it on Fourier series. Bor [4, 5] generalized the theorem dealing with factors affecting absolute Riesz summability and established two theorems using more general conditions for the infinite series. Bor [6, 7] enhance the result of Mazhar [13] on |C, 1|k summability. Özarslan [16] modified the result of Bor [8] by using generalized Cesàro summability. Özarslan [17] gave the more generalized result on absolute summability φ − |C, α|k . Özarslan [18] has proved the result on generalized Cesàro summability. Sonker and Munjal [23, 24] determined theorems with minimum conditions on generalized absolute summability for infinite series. S. Sonker (B) National Institute of Technology Kurukshetra, Kurukshetra, Haryana 136119, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_5

45

46

S. Sonker

To the best of the author’s knowledge, only a few mathematicians [10, 11, 21] studied about absolute product summability. Author’s interest is also to add something new in this theory and generalize the previous result obtained. In 2019, Sonker and Munjal [22] established a moderated result for Cesàro-Riesz product summability of infinite series. In this paper, I have extend the result of Sonker and Munjal [22] using the results [13, 15] to determine generalized Cesàro-Riesz product summability factor for infinite series.

2 Definitions and Notations The



an having partial sum {sn } and u n is defined as: un =

∞ 

u nk sk

k=0

before examining φ − |C1 R; δ|k summability, we introduce some fundamental summabilities which are extremely useful to work with the Cesàro-Riesz product summability. Definition 1 Let {sn } bewith BV if |s1 − s0 | + |s2 − s1 | + · · · + |sn − sn−1 | converges. Mathematically, ∞ n=1 |sn | < ∞.  Definition 2 [12]: an is s.t.b. absolutely summable if lim u n = s

n→∞



and

|u n − u n−1 | < ∞.

Definition 3 If Cesàro mean of {sn } is defined by tn , i.e. 1  sv n + 1 v=0 n

tn = and

∞ 

φnq−1 |tn |q < ∞,

n=1

then



an is |C, 1|q summable for q ≥ 1.

Cesàro-Riesz Product Summability …

47

Definition 4 The {tnR } be the nth (R, pn ) mean (Riesz mean) of {sn }, where tnR = If

∞ 

n 1  p v sv . Pn v=0

R q φnq−1 |tnR − tn−1 | < ∞,

n=1

then



an is φ − |R, pn |q summable for q ≥ 1, 0 ≤ δq < 1.

Definition 5 The nth (C1 R) i.e. Cesàro-Riesz sequence-to-sequence transformation of {sn } is denoted by tnC R , where 1  R t . n + 1 k=0 k n

tnC R = If

∞ 

CR q φnq−1 |tnC R − tn−1 | < ∞,

n=1

then If



an is φ − |C1 R|q summable for q ≥ 1. ∞ 

CR q φnl(δq+q−1) |tnC R − tn−1 | < ∞,

n=1

then



an is φ − |C1 R; δ|q for q ≥ 1 and 0 ≤ δq < 1.

3 Known-Result Mazhar [13] provide the result on Cesàro summability as follows: Theorem 1 [13] If λm = O(1), m 

n log n|2 λn | = O(1),

(1) (2)

n=1

and

m  |tv |q v=1

then



an λn is |C, 1|q summable.

v

= O(log m),

(3)

48

S. Sonker

Özarslan [15] has proved the result on Cesàro-summability using φ − |C, 1|q . He proved: Theorem 2 [15] Let {φn } be such that satisfying condition (1), (2), m q−1  φv |tv |q

vq

v=1

and

m q−1  φn n=v

Then



nq

= O(log m), 

=O

q−1

φv v q−1

(4)

 .

(5)

an λn is φ − |C, 1|q summable.

Sonker and Munjal [22] have proved the result on Cesàro-Riesz φ − |C1 R|q summability. They proved: Theorem 3 [22] Let {χn } be a ‘+ve’ non-decreasing which satisfies the condition (1) with ∞  nχn |2 λn | = O(1). (6) n=1 m q−1  φn |sn |q q−1

n=1

n q χn

= O(χm ) as m → ∞,

npn = O(Pn ), Pn = O(npn ) as n → ∞, m q−1  φn |t C R |q

= O(1),

n

n=1 m q−1  φn =O n q Pn−1 n=v

nq 

q−1

φv v q−1 Pv

m q−1  φn =O nq n=v+1

and

then



(8) (9)

 as m → ∞, q−1

φv v q−1

an λn is φ − |C1 R|q summable where q ≥ 1.

(10)



m q−1    φn = O φvq−1 , n n=v+1



(7)

,

(11)

(12)

Cesàro-Riesz Product Summability …

49

4 Lemma In order to demonstrate the main result, the following lemma [25] is significant. Lemma 1 [25] Let {χn } be a positive non-decreasing of reals which satisfies the conditions (1) and (6) Then, ∞  χn |λn | = O(1), (13) n=1

nχn |λn | = O(1),

(14)

χn |λn | = O(1).

(15)

and

5 Main Result The following result is an extended version of Sonker and Munjal [22] by using the result of Mazhar [13] and Özarslan [15] for the Cesàro-Riesz summability. Here, the result has been generalized from φ − |C1 R|q summability to φ − |C1 R; δ|q summability with the help of known results, which is further helpful to the researchers working on this field. The result can be stated as: Theorem 4 If {χn } be a ‘+ve’ non-decreasing which satisfies the conditions (1), (6), (8) with m δq+q−1  φn |sn |q = O(χm ) as m → ∞, (16) q−1 n q χn n=1 m δq+q−1 C R q  φn |t | n

nq

n=1 m δq+q−1  φn n=v

n q Pn−1

 =O

δq+q−1

φv v q−1 Pv

m δq+q−1  φn =O nq n=v+1

and

then

= O(1),



(17)

 as m → ∞, δq+q−1

φv v q−1

 ,

m δq+q−1    φn = O φvδq+q−1 , n n=v+1



(18)

an λn is φ − |C1 R; δ|q summable where q ≥ 1 and 0 ≤ δq < 1.

(19)

(20)

50

S. Sonker

Proof The (C1 R) mean tnC R or Yn of



1  R t n + 1 j=0 j

an λn is given by

n

Yn =

1  1 = n + 1 j=0 P j n

 j  (P j − Pv−1 )av λv .

(21)

v=0

Now, Yn = Yn − Yn−1 (where  denotes the forward deference) =

n n 1 1  1  a v λv − Pv−1 av λv n + 1 v=1 n + 1 Pn v=1

−

j n−1  1  1 (P j − Pv−1 )av λv n(n + 1) j=0 P j v=1

= Y1 + Y2 + Y3 (say).

(22)

For proving the main theorem, it will be enough to show ∞ 

φnδq+q−1 |Yn |q < ∞.

(23)

n=1

Using Minkowski’s inequality, |Y1 + Y2 + Y3 |q ≤ 3q (|Y1 |q + |Y2 |q + |Y3 |q ). Since q is finite, so we will prove that ∞ 

φnδq+q−1 |Yw |q = Tw < ∞ for w = 1, 2, 3.

(24)

n=1

So, we have,

n−1

q m δq+q−1 

 φn



T1 = O(1) λ s

v v

(n)q

n=1 m 

+ O(1)

n=1

v=1 δq+q−1 φn |λn sn |q (n)q

= I1 + I2 (say). Now, by using conditions (6), (13), (14), (16), (19) and (20), we have

(25)

Cesàro-Riesz Product Summability …

51

n−1 m δq+q−1   φn |sv |q |λv | q−1 q n χv n=1 v=1  n−1 q−1  × χv |λv |

I1 = O(1)

v=1

= O(1)

m δq+q−1  v|sv |q |λv | φv q−1

vχv

v=1

= O(1)

m−1 

(v|λv |)

v q−1 v δq+q−1  φr |sr |q

q−1 χr r q r =1 m δq+q−1  φv |sv |q O(1)m|λm | q−1 q χv v v=1 v=1

+

= O(1)

(26)

and I2 = O(1) = O(1)

m δq+q−1  φn

|s |q q−1 n

n=1 ∞ 

(n)q χn |λv |

v=1

= O(1)

∞ 

∞ 

|λv |

v=n

v δq+q−1  φn q−1 (n)q n=1 χn

|sn |q

|λv |χv = O(1).

(27)

v=1

Now, T2 = O(1)

m δq+q−1  φn q

n q Pn n=1

q



n−1



×

sv (Pv−1 λv ) + Pn−1 λn sn



v=1

= O(1)

m δq+q−1  φn q

n=1

n q Pn

|Pn−1 λn sn |q



q

m δq+q−1

n−1  

φn

+ O(1) (− pv λv sv )

q

qP n

v=1

n n=1

q

n−1

m δq+q−1  φn





+ O(1) (P λ s ) v v v q



q n Pn v=1

n=1 = L 1 + L 2 + L 3 (say).

Now, using conditions (6), (13), (14) and (18), we have

(28)

52

S. Sonker

L 1 = O(1)

∞ 

|λv |

v δq+q−1  φn

v=1

= O(1)

∞ 

nq

n=1

|sn |q

1 q−1 χn

|λv |χv = O(1),

(29)

v=1

δq+q−1 n−1  φn L 2 = O(1) | pv ||sv |q |λv |q n q Pn n=1 v=1 m 

= O(1)

 m 

 | pv ||sv |q |λv |q

v=1

⎛ n−1 ⎞q−1  pv ⎟ ⎜ ⎜ v=1 ⎟ ⎜ ⎟ ⎝ Pn ⎠

m δq+q−1  φn n q Pn

n=v+1 ∞ 

m δq+q−1  φv

|sv |q |λn | q q−1 n=v v=1 v χv ∞ n δq+q−1   φv |sv |q |λn | = O(1) = O(1) q−1 q v χv n=1 v=1 = O(1)

(30)

and L 3 = O(1)

m δq+q−1 n−1   Pvq |λv ||sv |q φn q

n=1

⎛

n−1 

·⎝

n q Pn v=1 ⎞q−1

q−1

χv

χv |λv |⎠

v=1

= O(1)

m  Pv |λv ||sv |q v=1 m 

q−1

χv

m δq+q−1  φn n q Pn

n=v+1 δq+q−1 φv = O(1) Pv |λv ||sv |q q−1 χ q−1 P v v v v=1

= O(1) as m → ∞.

(31)

Using condition (17) m δq+q−1  φn T3 = nq n=1

=



n−1 q

1  R

tq

n

j=0

m δq+q−1  φn |tnC R |q = O(1) as m → ∞. nq

(32)

n=1

Collecting conditions (25)–(27), we have T1 = O(1) as m → ∞.

(33)

Cesàro-Riesz Product Summability …

53

Collecting conditions (28)–(31), we have T2 = O(1) as m → ∞.

(34)

So, from (24), (32), (33) and (34), condition (23) holds. This proves the main result.

6 Corollaries From the main result, I have prove previous results as well as some new results in the form of corollaries. +ive real numbers such that the Corollary 1 Let {φn } and {χn } be sequences of  conditions (1), (6), (8), (16) and (18) hold, then an λn is φ − |R, pn ; δ|q where q ≥ 1, 0 ≤ δq < 1. Proof Use φ − |R, pn ; δ|q summability instead of φ − |C1 R; δ|q summability in Theorem 4. Corollary 2 By putting δ = 0 in main theorem, we get Theorem (3) of known result, Corollary 3 Let {φn } and {χn } be such that satisfying (1), (2) with m q−1  φn |sn |q n=1

and

m q−1  φn n=v

then



nq

= O((log m)q )

nq  =O

q−1

φv v q−1

 as m → ∞,

an λn is φ − |C, 1|q where q ≥ 1.

Proof This result is an advanced form of theorem (3) by using pn = 1, δ = 0 and χn = log n in corollary 1.

7 Conclusion This paper concentrated on the study of generalized absolute Cesàro-Riesz summability φ − |C1 R; δ|q factor obtained by the generalization of superimposition of summability (C, 1) on (R, pn ) summability. Through this examination, it has been concluded that the main result is an extended version of known and familiar results as explained in the corollary section. The importance of our result by generalization is to increase

54

S. Sonker

the rate of convergence of the series. So here, by adding extra variables in our main result, the rate of convergence of the series increases. The idea of summability of infinite series has been applied in almost all application areas of science like rectification of signals in FIR filter and IIR filter, to speed of the rate of convergence, orthogonal series and approximation theory. Using these techniques, the output of the waves can be made more balanced and to analyse the behavior of input. Acknowledgements The author offers her true thanks to the Science and Engineering Research Board for giving financial support through Project No.: EEQ/2018/000393.

References 1. Balci, M.: Absolute summability factors. Fac. Sci. Univ. Anqara Ser. A1(29), 63–80 (1980) 2. Bor, H.: On absolute weighted mean summability of infinite series and Fourier series. Filomat 30(10), 2803–2807 (2016) 3. Bor, H.: Generalized absolute Cesàro summability factors. Bull. Math. Anal. Appl. 8(1), 6–10 (2016) 4. Bor, H.: Some new results on infinite series and Fourier series. Positivity 19(3), 467–473 (2015) 5. Bor, H.: A new theorem on the absolute Riesz summability factors. Filomat 28(8), 1537–1541 (2014) 6. Bor, H.: On absolute summability factors. Proc. Anner. Soc. 118(1), 71–75 (1993) 7. Bor, H.: On absolute Riesz summability factors. Rocky Mount. J. Math. 24(4), 1263–1271 (1994) 8. Bor, H.: Absolute summability factors. Atti. Sem. Mat. Fis. Univ. Modena 39, 419–422 (1991) 9. Cesàro, E.: Sur la multiplication des series. Bull. Sci. Math. 14, 114–120 (1890) 10. Chandra, P., Jain, H.C.: Absolute product summability of the Fourier series and its applied series. Commun. Fac. Sci. Uni. Ank. Ser. A 37, 95–107 (1988) 11. Dikshit, H.P.: Absolute (C, 1)(N , pn ) summability of a Fourier series and its conjugate series. Pac. J. Math. 26(2), 245–256 (1968) 12. Flett, T.M.: On an extension of absolute summability and some theorems of Littlewood and Paley. Proc. London Math. Soc. 3(1), 113–141 (1957) 13. Mazhar, S.M.: On |C, 1|k summability factors of infinite series. Indian J. Math. 14(1), 45–48 (1972) 14. Mittal, M.L.: Absolute matrix summability factors of Fourier series. Int. J. Math. Game-Theory Algebra U.S.A. 13(5), 387–398 (2003) 15. Ozarslan, H.S.: On absolute Cesàro summability factors of infinite series. Commun. Math. Anal. 3(1), 53–56 (2007) 16. Ozarslan, H.S.: A note on absolute summability factors. Proc. Indian Acad. Sci. (Math. Sci.) 113(2), 165–169 (2003) 17. Ozarslan, H.S.: Factors for the φ − |C, α|k summability. Adv. Stud. Contemp. Math. 5(1), 25–31 (2002) 18. Ozarslan, H.S.: A note on generalized Cesàro summability. Adv. Pure Appl. Math. 5(1), 1–3 (2014) 19. Rhoades, B.E., Savas, E.: A summability factor theorem for generalized absolute summability. Real Anal. Exchange 31(2), 355–363 (2005) 20. Savas, E., Rhoades, B.E.: On summability factors. Acta Math. Hung. 112, 15–23 (2006) 21. Sulaiman, W.T.: Note on product summability of an infinite series. Int. J. Math. Sci. 2008(372604), 1–10 (2008)

Cesàro-Riesz Product Summability …

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22. Sonker, S., Munjal, A.: Sufficient conditions for infinite series by absolute φ-product summable factor. Kyungpook Math. J 23. Sonker, S., Munjal, A.: Absolute summability factor φ − |C, 1; δ|k of Infinite series. IJMA 10(23), 1129–1136 (2016) 24. Sonker, S., Munjal, A.: Application of quasi- f -power increasing sequences in absolute φ − |C, α; δ; l|k summability. In: Proc. ICCPMA (2017) 25. Sulaiman, W.T.: On some absolute summability factors of infinite series. Gen. Math. Notes 2(2), 7–13 (2011)

Ostrowski-Type Inequalities with Exponentially Convex Functions and Its Applications Anulika Sharma and Ram Naresh Saraswat

Abstract Ostrowski-type inequality is known as the estimate deviation of the function from its mean value using different characteristics of the functions. The inequalities have many applications in the area of Information Theory and Numerical analysis. In this paper, we have solved Ostrowski type inequality with the help of exponentially convex functions and exponentially s-convex function in the second sense. The applications and particular cases of proposed inequalities have also been presented. Keywords Convex function · Exponentially convex · Ostrowski’s inequality · New f-divergence measure

1 Introduction Let I = [k1 , k2 ] → R be a differentiable mapping on (k1 , k2 ), and its derivative I , : (k1 , k2 ) → R is bounded on (k1 , k2 ), i.e., |I , (l)| ≤ M∀l ∈ (K 1 , K 2 ). Then, the inequality is: 1 I (l) − k1 − k2

k2 k1

  M (l − k1 )2 + (k2 − l)2 , ∀l ∈ [k1 , k2 ], I (t)dt ≤ k2 − k1 2 (1.1)

This inequality was introduced by [1, 4–6, 12, 15]. Definition 1.1 The set K in H is said to be a convex set if u + f (v − u) ∈ K ∀u, v ∈ K , f ∈ [0, 1]

(1.2)

Definition 1.2 A positive function I is said to be exponentially convex if A. Sharma · R. N. Saraswat (B) Department of Mathematics and Statistics, Manipal University Jaipur, Jaipur, Rajasthan 303007, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_6

57

58

A. Sharma and R. N. Saraswat

I ( f k1 + (1 − f )k2 ) ≤ (1 − f )eβ I (k1 ) + f eβ I (k2 ) ∀k1 , k2 ∈ K , f ∈ [0, 1] and β ∈ R

(1.3)

Definition 1.3 Let s ∈ (0, 1] and K R0 A function I : K → R be said to be exponentially s-convex in the second sense if I ( f k1 + (1 − f )k2 ) ≤ f s eβ I (k1 ) + (1 − f )s eβ I (k2 ) ,

(1.4)

∀k1 , k2 ∈ K and f ∈ [0, 1]. This is introduced by [10]. Theorem 1.1 Let I : K R0 → R be an integrable exponentially s-convex in the second sense on k 0 ; then k1 , k2 ∈ K with k1 < k2 ; we have 

k1 + k2 I 2



1 ≤ k2 − k1

k2

(1.5)

k1

1 A1 ( f ) =

I (l)eβ I (k1 ) dl ≤ A1 ( f )eβ I (k1 ) + A2 ( f )eβ I (k2 )

fs eβ( f k1 +(1− f )k2

1 d f and A2 ( f ) )

0

(1 − f )s eβ( f k1 +(1− f )k2 )

0

2 Ostrowski-Type Inequalities Lemma 2.1 Let I : K → R be a differentiable mapping on k 0 for k1 , k2 ∈ K with k1 < k2 ; if I , ∈ L 1 [k1 , k2 ], then the following equation is 1 I (l) − k2 − k1 −

k2 k1

(k2 − l)2 k2 − k1

(1 − k1 )2 I (t)dt ≤ k1 − k2

1

1

f I , ( f l + (1 − f )k1 )d f

0

f I , ( f l + (1 − f )k2 )d f

(2.1)

0

for each l ∈ [k1 , k2 ]. Theorem 2.1 Firstly, we are using lemma given by [3]. Lemma 2.1 [3] Let I : K → R be a differentiable mapping on k 0 such that I , ∈ [K 1 , K 2 ] for K 1 , K 2 ∈ K with k1 < k2 . If I , is exponentially s-convex in second sense on [k1 , k2 ] for some s ∈ (0, 1] and |I , (l)| ≤ M, l ∈ [k1 , k2 ], then the following inequality is

Ostrowski-Type Inequalities with Exponentially Convex …

59

     k2   1 (l − k1 )2 βl ≤ M  I (l) − e I (t)dt  k −k  k1 − k2 (s + 2) 1 2   k1

(l − k1 ) (k2 − l)2 βl (k2 − l)2 eβ I (k1 ) + e + eβ I (k2 ) (s + 1)(s + 2) (s + 2) (s + 1)(s + 2) 2

+

 (2.2)

Proof Using Lemma 2.1     k2 1    (l − k1 )2 1   I (l) −  I (t)dt  ≤ f (I , ( f l − (1 − f )k1 ))d f  k2 − k1    k2 − k1 0

k1

+

(k2 − l)2 k2 − k1

≤

(l − k1 )2 k2 − k1

+

(k2 − l) k2 − k1

2

1

f (I , ( f l + (1 − f )k2 ))d f

0

1

 f f s |I , (l)|eβl + (1 − f )s |I , (k1 )|eβ I (K 1 ) d f

0

1



f f s |I , (l)|eβl + (1 − f )s |I , (k2 )|eβ I (k2 ) d f

0

 βl  e eβ I (K 1 ) (l − k1 )2 + ≤M k2 − k1 (s + 2) (s + 1)(s + 2)  βl  e eβ I (k1 ) (k2 − l)2 − +M k2 − k1 (s + 2) (s + 1)(s + 2)  βl e (l − k1 )2 eβ I (k1 ) (l − k1 )2 M (k2 − l)2 βl + + e = k2 − k1 (s + 2) (s + 1)(s + 2) (s + 2)  (k2 − l)2 eβ I (k2 ) + (s + 1)(s + 2) 1

1

0

0

1 f s+1 d f = s+2 and The proof is complete.

f (1 − f )s d f =

(2.3)

1 . (s+1)(s+2)

Remark By letting β = 0, we get inequality 2.1 of Theorem 2 in [1]. Corollary 2.1 By letting s = 1 in inequality 2.1, we get 1 I (l) − k2 − k1  ×

k2 I (t)dt ≤ k1

2 βl

(l − k1 ) e 3

+

M k2 − k1

(l − k1 )2 eβ I (k1 ) (k2 − l)2 eβl (k2 − l)2 eβk2 + + 6 3 6

 (2.4)

60

A. Sharma and R. N. Saraswat

Remark In Corollary 2.1, by assuming β = 0, we get inequality (1.1). Theorem 2.2 Let I : K → R be a differentiable mapping on k 0 such that I , ∈ [k1 , k2 ] for k1 , k2 ∈ K with k1 < k2 ; if |I , | y is exponentially s-convex in the second sense on [k1 , k2 ] for some s ∈ (0, 1], r, y > 1, r1 , 1y = 1 and |I , (l)| ≤ M, l ∈ [k1 , k2 ], then the following inequality is:     k2   M 1  I (l) − I (t)dt  ≤  1/r k − k 2 1  (k2 − k1 )(1 + r )  k1

 βl  βl 1/y 1/y  e e eβ I (k1 ) eβ I (k2 ) 2 2 + + × (l − k1 ) + (k2 − l) (s + 1) (s + 1) (s + 1) (s + 1) (2.5) for each l ∈ [k1 , k2 ]. Proof Let n > 1; now using Holder’s inequality, we obtain  1  k2 2   1 − k ) (l 1  I (l) − I (t)dt ≤ f |I , ( f l − (1 − f )k1 )k1 |d f  k − k k − k 2 1 2 1  0

k1

+

(k2 − l)2 k2 − k1

1

f |I , ( f l + (1 − f )k2 )k2 |d f

0

⎞ r1 ⎛ 1 ⎞ r1 ⎛ 1   (l − k1 )2 ⎝ ≤ f r d f ⎠ ⎝ |I , ( f l − (1 − f ))k1 |r d f ⎠ k2 − k1 ⎛ +

(k2 − l)2 ⎝ k2 − k1

0

1

⎞1/y f r⎠

0

⎛ 1 ⎞1/y  + ⎝ |I , ( f l + (1 − f )k2 )| y d f ⎠

0

(2.6)

0

Since |I , | y is exponentially s-convex in the second sense and |I , (l)| ≤ M, then we have 1

,

1

|I ( f l + (1 − f ))k1 | d f = 0

y

0

+ (1 − f )s |I , (k1 )| y eβ I (K 1 ) d f ≤ and

f s |I , (l)| y eβl M y βl M y β I (k2 ) e + e (s + 1) (s + 1)

(2.7)

Ostrowski-Type Inequalities with Exponentially Convex …

1

1

,

|I ( f l + (1 − f ))k2 | d f = y

0

61

f s |I , (l)| y eβl

0 ,

y β I (k2 )

+ (1 − f ) |I (k2 )| e s

df ≤

M y βl M y βk2 e + e (s + 1) (s + 1)

(2.8)

Remark Letting β = 0, we get inequality 2.2 of Theorem 3 in [1]. Corollary 2.2 Now taking s = 1 in Theorem 2.2, we get     k2   1 M  I (l) − I (t)dt  ≤  1/r k2 − k1   (k2 − k1 )(1 + r ) k1

  βl  βl β I (k1 ) 1/y β I (k2 ) 1/y e e e e + + × (1 − k1 )2 + (k2 − l)2 2 2 2 2

(2.9)

Remark Letting β = 0 in Corollary 2.2, we get result for convex function. Theorem 2.3 Let I : K → R be a differentiable mapping on k 0 such that I , ∈ [k1 , k2 ] for k1 , k2 ∈ K with k1 < k2 . If |I , | y is exponentially s-convex in second sense on [k1 , k2 ] for some s ∈ (0, 1], y1 and |I , (l)| ≤ M, l ∈ [k1 , k2 ], then the inequality is:     k2   1 M  I (l) − I (t)dt  ≤  1−1/y k − k − k (k 2 1 2 1 )2   k1

 βl 1/y e eβ I (k1 ) + × (l − k1 )2 (s + 2) (s + 2)(s + 1)  βl 1/y  β I (k2 ) e e + +(k2 − l)2 (s + 1) (s + 2)(s + 1) for each l ∈ k1 , k2 . Proof Let r > 1, and using power mean inequality,     k2   1  I (l) − I (t)dt   k − k 2 1   k1

+

(k2 − l)2 k2 − k1

1 0

f |I , ( f l + (1 − f )k2 )|d f

(2.10)

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A. Sharma and R. N. Saraswat

⎞ 1y ⎞1− 1y ⎛ 1 ⎛ 1    ,  (l − k1 )2 ⎝ ≤ f d f ⎠ ⎝ f  I ( f l + (1 − f )k1 ) y d f ⎠ k2 − k1 0

0

⎞1−1/y ⎛ 1 ⎛ 1 ⎞1/y   (k2 − l)2 ⎝ ⎝ f |I , ( f l + (1 − f )k2 )| y ⎠ + fdf⎠ k2 − k1 0

(2.11)

0

Since |I , | y is exponentially s-convex in the second sense and |I , (l)| ≤ M, then 1

f |I , ( f l + (1 − f )k1 )| y d f

0

1 =



f f s |I , (l)| y eβl + (1 − f )s |I , (k1 )|s eβ I (k1 ) d f

0

≤

M y βl My e + eβ I (k1 ) (s + 2) (s + 1)(s + 2)

(2.12)

and 1

f |I , ( f l + (1 − f )k2 )| y d f

0

1 =

 f f s |I , (l)| y eβl + (1 − f )s |I , (k2 )| y eβ I (k2 ) d f

0

≤

M y βl My e + eβ I (k2 ) s+2 (s + 1)(s + 2)

(2.13)

Remark Letting β = 0, we get inequality (2.3). Corollary 2.3 By taking s = 1, we get     k2   1 M  I (l) − I (t)dt  ≤  1−1/y k2 − k1   (k2 − k1 )2 k1

  βl  βl β I (k1 ) 1/y β I (k2 ) 1/y e e e e + + × (l − k1 )2 + (k2 − l)2 3 6 3 6 Remark Letting β = 0, in corollary, we get result for convex function. Remark By setting l =

k1 +k2 , 2

we get several types of inequalities.

(2.14)

Ostrowski-Type Inequalities with Exponentially Convex …

63

Theorem 2.4 Let I : K → R be a differentiable mapping on K 0 such that I , ∈ [k1 , k2 ] with k1 < k2 ; if |I , | y is exponentially s-convex on [k1 , k2 ], r, y > 1 r1 + 1y = 1, then the inequality is     k2   1 2s−1/y  I (l) − ≤ I (t)dt   1/r k2 − k1   (k2 − k1 )(1 + r ) k1           l + k1  2  , l + k2  + I × (l − k1 )2  I , − l) (k 2    2 2

(2.15)

for each l ∈ [k1 , k2 ]. Proof Let r > 1; now using Holder’s inequality, we get     k2 1   (l − k1 )2 1  I (l) −  I (t)dt  ≤ f |I , ( f l + (1 − f )k1 )|d f  k2 − k1 k2 − k1   k1

(k2 − l)2 + k2 − k1 ⎛

1 0

1

0

⎛ 1 ⎞1/r 2  − k (l ) 1 ⎝ f |I . ( f l + (1 − f )k1 )|d f ≤ f rd f ⎠ k2 − k1 ⎞1/y

  × ⎝  I , ( f l + (1 − f )k1 ) y d f ⎠ ⎛

0

0

⎞1/r ⎛ 1  (k2 − l)2 ⎝ + f rd f ⎠ k2 − k1 0

⎞1/y

1

 y × ⎝  I , ( f l + (1 − f )k2 ) y  ⎠

(2.16)

0

Since from inequality (1.4), |I , | is exponentially s-convex     l + k1  y  I 2

(2.17)

   y    ,   I ( f l + (1 − f )k2 ) y  y d f ≥ 2s−1  I , l + k2    2

(2.18)

1

,

|I ( f l + (1 − f )k1 )| d f ≥ 2 y

s−1  ,

0

and 1 0

Remark Using inequalities (2.18) and (2.17) in (2.16), we get (2.15). 2 in (2.15), then we get inequality (2.8) in [1], and Remark If we take l = k1 +k 2 also letting s = 1 in (2.15), then we get inequality (2.9) in [1].

64

A. Sharma and R. N. Saraswat

3 Applications to Means Assume that k1 , k2 be positive numbers and k1 < k2 : (i) (ii)

The arithmetic mean A(k1 , k2 ) = The identic mean

k1 +k2 2

1   k −k 1 k2k2 2 1 I (k1 , k2 ) = , k1 = k2 , k1 , k1 = k2 , where k1 , k2 > 0 e k1k1

(iii)

The p-logarithmic mean L p (k1 , k2 ) =



p+1

p+1

k2 −k1 ( p+1)(k2 −k1 )

1/ p

,

p ∈ R{−1, 0}

Proposition 3.1 Let 0 < k1 < k2 , y ≥ 1 and 0 < s < 1. Then, we have   s  A (k1 , k2 ) − L s (k1 , k2 ) ≤ s



M 1

(k2 − k1 )21− y

 1y eβl eβ I (k1 ) + × (l − k1 ) (s + 2) (s + 1)(s + 2)  βl  1y  e eβ I (k2 ) 2 + +(k2 − l) (s + 2) (s + 1)(s + 2) 2

2 Proof The result is obtained by letting l = k1 +k in (2.10) with exponentially s2 convex function in second sense I : (0, ∞) → R, I (l) = l s ∀ β ≤ −1.

4 Conclusion In Corollaries 2.1 and 2.3, we have obtained new Ostrowski’s type inequalities for exponentially convex function Theorems 2.1–2.4, also obtained new Ostrowski’s type inequalities for exponentially s-convex function. Application obtained by Proposition 3.1.

References 1. Alomari, M., Darus, M., Dragomir, S.S., Cerone, P.: Ostrowski type inequalities for functions whose derivatives are s-convex in the second sense. Appl. Math. Lett. 23(9), 1071–1076 (2010) 2. Awan, M.U., Noor, M.A., Noor, K.I.: Hermite-Hadamard inequalities for exponentially convex functions. Appl. Math. Inf. Sci. 12(2), 405–409 (2018)

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3. Cerone, P., Dragomir, S.S.: Ostrowski type inequalities for functions whose derivatives satisfy certain convexity assumptions. Demonstratio Math. 37(2), 299–308 (2004) 4. Dragomir, S.S., Rassias, Th.M.: Ostrowski Type Inequalities and Applications in Numerical Integration. Kluwer Academic Publishers, Dordrecht, Boston, London (2002) 5. Dragomir, S.S.: Some companions of Ostrowski’s inequality for absolutely continuous functions and applications. Facta Univ. Ser. Math. Inform. 19, 1–16 (2004) 6. Dragomir, S.S., Wang, S.: Applications of Ostrowski’s inequality to the estimation of error bounds for some special means and for some numerical quadrature rules. Appl. Math. Lett. 11(1), 105–109 (1998) 7. Jain, K.C., Saraswat, R.N.: A new information inequality and its application in establishing relation among various f-divergence measure. J. Appl. Math. Stat. Inf. 8, 17–32 (2012) 8. Jain, K.C., Saraswat, R.N.: Some well-known inequalities and its applications in information theory. Jordan J. Math. Stat. 6(2), 157–167 (2013) 9. Jain, K.C., Saraswat, R.N.: Some bounds of information divergence measure in terms of relative-arithmetic divergence measure. Int. J. Appl. Math. Stat. 32, 48–58 (2013) 10. Mehreen, N., Anwar, M.: Hermite–Hadamard type inequalities for exponentially p-convex functions and exponentially s-convex functions in the second sense with applications. J. Inequal. Appl. 2019(1) (2019) 11. Mehreen, N., Anwar, M.: Ostrowski type inequalities via some exponentially convex functions with applications. AIMS Math. 5(2), 1476–1483 (2020) 12. Ozdemir, M.E., Kavurmacı, H., Set, E.: Ostrowski’s type inequalities for (m; M)-convex functions. Kyungpook Math. J. 50(3), 371–378 (2010) 13. Pachpatte, B.G.: On an inequality of Ostrowski type in three independent variables. J. Math. Anal. Appl. 249(2), 583–591 (2000) 14. Saraswat, R.N., Ajay, T.: New F-divergence and Jensen-Ostrowski’s type inequalities. Tamkang J. Math. 50(1), 111–118 (2019) 15. Saraswat, R.N., Ajay, T.: Ostrowski inequality and applications in information theory. Jordan J. Math. Stat. 11(4), 309–323 (2018). Author, F.: Article title. Journal 2(5), 99–110 (2016)

Energy of 2-Corona of Graphs J. Veninstine Vivik and P. Xavier

Abstract This paper develops the corona product to 2 times corona product of graphs and calibrates its energy. The proposed model extends as a graph with two successive generations of complexity, whose structure is constructed as a matrix based on its adjacency. The energy is measured from totalling the modulus of latent roots of the adjacency matrix obtained from graph G, and its largest eigen value is known to be spectral radius. The energy upper bound for 2-corona product of same graphs and different graphs in subsequent products are acquired. Keywords Eigen values · Graph energy · Corona product

1 Introduction Admist the various graph products, the corona product has multiplex ties of graphs and has complex network structure. This paper extends the idea of corona product initiated by Frucht and Harary [1] to 2 times the corona product of graphs. Nada et al. [2] investigated the corona between cycles and paths. Tavakoli et al. [3] studied the corona product of median graphs. The complexity behavior of corona product of some classes of graphs has been broadly explored by Kaliraj et al. [4]. Further Furma`ncyzk and Kubale analyzed corona product of cubic graphs [5] and multiproduct graph G ◦l H [6]. The model introduced in this paper is almost the extended idea of lcorona product graph. A graph is usually modeled with nodes V and their links between nodes with edges E as G(V, E). In particular for corona product, the first graph is hosted as the center graph G and is taken product with other graph H by connecting |V (G)| duplication of H , such that each node of G is connected to all nodes in one corresponding duplication of H . This sort of product is extended one more time with another graph I and is said to be 2 corona product of graphs. Here, the adjacency matrix of these J. V. Vivik (B) · P. Xavier Department of Mathematics, Karunya Institute of Technology and Sciences, Coimbatore 641114, Tamil Nadu, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_7

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graphs is considered as the matrix of successive generations. The proposed model in this paper is narrowed to the study of 2-corona product of same type of graphs and different type of graphs. Both these provide the possibility of generating graphs with different random order. The energy of a graph emerges from chemistry owing to importance of total π electron energy of carbon compounds. It is described as the aggregate of modulus of the characteristic roots of the matrix A(G) and was introduced by Gutman [7]. The study on various graph energies can be spotted in several papers, refer, e.g. [8, 9]. Liu et al. [10] performed the study on upper bounds for energy of graphs. In another work, Das et al. [11] and Sridhara et al. [12] investigated the improved bounds for graph energy. Barik et al. [13] initiated the spectrum of corona between two graphs and also studied by McLeman and McNicholas [14]. Various research works are carried out in the field of spectra of corona graphs. This paper establishes the concept of 2-corona product (2-c.p) of graphs. The energies for three different cycle, complete graphs and combination of complete, cycle and path graphs are determined, and its nearest bounds are calibrated. Also, the spectral radius is obtained from the dominant eigen value of the adjacency matrix of these graphs.

2 2-Corona Product of Graphs Definition 1 [3] The corona product of graphs H1 ,H2 of order n 1 and n 2 , respectively, is the H1 ◦ H2 graph obtained by considering H1 , as the middle graph, n 1 copies of H2 , as the exterior graph and making the kth vertex of H1 connected to the whole vertices in kth copy of H2 , where 1 ≤ k ≤ n 1 . Definition 2 (Energy) [9] Let the graph G consists of nodes {u 1 , u 2 , . . . , u n } and edges {e1 , e2 , . . . , em }. The adjacency matrix A(G) is a square matrix of order n whose (i, j)th-entry is  1, if u i and u j are neighbors ai j = 0, otherwise. Let μ1 , μ2 , . . . , μn be the eigenvalues of A(G). The aggregate of modulus of the eigenvalues is set out nas the energy of the graph. |μi |. It is also known as ordinary energy of a graph. Hence E(G) = i=1 Definition 3 Let G 1 , G 2 and G 3 be three distinct graphs. The 2-corona product of these graphs is constructed by considering one copy of G 1 as the middle graph and join |V (G 1 )| duplications of G 2 , such that the ith vertex of G 1 is adjacent to each vertex in the ith duplication of G 2 , is called the first corona and the corona product of the next generation of any graph G 3 is obtained by joining successively k copies of G 3 where k = |V (G 1 )|.|V (G 2 )|. Accordingly, the jth vertex of G 2 is connected to

Energy of 2-Corona of Graphs

69

Fig. 1 2-Corona product of K 4 , C3 and P2

each vertex in jth copy of G 3 , where 1 ≤ i ≤ n, 1 ≤ j ≤ k is known as the 2-corona product of graphs and is denoted by G 1 ◦ G 2 ◦ G 3 or G r −1 ◦2 G r , 2 ≤ r ≤ 3 (Fig. 1). Theorem 1 [7] For a graph H having ν vertices and  edges, it is shown that     2

2  2  + (ν − 1) 2 − E(H ) ≤ ν ν while if H is k-regular, E(H ) ≤ k +

√

k(ν − 1)(ν − k).

Theorem 2 [8] For every  > 0, ∃ infinitely many α for every of it, ∃ a r -regular E(Y ) graph Y with order α where r < α − 1 and r +√r (α−1)(α−r < . ) 



Lemma 1 [15] Let ξ = (ξi ) be a real non null n × 1 vectors, m = ξn , s 2 = ξ nCξ  where  is the n × 1 vector of ones, C = I − n is the center matrix,  is the transpose of  and ξ1 ≥ ξ2 ≥ . . . ≥ ξn then ξn ≤ m − s 1 ≤ m + s 1 ≤ ξ1 . (n−1) 2

(n−1) 2

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3 Energy of 2-Corona Product of Graphs The energy of 2-c.p of graphs is obtained by means of multiplying same three graphs of different order, and three different graphs with different order are examined. More precisely, the complete, cycle and path model network of graphs are structured, and its energy values are determined in this section.

3.1 Energy of 2-c.p of Same Graphs Theorem 3 Let G = K r −1 ◦2 K r , 2 ≤ r ≤ 3 be the 2-c.p of complete graphs with p vertices and q edges then its energy is ε(G) ≤ 2 p. Proof Consider the complete graphs of different orders as K l , K m and K n . The 2corona product of these three complete graphs K r −1 ◦2 K r , 2 ≤ r ≤ 3 consists of p = l + lm(1 + n) vertices and q = 21 [l(l − 1) + lm(m + 1) + lmn(n + 1)] edges. The adjacency situations of K r −1 ◦2 K r are with 1’s on υ ≤ 1, ω ≤ l, υ = ω υ ≤ 1, (υ − 1)m + l + 1 ≤ ω ≤ υm + l ω ≤ 1, (ω − 1)m + l + 1 ≤ υ ≤ ωm + l

for 1 ≤ k ≤ l l + (k − 1)m + 1 ≤ υ ≤ km + l, l + (k − 1)m + 1 ≤ ω ≤ km + l, υ = ω

for 1 ≤ p ≤ lm υ = l + p, ( p − 1)n + lm + l + 1 ≤ ω ≤ pn + lm + l ω = l + p, ( p − 1)n + lm + l + 1 ≤ υ ≤ pn + lm + l l+lm + ( p − 1)n+1 ≤ υ ≤ pn + l + lm, l + lm + ( p − 1)n + 1 ≤ ω ≤ pn + l + lm, υ = ω The non-adjacency situations are with 0’s on υ = ω and elsewhere. Based on this situations the neighborhood matrix of 2-corona graph G is formulated as follows.

Energy of 2-Corona of Graphs

v1 v2 v3 0 1 1 ⎜1 0 1 ⎜ ⎜1 1 0 ⎜ . . . ⎜ . . . ⎜ . . . ⎜ ⎜1 1 1 ⎜ ⎜1 0 0 ⎜ ⎜1 0 0 ⎜ . . . ⎜ . . . ⎜ . . . ⎜ ⎜0 0 0 ⎜ ⎜ . . . ⎜ .. .. .. ⎜ ⎜ ⎜0 0 0 ⎝ ⎛

v1 v2 v3 .. .

vl vl+1 vl+2 .. . vl+lm .. . v p−1 vp

0

0

71

· · · vl vl+1 vl+2 ... 1 1 1 ... 1 0 0 ... 1 0 0 .. .. .. .. . . . . ... 0 0 0 ... 0 0 1 ... 0 1 0 . .. .. .. . .. . .

0

... 0 . .. . ..

0 .. .

0 .. .

... 0

0

0

... 0

0

0

· · · vl+lm ... 0 ... 0 ... 0 .. .. . . ... 0 ... 0 ... 0 .. .. . . .. . 0 .. .. . . .. . 1 .. . 1

· · · v p−1 v p ⎞ ... 0 0 ... 0 0 ⎟ ⎟ ... 0 0 ⎟ .. .. ⎟ .. ⎟ . . . ⎟ ⎟ ... 0 0 ⎟ ⎟ ... 0 0 ⎟ ⎟ ... 0 0 ⎟ .. .. ⎟ .. ⎟ . . . ⎟ ⎟ ... 1 1 ⎟ ⎟ .. .. ⎟ .. . . . ⎟ ⎟ ⎟ ... 0 1 ⎟ ⎠ ...

1

0

The characteristic equation of this adjacency matrix with order p is of the form φ(G, μ) = 0 which has exactly p roots. Therefore, from Lemma 1, μ1 ≥ μ2 ≥ · · · ≥ μ p . Here, the spectral radius μ1 is considered as the dominant eigen value. p The energy E = κ=1 |μκ |. By Cauchy Schwarz inequality  p  κ=1

 p−1  κ=2

2 |μκ |

≤

p  κ=1

|1|

p 

|μκ |2

κ=1 2

|μκ | − |μ1 | − |μ p |

≤

 p−1 

|1| − 2

  p−1 

κ=2

 |μκ | − |μ1 | − |μ p | 2

κ=2

 2   ε(G) − |μ1 | − |μ p | ≤ ( p − 2) (2 p)2 − |μ1 |2 − |μ p |2      ε(G) ≤ |μ1 | + |μ p | + ( p − 2) 4 p 2 − |μ1 |2 − |μ p |2 Now let |μ1 | = r and |μ p | = s      2 2 2 ε(G) ≤ r + s + ( p − 2) 4 p − r − s Construct the maximizing function of the form   1  f (r, s) = √ r + s + ( p − 2) (4 p 2 − r 2 − s 2 ) p The first and second order derivatives of f (r, s) are

2

2

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r ( p − 2) 1  fr = √ − ,  √ p p ( p − 2) 4 p 2 − r 2 − s 2 s( p − 2) 1  fs = √ − ,  √ p p ( p − 2) 4 p 2 − r 2 − s 2   p − 2 4 p2 − s 2 =− √  3 , p 4 p2 − r 2 − s 2 2 √

frr

  p − 2 4 p2 − r 2 f ss = − √  3 , p (4 p 2 − r 2 − s 2 2 √

and

√ rs p − 2 fr s = f sr = − √  3 . p 4 p2 − r 2 − s 2 2

The Hessian matrix is defined by ⎡

√ √

−

√

⎢

2 f (r, s) = ⎣

−

p−2[4 p2 −s 2 ]

p [4 p2 −r 2 −s 2 ] √ r s p−2

3 2 3

p [4 p2 −r 2 −s 2 ] 2

− −

√ r s p−2 3 √ 2 2 2 2 √p [4 p −r 2−s 2] p−2[4 p −r ] 3 √ p [4 p2 −r 2 −s 2 ] 2

⎤ ⎥ ⎦

Equating the partial derivatives fr and f s of the function to zero. r 2 ( p − 1) + s 2 = 4 p 2 r 2 + s 2 ( p − 1) = 4 p 2 solving the above two equations √ r = s = ±2 p.  √ √  Therefore, the stationary points are ±2 p, ±2 p . The Hessian at the critical point is 

−

f (r, s) = − 2

( p−1) 2 p( p−2) 1 2 p( p−2)

− −

1 2 p( p−2) ( p−1) 2 p( p−2)

which isa negative definite and has a local maximum. √  √ So f ±2 p, ±2 p ≥ f (r, s) which implies



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Table 1 E, ε and μ1 of 2-c.p of complete graphs Graphs Vertices Edges Energy Kl ◦ K m ◦ K n p q E K3 ◦ K3 ◦ K5 K 6 ◦ K 15 ◦ K 9 K 11 ◦ K 8 ◦ K 12 K 25 ◦ K 15 ◦ K 10 K 20 ◦ K 10 ◦ K 30

75 906 1155 4150 6220

213 4785 7315 23925 94290

137.4697 1780.5 2272.8 8187 12385

Table 2 E, ε and μ1 of 2-corona product of cycle graphs Graphs Vertices Edges Energy Cl ◦ C m ◦ C n p q E C3 ◦ C4 ◦ C5 C7 ◦ C17 ◦ C6 C10 ◦ C8 ◦ C13 C12 ◦ C22 ◦ C15 C16 ◦ C14 ◦ C25

75 840 1130 4236 5840

147 1673 2250 8460 11664

124.1938 1409 1808.3 6737.7 9050.8

Energy bound ε

Spectral radius μ1

150 1812 1812 8300 12440

6.2111 16.3897 13.6713 25.3913 30.4574

Energy bound ε

Spectral radius μ1

150 1680 2260 8472 11680

5 6.7958 6.5826 8.0828 8.2450

f (r, s) ≤ 2 p. it is successfully bounded above function with a point of local maximum Thus, √  √ at ±2 p, ±2 p . Hence, ε(G) ≤ 2 p. Illustration: Table 1 illustrates the energy bounds and spectral radius for 2-corona of complete graphs. Theorem 4 Let G = Cl ◦ Cm ◦ Cn be the 2-corona product of cycle graphs, which contains p vertices and q edges then its energy is ε(G) ≤ 2 p. Proof Consider the cycle graphs of different orders as Cl , Cm and Cn . The 2-corona product of these three cycle graphs Cl ◦ Cm ◦ Cn consists of vertices p = l + lm(1 + n) and q = l + 2lm(1 + n) edges. The connection matrix of G is formulated and is obvious that the remaining proof is much the same to Theorem 3. Illustration: Table 2 illustrates the energy bounds and spectral radius for 2-corona of cycle graphs.

3.2 Energy of 2-Corona Product of Different Graphs Theorem 5 Let ψ = G l ◦ G m ◦ G n be the 2-corona product of three different graphs having p vertices and q edges then its energy is ε(ψ) ≤ 2 p.

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Table 3 E, ε and μ1 of 2-corona product of three graphs Graphs Vertices Edges Energy Gl ◦ G m ◦ G n p q E K 4 ◦ C4 ◦ P3 K 10 ◦ C7 ◦ P12 K 20 ◦ C12 ◦ P5 C4 ◦ K 4 ◦ P3 C6 ◦ K 10 ◦ P5 C16 ◦ K 7 ◦ P11 C15 ◦ K 25 ◦ P10 P4 ◦ C3 ◦ K 4 P10 ◦ C15 ◦ K 5 P7 ◦ C14 ◦ K 10

68 920 1460 68 366 1360 4140 64 910 1830

118 1795 2830 124 876 2816 21390 148 2560 11210

99.6752 1434.9 2251.1 101.9951 592.5556 2144.6 7764.9 112.4155 1611.7 3476.3

Energy bound ε

Spectral radius μ1

136 1840 2920 136 732 2720 8280 128 1820 3660

4.9235 10.0607 19.6892 5.0964 10.7002 8.6621 10.7091 4.9507 6.8261 12.3347

Proof Consider three different graphs namely complete, cycle and path graphs as K l , Cm and Pn . Out of these, any graphs can be taken as G 1 , G 2 and G 3 . The 2-corona product of these three different graphs ψ = G l ◦ G m ◦ G n consists of vertices p = l + lm(1 + n) and q = η(G 1 ) + lm (η(G 2 )) + lmn (η(G 3 )) edges, where η(G) represents the total number of edges in the graph. The adjacency matrix of ψ is constructed, and the remaining proof is similar to Theorem 3. Illustration: Table 3 illustrates the energy bounds and spectral radius for 2-Corona of graphs. Remark If l, m and n are the orders of three graphs G 1 , G 2 and G 3 , respectively, then G l ◦ G m ◦ G n = G m ◦ G n ◦ G l = G n ◦ G l ◦ G m and E(G l ◦ G m ◦ G n ) = E(G m ◦ G n ◦ G l ) = E(G n ◦ G l ◦ G m ).

4 Conclusion The energy of a graph has extensive applications in the field of mathematics and chemical sciences. It is hard to fix the energy bounds due to complex structure of graphs. This work introduces the concept of 2-c.p of graphs and the energy bounds using this product for three different combinations of graphs are studied. This product emerges as a graph with two successive generations which helps in the modeling of networks. Further this idea can be explored for multi-corona product of various random graphs.

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References 1. Frucht, R., Harary, F.: On the corona of two graphs. Aequationes Mathematicae 4, 322–325 (1970) 2. Nada, S., Elrokh, A., Elsakhawi, E., Sabra, D.: The corona between cycles and paths. J. Egypt. Math. Soc. 25, 111–118 (2017) 3. Tavakoli, M., Rahbarnia, F., Ashrafi, A.R.: Studying the corona product of graphs under some graph invariants. Trans. Comb. 3(3), 43–49 (2014) 4. Kaliraj, K., Veninstine, V.J., Vernold, V.J.: Equitable coloring on corona graph of graphs. J. Combin. Math. Combin. Comput. 81, 191–197 (2012) 5. Furma`nczyk, H., Kubale, M.: Equitable coloring of corona products of cubic graphs is harder than ordinary coloring. Ars Mathematica Contemporanea 10, 333–347 (2016) 6. Furma`nczyk, H., Kubale, M.: Equitable colorings on corona multiproducts of graphs. Discussiones Mathematicae Graph Theory 37, 1079–1094 (2017) 7. Gutman, I.: The energy of a graph. Ber. Math. Stat. Sekt. Forschungszentrum Graz 103, 1–22 (1978) 8. Balakrishnan, R.: The energy of a graph. Linear Algebra Appl. 387, 287–295 (2004) 9. Samir Vaidya, K., Kalpesh Popat, M.: Some new results on energy of Graphs. MATCH Commum. Math. Comput. Chem. 7, 589–594 (2017) 10. Liu, H., Lu, M., Tian, F.: Some upper bounds for the energy of graphs. J. Math. Chem. 41(1), 45–57 (2007) 11. Das, K.C., Mojallal, S.A., Gutman, I.: Improving the McCelland inequality for energy. MATCH Commun. Math. Comput. Chem. 70, 663–668 (2013) 12. Sridhara, G., Rajesh Kanna, M.R., Jagadeesh, R., Cangul, I.N.: Improved McClelland and Koolen-Moulton bounds for the energy of graphs. Sci. Magna 13(1), 1–10 (2018) 13. Barik, S., Pati, S., Sarma, B.K.: The spectrum of the corona of two graphs. SIAM J. Discrete Math. 21, 47–56 (2007) 14. McLeman, C., McNicholas, E.: Spectra of coronae. Linear Algebra Appl. 435, 998–1007 (2011) 15. Wolkowicz, H., George Styan, P.H.: Bounds for eigen values using traces. Linear Algebra Appl. 29, 471–506 (1980)

Generalized KKM Mapping Theorems Bhagwati Prasad Chamola, Ritu Sahni, and Manoj Sahni

Abstract The theorems regarding KKM maps are considered one of the most significant findings in the fixed-point theory. It is useful in the study of minimax theorem, coincidence theorems, and saddle point theorems. The present paper deals with some generalized KKM mapping theorems using a pair of multivalued mappings in Hausdorff topological vector space. Some important and familiar results are also derived as particular cases. Keywords KKM maps · Multivalued mappings · Hausdorff topological vector space · Finite intersection property · Brouwer fixed-point theorem

1 Introduction In game theory, fixed-point theorems are useful techniques for proving equilibrium and existence theorems. For example, Brouwer’s, Kakutani, and Fan’s fixed-point theorems have been most often utilized results in the theory of game and economics. These theorems have been further used to demonstrate the minimax and existence of saddle points theorems. In game theory, minimax theorems are commonly exploit. The first theorem based on minimax results was presented in the year 1928 by Von Neumann [1]. Many articles that verified the Neumann inequality then appeared in the literature. It is observed that most minimax theorems are established in diverse situations by using fixed-point theorems in different settings on the functions/mappings or the spaces. Von Neumann improved the fundamental minimax result of game theory by establishing an intersection lemma in 1937 [2]. Kakutani [3] developed a fixed-point theorem, which is an extension of the fixed-point theorem of Browder for multimaps, in 1941, in order to offer simpler proofs of lemma and minimax theorem

B. P. Chamola Jaypee Institute of Information Technology, Noida, India R. Sahni · M. Sahni (B) Pandit Deendayal Energy University, Gandhinagar, Gujarat, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_8

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established by von Neumann. This marked the start of the multimaps fixed-point theory, which has a close relationship to the minimax theory in games. After that Sion [4] also extended result of von Neumann based on the Knaster, Kuratowski, and Mazurkiewicz theorem [5] on the n-dimensional simplex, which was published in 1929. This theorem is an analog of fixed-point theorem proved by Brouwer in the year 1912. Initially, Ky Fan [6–8] developed the findings of the KKM theory for subsets of topological vector spaces, which are convex in nature. In these types of topological vector spaces, many intersection theorems and applications were investigated. By establishing many variations of KKM theorems for convex spaces, Lassonde [9] expanded the KKM theory to convex spaces and provided a systematical advancement of the KKM theorems. Horvath [10, 11] further developed the KKM theorem in contractible spaces, H-space, whose subsets are contractible according to the suitability or pseudo-convex spaces. In these articles, most of Fan’s KKM theory findings are proved in H-spaces by changing the convexity criterion with contractibility. Park [12] defined novel variants of KKM theorems, Fan-Browder kind of theorems, minimax theorems, and other H-space theorems using these terminologies. These findings were extended to even more generic spaces, for example, KKM spaces, G-convex, L-spaces, abstract convexity spaces, geodesic spaces, hyperbolic spaces (see for instance, [13–28]). In this way, the KKM theory is constantly improved. There has been a lot of work done in the literature to generalize the KKM theory to make it more applicable to a vast range of problems, such as minimax theorems, saddle point theorems in game theory, economics and optimization theory, variational inequalities, [6, 7, 29–37] are some examples. The goal of this work is to propose a KKM theorem in a more extended form by utilizing Chang and Zhang’s idea [6]. An extended version of minimax inequality is established, and as an application, a saddle point existence result in generic contexts is proved. As a consequence, for two-person zero sum parametric games, a saddle point theorem is established, and a number of well-known findings as special cases are derived.

2 Preliminaries First, we provide some fundamental concepts that will be employed in our results. For notations and preliminaries, we use [5, 16–18, 29–37]. Definition 2.1 [35] A saddle point of f in X × Y is a point (x, y) ∈ X × Y with the condition f (x, y) ≤ f (x, y) ≤ f (x, y), ∀ (x, y) ∈ X × Y. A point (x, y) ∈ X × Y is termed as game solution iff it is also proven to be a saddle point of f in X × Y.

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Definition 2.2 [5] Consider F to be a Hausdorff topological vector space and A a nonempty subset of F. A KKM mapping is a multivalued mapping X : A → 2 F , that is, mapping having values X (a) ⊂ F, for each a in A if co{a1 , a2 , . . . , an } ⊂  n i=1 X (ai ) for any finite subset {a1 , a2 , ..., an } ⊂ A where co{a1 , a2 , . . . , an } represents the convex hull of the set {a1 , a2 , . . . , an }. Definition 2.3 [32] Consider F to be a topological vector space and A a nonempty subset of F. A multivalued mapping X : A → 2 F is known as generalized KKM exists a finite subset mapping; if for a given finite set {a1 , . . . ,an } ⊂ A, there  {b1 , . . . , bn } ⊂ F such that for each subset bi1 , . . . , bik ⊂ {b1 , . . . , bn }, 1 ≤ k ≤      n, we have co bi1 , ..., bik ⊂ kj=1 X ai j . For two mappings, this definition is defined as follows: Definition 2.4 Consider F to be a topological vector space, A a nonempty subset of F, and X, Y : A → 2 F . Then, Y is called generalized X-KKM mapping if we have for any finite set {a1 , . . . , an } ⊂ A and each {i 1 , . . . , i k } ⊂ {1, 2, . . . , n}, ⎧ ⎫ k k ⎨

 ⎬

  co ⊂ X ai j Y ai j . ⎩ ⎭ j=1

j=1

It is noted that Definition 2.3 becomes Definition 2.2 when X : A → F and X (ai ) = bi for each i ∈ {1, 2, . . . , n}. Definition 2.5 [32] Consider F to be a topological vector space and A a nonempty subset of F. A function ϕ : A × A → (−∞, ∞) is called γ -generalized quasiconvex in b for some γ ∈ (−∞, ∞); if for any finite subset {b1 , b2 , . . . , bn } ⊂  A, there exists such that, for any subset a finite subset {a1 ,. . . , an } ⊂ A , . . . , a i i   k ⊂ {a1 , . . . , an } 1 and any a0 ∈ co ai1 , . . . , aik , we have γ ≤ maxi≤ j≤k ϕ a0 , bi j . Definition 2.6 Consider F to be a topological vector space, A a nonempty subset of F, γ ∈ (−∞, ∞), and X : A → 2 B , here, B ∈ F. In this case, the function ϕ : A × A → (−∞, ∞) is called X − γ generalized quasiconvex in b if for any any a0 ∈ finite subset{b1,b2 , . . . , bn } ⊂ A, each{i 1 ,. . . , i k } ⊂ {1, 2, . . . , n}, and   k ⊂ B and any bi j ∈ X ai j , we have γ ≤ maxi≤ j≤k ϕ a0 , bi j . co j=1 X ai j It is observed that when X : A → B and X (ai ) = bi for each i ∈ {1, 2, . . . , n}, the function ϕ, then, turns into γ -generalized quasi convex in b. Definition 2.7 [37] Let us consider A and B as two topological spaces. A multivalued mapping X : A → 2 B is considered to be transfer closed valued on A if, ∩ X (a) = ∩ X (a).

a∈A

a∈A

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Definition 2.8 [34] Let us consider A and B as two topological spaces. A multivalued mapping X : A → 2 B is considered to be intersectionally closed valued on A if, ∩ X (a) = ∩ X (a).

a∈A

a∈A

Definition 2.9 [30, 34] Let us consider A and B as two topological spaces. The function f : A × B → R = R ∪ {±∞} is called: (i) (ii) (iii)

(iv)

quasiconcave in a if the set {a ∈ A : f (a, b) ≥ λ} is convex, for each b ∈ B and λ ∈ R; quasiconvex in b if the set {b ∈ B : f (a, b) ≤ λ} is convex, for each a ∈ A and λ ∈ R; upper semicontinuous (resp., generally upper semicontinuous) in a if the set {a ∈ A : f (a, b) ≥ λ} is closed (resp., intersectionally closed), for each b ∈ B and λ ∈ R; lower semicontinuous (resp., generally lower semicontinuous) in b if the set {b ∈ B : f (a, b) ≤ λ} is closed (resp., intersectionally closed), for each a ∈ A and λ ∈ R.

Definition 2.10 [30] Let us consider A and B as two topological spaces. The function f : A × B → R = R ∪ {±∞} is then stated to be: (i)

(ii)

transfer upper semicontinuous in a, if ∀λ ∈ R, and ∀ a ∈ A, b ∈  f (a, b)  there exists a neighborhood N (a) of a and a point b ∈  < λ,  that f v, b < λ, ∀ v ∈ N (a). transfer lower semicontinuous in b if ∀λ ∈ R, and ∀ a ∈ A, b ∈  f (a, b) λ,  there exists a neighborhood N (b) of b and a point a ∈  >  that f a , u > λ for all u ∈ N (b).

B with B such B with A such

Definition 2.11 [35] Let us consider A and B as two topological spaces. A function ϕ : A × B → (−∞, ∞) is called: (i)

(ii)

γ -transfer lower semicontinuous function in a for some γ ∈ (−∞, ∞) if ∀  a ∈ A and b ∈ B with ϕ(a, b) > γ , there  exists some b ∈ B and some  neighborhood N (a) of a such that ϕ u, b > γ , ∀ u ∈ N (a). γ -transfer upper semicontinuous function in b for some γ ∈ (−∞, ∞) if ∀  a ∈ A and b ∈ B with ϕ(a, b) < γ , there  exists some a ∈ B and some  neighborhood N (b) of b such that ϕ a , v < γ ∀ v ∈ N (b).

Definition 2.12 Let us consider A and B as two nonempty sets. The mapping X : A → 2 B is called surjective if X ( A) = ∪a∈A X (a) = B. It is noted that every function which is upper semicontinuous in x (resp., lower semicontinuous in y) is transfer upper semicontinuous in x (resp., transfer lower semicontinuous in y); however, this is not always the case, as shown in Example 2.1.

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Example 2.1 Let A = B = (0, ∞) and f : A × B → R given as  f (a, b) =

2(a + b) + 1/2 if a + b ≥ 1 a + 2b if a + b < 1.

Then, f (a, b) is transfer lower semicontinuous in b but not lower semicontinuous in b.

3 Main Results We begin by proving the section’s main result, which will be utilized in the next part. Theorem 3.1 Consider F to be a Hausdorff topological vector space and A a nonempty subset of F. Suppose Y : A → 2 F be any multivalued mapping such that ∀ a ∈ A, Y (a) is finitely closed. The family of sets {Y (a) : a ∈ A}, thus, has a finite intersection property iff the mapping Y is a generalized X -KKM mapping for some mapping X : A → 2 F . Proof Consider a family of sets {Y (a)}a∈A with finite intersection property. Then, n Y (ai ) = ϕ, for any finite subset {a1 , a2 , . . . , an } ⊂ A. Taking we have, ∩i=1 n a∗ ∈ ∩i=1 Y (ai ) and define X : A → 2 F by X (a) = {a∗ }, ∀a ∈ A. Therefore, ∀ {i 1 , . . . , i k } ⊂ {1, 2, . . . , n}, we have ⎧ ⎫ k ⎨

 ⎬   k co = co{a∗ } = {a∗ } ⊂ ∪ Y ai j . X ai j ⎩ ⎭ j=1 j=1

This shows that Y : A → 2 F is a generalized X -KKM mapping. Let us consider Y : A → 2 F to be a generalized X -KKM mapping, X : A → 2 F and family {Y (a) : a ∈ A} does not possess finite intersection property; i.e., there n Y (ai ) = ϕ. As we must be some finite subset {a1 , a2 , . . . , an } ⊂ A such that ∩i=1 } ⊂ {1, 2, . . . , n}, {i , . . . , i know that Yis a generalized X -KKM mapping, then ∀ 1 k      we have co kj=1 X ai j ⊂ ∪kj=1 Y ai j . Particularly, co{X (ai )} ⊂ Y (ai ) ∀i ∈ {1, 2, . . . , n}. Let S = co{b1 , b2 , . . . , bn }, L = span{b1 , b2 , . . . , bn }. Then S ⊂ L . Since Y is finitely closed, Y (ai ) ∩ L is also a closed set. Assume d is the Euclidean metric on L. It is clear that / L ∩ Y (ai ) d(a, L ∩ Y (ai )) > 0 ⇔ a ∈ We now define a function f : S → [0, ∞) as follows:

(3.1)

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f (c) =

n 

d(c, L ∩ Y (ai ))bi

i=1 n It follows from (1) and ∩i=1 Y (ai ) = ϕ that for each c ∈ S, f (c) > 0. Let

R(c) =

n  1 d(c, L ∩ Y (ai ))bi . f (c) i=1

(3.2)

Therefore, R is a continuous function that maps from S into S. According to the Brouwer fixed-point theorem, there is an element c∗ ∈ S such that n  1 d(c∗ , L ∩ Y (ai ))bi c∗ = F(c∗ ) = f (c) i=1

(3.3)

I = {i ∈ {1, 2, . . . , n} : d(c∗ , Y (ai ) ∩ L > 0).

(3.4)

Denote

/ Y (ai ) ∩ L, for each i ∈ I. We have Then, c∗ ∈ / ∪ Y (ai ) c∗ ∈ i∈I

(3.5)

because c∗ ∈ L , c∗ ∈ / Y (ai ), ∀i ∈ I. From (3.3) and (3.4), we have c∗ =

 i∈I

1 d(c∗ , L ∩ Y (ai ))bi ∈ co(bi : i ∈ I }. f (c∗ )

However, given X -KKM mapping from A into 2 F , it  that Y isa generalized   k k ⊂ j=1 Y ai j . Thus, we have follows that co j=1 X ai j c∗ ∈ co(bi : i ∈ I } ⊂ ∪ Y (ai ). i∈I

(3.6)

This contradicts (3.5). This implies that {Y (a) : a ∈ A} has the finite intersection property. The proof of the theorem is now completed. If X : A → F is defined as a single valued map with X (ai ) = bi for each i ∈ {1, 2, . . . , n}, then the preceding result entails following Chang and Zhang’s results [34, Theorem 3.1]. Corollary 3.1 [32] Consider F to be a Hausdorff topological vector space and A a nonempty subset of F. Suppose Y : A → 2 F be any multivalued mapping such that

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∀ a ∈ A, Y (a) is finitely closed. The family of sets {Y (a) : a ∈ A}, thus, has a finite intersection property iff the mapping Y is a generalized KKM mapping. The KKM mapping theorem is extended as follows: Theorem 3.2 Consider F to be a Hausdorff topological vector space and B a nonempty convex subset of F. Suppose ϕ = A ⊂ B and maps X, Y : A → 2 F where Y is an intersectionally closed valued map of B. Assume that some precompact convex subset B0 of B contains a nonempty subset A0 ⊂ A such that Y (a0 ) is compact subset of B for at least one a0 ∈ A0 and let Y : A → 2 F denotes a generalized X-KKM mapping. Therefore, ∩a∈A Y (a) = ϕ. Proof Let A1 = A0 ∪ {a1 , a2 , . . . , an }, for any finite subset {a1 , a2 , . . . , an } of A. Because B0 is a precompact convex subset of B, the convex hull of B0 ∪ {a1 , a2 , . . . , an } is likewise a compact convex subset of B, and we will refer to it as K. Let T (b) = Y (b) ∩ K , for each b ∈ A1 . Here, each T (b) is also compact because Y(b) is an intersectionally closed valued map of B, and K is a compact subset of B. Moreover, because Y : A → 2 F is defined by Y (a) = Y (a) for each a ∈ A, we know that Y is a closed valued generalized X-KKM mapping. As a result, we can simply demonstrate that T is likewise an X-KKM map. Therefore, we have ∩ T (b) = ϕ by Theorem 3.1. Thus, we have a∈A1

ϕ = ∩ T (a) = ∩ Y (a) ∩ K a∈A1

a∈A1

= ∩ Y (a0 ) ∩ Y (a1 ) ∩ · · · ∩ Y (an ) a0 ∈A0

⊂ ∩ Y (a0 ) ∩ Y (a1 ) ∩ · · · ∩ Y (an ). a0 ∈A0

Let C represents compact set ∩ Y (a0 ). Then, for every finite subset n a0 ∈A0 {a1 , a2 , . . . , an } ⊂ A, we have i=1 Y (ai ) ∩ C = ϕ. Since C is a compact subset of Y and each Y (a) is an intersectionally closed valued map of B, this implies that each Y (a) ∩ C is also a compact subset of B. Since the family {Y (a) ∩ C|a ∈ A} has the finite intersection property, we have ∩ Y (a) ∩ C = ∩ Y (a) = ϕ.

a∈A

a∈A

The proof is now complete. The particular case of Theorem 3.1 is the following corollary: Corollary 3.2 Consider F to be a Hausdorff topological vector space and A a nonempty convex subset of F. Consider two multi maps X, Y : A → 2 F , where Y is an intersectionally closed valued map such that Y (a0 ) = K is compact for at least one a0 ∈ A and let Y : A → 2 F represents a generalized X-KKM mapping. Then, we have ∩ Y (a) = ϕ. a∈A

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Proof Given Y : A → 2 F , which is defined as Y (a) = Y (a), ∀a ∈ A, we know that Y is a generalized X-KKM mapping with closed values. According to Theorem 3.1, the family of sets {Y (a) : a ∈ A} satisfies the finite intersection property. We have ∩a∈A Y (a) = ϕ, since Y (a0 ) is compact. Because Y is an intersectionally closed valued mapping, ∩ Y (a) = ∩ Y (a) = ϕ.

a∈A

a∈A

Note that, if X : A → F is defined as a single valued map with X (ai ) = bi for each i ∈ {1, 2, . . . , n}, then Corollary 3.3 incorporates the result of Ansari et al. [31, Th. 2.1]. Corollary 3.3 [29] Consider F to be a Hausdorff topological vector space and A a nonempty convex subset of F. Consider the multi map Y : A → 2 F , where Y is transfer closed valued map such that Y (a0 ) = K is compact for at least one a0 ∈ A and let Y : A → 2 F represents a generalized KKM mapping. Then, we have ∩ Y (a) = ϕ. a∈A

X − γ -generalized quasi-convexity (quasi-concavity) and generalized X-KKM mappings have the following relationship: Theorem 3.2 Consider F to be a topological vector space and A a nonempty convex subset of F. Let ϕ : A × A → (−∞, ∞), γ ∈ (−∞, ∞) and X : A → 2 A . The following are then equivalent: (a)

The mapping Y : A → 2 A is defined as Y (b) = {a ∈ A : ϕ(a, z) ≤ γ , ∀z ∈ X (a)},   r esp.Y (b) = {a ∈ A : ϕ(a, z) ≥ γ , ∀z ∈ X (a)} , a generalized X-KKM mapping, for each b ∈ A. (b) ϕ(a, b) is X − γ -generalized quasiconcave, [resp., X − γ -generalized quasiconvex] in b.

Proof We simply prove the conclusion solely for the first case in (a) and (b), for the purpose of simplicity. In the same way, the other may be established. ⇒ (b). As Y : B → 2 A is a generalized X-KKM mapping, we have (a)     k ⊂ ∪kj=1 Y ai j , for any finite set {a1 , a2 , . . . , an } ⊂ A, and co j=1 X ai j    each {i 1 , . . . , i k } ⊂ {1, 2, . . . , n}. As a result, for any b0 ∈ co kj=1 X ai j , b0 ∈     ∪kj=1 Y ai j . So, there exists some m ∈ {1, 2, . . . , k}, such that b0 ∈ Y bim , and we     have φ b0 , z im ≤ γ , ∀z im ∈ X aim . Therefore, we have     min a0 , z i j ≤ γ for all z im ∈ X aim .

1≤ j≤k

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In other words, φ is X-γ -generalized quasiconcave in b. (b) ⇒ (a). It is known that φ is X-γ -generalized quasiconcave in b, so for any finite set {a1 , a2,. . . , an } ⊂ A, each {i 1 , . . . , i k } ⊂ {1, 2, . . . , n}, and any b0 ∈ co kj=1 X ai j ,     min φ b0 , z i j ≤ γ for all z im ∈ X aim .

1≤ j≤k

    Therefore, for some   m : 1 ≤ m ≤ k, wek have φ b0 , z im ≤ γ , ∀z im ∈ X aim . It shows that b0 ∈ Y aim and, hence, b0 ∈ ∪ j=1 Y ai j .    By the arbitrariness of b0 ∈ co kj=1 X ai j , we have ⎧ ⎫ k ⎨

 ⎬   k co ⊂ ∪ Y ai j . X ai j ⎩ ⎭ j=1 j=1

As a result, the Y : B → 2 A is a generalized X-KKM mapping. Assume A = F is a topological vector space, Y : A → 2 F is given as Y (b) = {a ∈ F : ϕ(a, b) ≤ γ }(r esp., Y (b) = {a ∈ F : ϕ(a, b) ≥ γ }), ∀ b ∈ F is a generalized KKM mapping, X : A → F be a single valued map with X (ai ) = bi for each i ∈ {1, 2, . . . , n}, and there is a function ψ(a, b), which is γ -generalized quasiconcave (respectively, convex) in b, Theorem 3.2, thus, entails Proposition 3.1 of Ansari et al. [29] given in the form of following corollary. Corollary 3.4 [29]. Consider F to be a topological vector space and A a nonempty convex subset of F. Let ϕ : A × A → (−∞, ∞) and γ ∈ (−∞, ∞). The following are then equivalent: The mapping Y : A → 2 A is defined by

(i)

  Y (b) = {a ∈ A : ϕ(a, b) ≤ γ } r esp., Y (b) = {a ∈ A : ϕ(a, b) ≥ γ } ,

(ii)

a generalized KKM mapping. ϕ(a, b) is γ -generalized quasiconcave, [resp., γ -generalized quasiconvex] in b.

The following general minimax inequality is now stated and proven. Theorem 3.3 Consider F to be a Hausdorff topological vector space, A a nonempty closed convex subset of F, and X : A → 2 A . Let us consider two maps ϕ, ψ : A × A → (−∞, ∞) and a number γ ∈ (−∞, ∞) satisfying following conditions: (i) (ii)

ϕ(a, b) is a γ -generalized lower semicontinuous function in a, for any fixed b ∈ A; ψ(a, b) is a X-γ -generalized quasiconcave function in b, for any fixed a ∈ A;

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For all (a, b) ∈ A × A, ϕ(a, b) ≤ ψ(a, b); For at least one b0 ∈ A, the set {a ∈ A : ϕ(a, b0 ) ≤ γ } is precompact. Then, there exists a ∈ A such that inf a∈A, supz∈X (a) ϕ(a, z) ≤ γ .

Proof Define the maps T, S : A → 2 A by {a ∈ A : ψ(a, z) ≤ γ , ∀z ∈ X (a)} and S(b) = T (b) = {a ∈ A : ϕ(a, z) ≤ γ , ∀z ∈ X (a)}. S is an intersectionally closed valued mapping on A, from condition (i). In fact, if a ∈ / S(b), then ϕ(a, z) > γ . As ϕ(a, b) is given to be γ -generalized lower semicontinuous function in a; so, there exists a b ∈ A and a neighborhood N(a) of a such that   ϕ r, b > γ , ∀r ∈ N (a).   Therefore, S b ⊂ A\N (a) and a ∈ / S(b ). As a result, S is intersectionally closed valued and T is a generalized X-KKM mapping based on condition (ii) and Theorem 3.1. We get T (b) ⊂ S(b), from condition (iii) thus, S is likewise a generalized X-KKM mapping. As a result, S is also generalized X-KKM mapping. Also, S(b0 ) is precompact, using condition (iv). Thus, S(b0 ) is compact. Now, according to Theorem 3.2, ∩a∈A S(a) = ϕ. Thus, there exists a ∈ A such that ϕ(a, z) ≤ γ , ∀z ∈ X (A). Particularly, we have inf a∈A, supz∈X (a) ϕ(a, z) ≤ γ . This proof is now complete. It is noted that Theorem 3.3 entails Theorem 3.1 of Ansari et al. [29] if X : A → A, X (ai ) = bi for each i ∈ {1, 2, . . . , n} and ψ(a, b) is a γ −generalized quasiconcave function in b. Corollary 3.5 [29] Consider F to be a Hausdorff topological vector space and A a nonempty closed convex subset of F. Let us consider two maps ϕ, ψ : A × A → (−∞, ∞) and a number γ ∈ (−∞, ∞) satisfying following conditions: (i) (ii) (iii) (iv)

ϕ(a, b) is a γ -transfer lower semicontinuous function in a, for any fixed b ∈ A; ψ(a, b) is a γ -generalized quasiconcave function in b, for any fixed a ∈ A; For all (a, b) ∈ A × A, ϕ(a, b) ≤ ψ(a, b); For at least one b0 ∈ A, the set {a ∈ A : ϕ(a, b0 ) ≤ γ } is precompact. Then, there exists a ∈ A such that inf sup ϕ(a, b) ≤ γ , ∀b ∈ A.

a∈A, z∈X (a)

As an application of the aforementioned results, we present the saddle point results below. Theorem 3.4 Consider F to be a Hausdorff topological vector space, A a nonempty closed convex subset of F, and X : A → 2 A . Let X : A → 2 A a surjective mapping and ϕ : A × A → (−∞, ∞) with γ ∈ (−∞, ∞) a given number, satisfying following conditions:

Generalized KKM Mapping Theorems

(i) (ii) (iii)

87

ϕ(a, b) is a γ -generally lower semicontinuous function in a and X-γ generalized quasiconcave function in b. ϕ(a, b) is a γ -generally upper semicontinuous function in b and X-γ generalized quasiconvex function in a. There exists a1 , b1 ∈ A such that the sets {a ∈ A : ϕ(a, b1 ) ≤ γ } and {b ∈ A : ϕ(a1 , b) ≥ γ } are both  precompact. Then, there exists a saddle point of ϕ(a, b); i.e., ∃ a, b ∈ A × A such that     ϕ(a, b) ≤ ϕ a, b ≤ ϕ a, b , ∀a, b ∈ A. Furthermore, we have got inf sup ϕ(a, b) = sup inf ϕ(a, b) = γ .

a∈A, b∈A

b∈A, a∈A

Proof Let us consider ϕ = ψ, then by Theorem 3.3, there exists a ∈ A such that ϕ(a, z) ≤ γ , ∀z ∈ X (b). Since X is surjective, so we have sup inf ϕ(a, z) = sup inf ϕ(a, b), and thus, b∈A, Z ∈X (b)

b∈A, a∈A

ϕ(a, b) ≤ γ , ∀b ∈ A.

(3.7)

Now, consider f : A × A → (−∞, ∞) which is defined as f (b, a) = −ϕ(a, b). Then, f (a, b) is γ -generalized lower semicontinuous function in a and X − γ generalized quasiconcave function in b, by condition (ii). Now, as a result of Theorem 3.3, ∃b ∈ A such that     sup inf ϕ z, b = − inf sup f b, z

b∈A, z∈X (a)

z∈X (a), b∈A

It is noted that X is surjective. So,     inf sup ϕ z, b = inf sup ϕ a, b

z∈X (a), b∈A

a∈A, b∈A

And thus,   ϕ a, b ≥ γ , ∀a ∈ A.

(3.8)

  Joining (3.7) and (3.8), we get ϕ a, b = γ and     ϕ(a, b) ≤ ϕ a, b ≤ ϕ a, b ,

∀a, b ∈ A.

(3.9)

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Finally, again using (3.7) and (3.8), we have that sup inf ϕ(a, b) ≤ inf sup ϕ(a, b)

b∈A, a∈A

a∈A, b∈A

    ≤ sup ϕ(a, b) ≤ ϕ a, b ≤ inf ϕ a, b by (3.9) a∈A

b∈A

≤ sup inf ϕ(a, b). b∈A, a∈A

Consequently, inf sup ϕ(a, b) = sup inf ϕ(a, b) = γ .

a∈A, b∈A

b∈A, a∈A

The proof is now complete. Theorem 3.4 entails Theorem 3.2 of Ansari et al. [29] if X : A → A, X (ai ) = bi for each i ∈ {1, 2, . . . , n} and ϕ(a, b) is a quasiconcave function in b and γ generalized quasiconvex function in a. Corollary 3.6 [29] Consider F to be a Hausdorff topological vector space and A a nonempty closed convex subset of F. Let ϕ : A× A → (−∞, ∞) and γ ∈ (−∞, ∞) be given number, satisfying the following conditions: ϕ(a, b) is a γ -transfer lower semicontinuous function in a and γ -generalized quasiconcave function in b. (ii) ϕ(a, b) is a γ -transfer upper semicontinuous function in b and γ -generalized quasiconvex function in a. (iii) There exists a1 , b1 ∈ A such that the sets {a ∈ A : ϕ(a, b1 ) ≤ γ } and {b ∈ A : ϕ(a1 , b) ≥ γ } are both precompact.   Then, there exists a saddle point of ϕ(a, b); i.e., ∃ a, b ∈ A × A such that (i)

    ϕ(a, b) ≤ ϕ a, b ≤ ϕ a, b , ∀a, b ∈ A. Furthermore, we have got inf sup ϕ(a, b) = sup inf ϕ(a, b) = γ .

a∈A, b∈A

b∈A, a∈A

The following saddle point theorem for parametric games in topological vector spaces is now presented: Theorem 3.5 Consider E and F to be a two Hausdorff topological vector spaces with A ⊂ E and B ⊂ F are two nonempty closed convex subsets and A a nonempty closed convex subset of F. Let X : B → 2 A be a surjective mapping, γ ∈ (−∞, ∞) a given number, and G δ = f − δg : A × B → R with the following conditions:

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f (a, b) is a γ -generally lower semicontinuous function in a and X-γ generalized quasiconvex function in b. (ii) f (a, b) is a γ -generally upper semicontinuous function in b and X-γ generalized quasiconcave function in a. (iii) g(a, b) is a γ -generally lower semicontinuous function in b and X-γ generalized quasiconvex function in a. (iv) g(a, b) is a γ -generally upper semicontinuous function in a and X-γ generalized quasiconcave function in b. (v) There exists a1 , b1 ∈ A such that the sets {a ∈ A : G δ (a, b1 ) ≤ γ } and {b ∈ A : G δ (a1 , b) ≥ γ } are both precompact.   Then, there exists a saddle point of G δ (a, b); i.e., ∃ a, b ∈ A × A such that (i)

    G δ (a, b) ≤ G δ a, b ≤ G δ a, b , ∀a, b ∈ A. Furthermore, we have got inf sup G δ (x, y) = sup inf G δ (x, y) = γ .

x∈X y∈X

y∈X x∈X

Proof The function G δ (a, b) is γ -generally upper semicontinuous function in b and X-γ -generalized quasiconcave function in a, based on conditions (ii) and (iii) of the theorem and because of δ ≥ 0. In a similar way, G δ (a, b) is γ -generalized lower semicontinuous function in a and X-γ -generalized quasiconvex in b. Then,   function according to Theorem 3.4, G δ (a, b) possesses a saddle point a, b ∈ A × B. From Theorem 3.5, we have the following result. Corollary 3.7 Consider E and F to be a two Hausdorff topological vector spaces with A ⊂ E and B ⊂ F are two nonempty closed convex subsets and A a nonempty closed convex subset of F. Let γ ∈ (−∞, ∞) be a given number and G δ = f − δg : A × B → R satisfies following conditions: (i) (ii) (iii) (iv) (v)

f (a, b) is a γ -generally lower semicontinuous function in a and γ -generalized quasiconvex function in b. f (a, b) is a γ -generally upper semicontinuous function in b and γ generalized quasiconcave function in a. g(a, b) is a γ -generally lower semicontinuous function in b and γ -generalized quasiconvex function in a. g(a, b) is a γ -generally upper semicontinuous function in a and γ -generalized quasiconcave function in b. There exists a1 , b1 ∈ A such that the sets

{a ∈ A : G δ (a, b1 ) ≤ γ } and {b ∈ A : G δ (a1 , b) ≥ γ } are  both precompact. Then, there exists a saddle point of G δ (a, b); i.e., ∃ a, b ∈ A × A such that

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    G δ (a, b) ≤ G δ a, b ≤ G δ a, b , ∀a, b ∈ A. Furthermore, we have got inf sup G δ (x, y) = sup inf G δ (x, y) = γ .

x∈X y∈X

y∈X x∈X

References 1. von Neumann, J.: Zur Theorie der Gesellschaftsspiele. Math. Ann. 100, 295–320 (1928) 2. von Neumann, J.: Über ein ökonomische Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes. In: Menger, K. (ed.) Ergebnisse eines Mathematischen Kolloquiums, pp. 73–83. Wien (1937) 3. Kakutani, S.: A generalization of Brouwer’s fixed point theorem. Duke Math. J. 8, 457–459 (1941) 4. Sion, M.: On general minimax theorem. Pac. J. Math. 8, 171–176 (1958) 5. Knaster, B., Kuratowski, K., Mazurkiewicz, S.: Ein Beweis des Fixpunktsatzes fÄur nDimensionale Simplexe. Fund. Math. 14, 132–137 (1929) 6. Fan, K.: A minimax inequality and application. In: Shisha, O. (ed.) Inequalities, III (Proc. Third Sympos., UCLA, 1969. Dedicated to the Memory of T. S. Motgkin, pp. 103–113. Academic, New York (1972) 7. Fan, K.: Fixed point and related theorems for noncompact convex sets. In: Moeschlin, O., Pallaschke, D. (eds.) Game Theory and Related Topics, pp. 151–156. North-Holland (1979) 8. Fan, K.: Some properties of convex sets related to fixed point theorems. Math. Ann. 266, 519–537 (1984) 9. Lassonde, M.: On the use of KKM-multifunctions in fixed point theory and related topics. J. Math. Anal. Appl. 97, 151–201 (1983) 10. Horvath, C.: Contractibility and generalized convexity. J. Math. Anal. Appl. 156(2), 341–357 (1991) 11. Horvath, C.: Extension and selection theorems in topological spaces with a generalized convexity structure. Ann. Fac. Sci. Toulouse Math. 6(2), 253–269 (1993) 12. Park, S.: Some coincidence theorems on acyclic multifunctions and applications to KKM theory. In: Tan, K.-K. (ed.) Fixed Point Theory and Applications, pp. 248–277. World Sci. Publ., River Edge, NJ (1992) 13. Park, S.: On minimax inequalities on spaces having certain contractible subsets. Bull. Austral. Math. Soc. 47(1), 25–40 (1993) 14. Park, S.: Continuous selection theorems in generalized convex spaces. Numer. Funct. Anal. Optim. 20, 567–583 (1999) 15. Park, S.: Various subclasses of abstract convex spaces for the KKM theory. Proc. Natl. Inst. Math. Sci. 2(4), 35–47 (2007) 16. Park, S.: Remarks on some basic concepts in the KKM theory. Nonlinear Anal. 74, 2439–2447 (2011) 17. Park, S.: New generalizations of basic theorems in the KKM theory. Nonlinear Anal. 74, 3000–3010 (2011) 18. Park, S.: A genesis of general KKM theorems for abstract convex spaces. J. Nonlinear Anal. Optim. 2(1), 133–146 (2011) 19. Park, S.: Equilibrium existence theorems in KKM spaces. Nonlinear Anal. 69, 4352–4364 (2008) 20. Park, S., Kim, H.: Foundations of the KKM theory on generalized convex spaces. J. Math. Anal. Appl. 209(2), 551–571 (1997)

Generalized KKM Mapping Theorems

91

21. Park, S., Kim, H.: Generalizations of the KKM type theorems on generalized convex spaces. Indian J. Pure Appl. Math. 29(2), 121–132 (1998) 22. Park, S., Kim, H.: Coincidence theorems on a product of generalized convex spaces and applications to equilibria. J. Korean Math. Soc. 36(4), 813–828 (1999) 23. He, R.H.: KKM-theorems in L-spaces. In: Recent Advances in Computer Science and Information Engineering, vol. 124, pp. 57–62 (2012) 24. Xiang, S., Xia, S., Chen, J.: KKM lemmas and minimax inequality theorems in abstract convexity spaces. Fixed Point Theory Appl. 209, 1–12 (2013) 25. Nikbakhtsarvestani, F., Vaezpour, S.M., Asadi, M.: A characterization of the generalized KKM mappings via the measure of noncompactness in complete geodesic spaces. J. Nonlinear Funct. Anal. 2017, 1–18 (2017) 26. Park, S.: Generalizations of some KKM type results on hyperbolic spaces. Nonlinear Funct. Anal. Appl. 23(4), 805–818 (2018) 27. Khanh, P.Q., Long, V.S.T.: General theorems of the Knaster-Kuratowski-Mazurkiewicz type and applications to the existence study in optimization. J. Math. Progr. Oper. Res. 69(12), 2695–2717 (2020) 28. Park, S.: Improving KKM theory on general metric type spaces. Topol. Algebra Appl. 9, 1–12 (2021) 29. Ansari, Q.H., Lin, Y.C., Yao, J.C.: General KKM theorem with applications to minimax and variational inequalities. J. Optim. Theory. Appl. 104(1), 41–57 (2000) 30. Balaj, M., Muresan, S.: Generalizations of the Fan-Browder fixed point theorem and minimax inequalities. Archivum Mathematicum (BRNO) Tomus 41, 399–407 (2005) 31. Camona, G., Carvalho, L.: Repeated two-person zero-sum games with unequal discounting and private monitoring. J. Math. Econ. 63(C), 131–138 (2016) 32. Chang, S., Zhang, Y.: Generalized KKM theorem and variational inequalities. J. Math. Anal. Appl. 159, 208–223 (1991) 33. Lai, H.C.: On a dynamic fractional game. J. Math. Anal. Appl. 294(2), 644–654 (2004) 34. Luc, D.T., Sarabi, E., Soubeyran, A.: Existence of solutions in variational relation problems without convexity. J. Math. Anal. Appl. 364, 1028–1042 (2010) 35. Nessah, R., Tian, G.: Existence of equilibrium in minimax inequalities, saddle points, fixed points, and games without convexity and compactness Assumptions. J. Optim. Theory Appl. 157, 75–95 (2013) 36. Sawasaki, Y., Kimura, Y., Tanaka, K.: A two-person zero-sum game with fractional loss function. J. Oper. Res. Soc. Jpn. 43(1), 209–218 (2000) 37. Tian, G.Q.: Generalizations of the FKKM theorem and the Ky Fan minimax inequality, with applications to maximal elements, price equilibrium, and complementarity. J. Math. Anal. Appl. 170, 457–471 (1992)

Optimal Mild Solutions of Time-Fractional Stochastic Navier-Stokes Equation with Rosenblatt Process in Hilbert Space K. Anukiruthika and P. Muthukumar

Abstract This manuscript aims to establish the optimal mild solutions of timefractional stochastic Navier-Stokes equation governed by the Rosenblatt process. The basic deterministic nonlinear time-fractional partial differential system is enriched with a stochastic term. The solvability is discussed with the aid of stochastic analysis, fractional calculus, fixed point theorem of condensing maps, and by establishing an appropriate measure of non-compactness. Further, an optimal mild solution is established. Keywords Existence of mild solutions · Hilbert space · Measure of non-compactness · Optimal mild solutions · Rosenblatt process · Time-fractional Navier-Stokes equation

1 Introduction Navier-Stokes Equations (NSEs) play a vital role in modeling incompressible Newtonian fluid flow. It serves as a great tool in domains like aeronautical sciences, plasma physics, thermo-hydraulics, and meteorology [1, 2]. Also, fractional differential equations are widely applied in fluid mechanics, finance, physical, and biological systems [3, 4]. The long memory processes of a system can be characterized using the time-fractional differential equations. The study of NSEs with time-fractional derivative is termed as time-fractional NSEs. A strong motivation for investigating this model arises from the fact that the phenomenon of anomalous diffusion in fractal media is simulated by NSE with time-fractional derivative. Some recent contributions regarding the mild solutions of time-fractional NSEs are established in [2, 5]. Stochastic Differential Equations (SDEs) have achieved considerable acceptance with vast applications in physics, chemistry, and engineering domains [6, 7]. The noise term will elaborate the important characteristic like ergodic behavior, and its K. Anukiruthika · P. Muthukumar (B) The Gandhigram Rural Institute (Deemed to be University), Gandhigram 624302, Tamil Nadu, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_9

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invariant measure which cannot be found in deterministic NSEs. The mild solution of stochastic NSEs with jump noise in L p -space is studied in [8]. Rosenblatt process is applied in the models which are non-Gaussian but selfsimilar with increments which are stationary. It has various applications in the domains such as telecommunications, turbulence, and economics [9, 10]. The study on Optimal Mild Solutions (OMSs) is attracted by many researchers [11, 12]. The existence of OMSs is studied with Lipschitz continuous conditions for a partial differential system by the authors in [12]. Recently, the authors in [13] studied the existence of OMSs for stochastic delay differential system. The novelty in this work is the study on existence of OMSs for time-fractional stochastic NSE driven by Rosenblatt process. The results are derived with aid of the theorem of condensing maps, stochastic analysis, fractional calculus, and semi-group properties. The major contributions in this manuscript are as follows: • Rosenblatt process is newly introduced for the time-fractional stochastic NSE in Hilbert space. • By relaxing the stronger conditions like Lipschitz continuous and compactness of the operator, the authors in this paper studied the solvability of the proposed system using the Hausdorff Measure of Non-Compactness (HMNC). • Further, the OMSs for the time-fractional stochastic NSE is obtained.

2 Model Description of Time-Fractional Stochastic NSE Consider the nonlinear time-fractional stochastic NSE driven by Rosenblatt process on a domain D ⊆ Rn , n ≥ 1 and its smooth boundary be ∂D: c α ∂s (s, x)

− ν(s, x) + ((s, x) · )(s, x) = − p(s, x) + g(s, x) ∂ Z H (s) , x ∈ D, s > 0, + (s, x) ∂s  · (s, x) = 0, x ∈ D,   (s, x) = 0, x ∈ ∂D, (0, x) = a, x ∈ D, (1) ∂D

where 21 < α < 1, c ∂sα is the Caputo derivative of order α, (s, x) denotes the velocity field at a point x ∈ D and at the time s > 0, p(s, x) is the associated pressure field,  denotes the Laplace operator, ν is the viscosity coefficient, g(s, x) represents external force and Z H (s) is the Rosenblatt process defined in Sect. 3. Let (, F, P) be the complete probability space with the normal filtration {Fs , s ≥ 0}. Here  is an L2F (, Rn )-valued function with the initial value a which is F0 -measurable square integrable Rn -valued random variable. Initially for removing the pressure term, Helmholtz projection P is applied to each term of (1), for s > 0. Hence, (1) is given by,

Optimal Mild Solutions of Time-Fractional Stochastic …

95

c α ∂s (s, x)

− ν P(s, x) + P((s, x) · )(s, x) = Pg(s, x) ∂ Z H (s) + P(s, x) , ∂s  · (s, x) = 0,   (s, x) = 0, (0, x) = a. ∂D

The operator −ν P = A with Dirichlet boundary conditions is meant to be the stokes operator A in the divergence-free function space under consideration. The abstract form of (1) is (see, [2]): c

Dsα z(s) = −Az(s) + F(z(s), z(s)) + Pg(s) + P(s)

dZ H (s) , ds

z(0) = a.

(2)

Here, the nonlinear operator F(z, z) = −P(z · )z, with a slight abuse of notation F(z) = F(z, z) and we shall also use the notations g(z(s)) instead of Pg(s) and (s) instead of P(s). Helmholtz operator is applied to each term in (2) and rewritten as: c

Dsα z(s) = −Az(s) + F(z(s)) + g(z(s)) + (s)

dZ H (s) , s ∈ I := [0, b], ds

z(0) = a.

(3)

This manuscript is characterized as follows: The preliminary results, some notations, and basic definitions are introduced in Sect. 3. The solvability of time-fractional stochastic NSE is given in Sect. 4, with aid of the properties of Stokes operator, fractional calculus, and fixed point theorem using HMNC. Further, Sect. 5 contributes the existence of an OMS for the proposed system (3).

3 Preliminaries This section deals with some basic definitions and preliminaries: • Let the two real separable Hilbert spaces be H and K with the norm  ·  and  · K , respectively. • An H-valued random variable is an Fs -measurable function z(s) :  → H and a collection of random variable S p := {z(s, ω) :  → H, s ∈ I} is known as stochastic process. • Let the Stokes operator A be self-adjoint and there exists √ eigenvectors eς corresponding to the eigenvalues γς with Aeς = γς eς , eς = 2 sin(ς π ), γς = π 2 ς 2 , β

β

ς ∈ N. For any β > 0, A 2 eς = γς2 eς , ς = 1, 2, . . . , and the domain of the frac-

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K. Anukiruthika and P. Muthukumar β tional power is denoted by H˜ β = D(A 2 ) = {q ∈ L2 (D) : q2H˜ β = β 2

∞  ς=1

β

γς2 qς2
0, z ∈ C. (αn + β)

If β = 1, denote Eα,1 (z) as Eα (z). For more details on the properties of the operators, Eα (s) and Eα,α (s) see, [14]:

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97

Lemma 2 [14] The linear operators Eα (s) and Eα,α (s) are bounded for s > 0. Also, if 0 < α < 1 and 0 ≤ β < 2, then there exists ϑ > 0 with αβ

αβ

Eα (s)X  H˜ β ≤ ϑs − 2 X  and Eα,α (s)X  H˜ β ≤ ϑs − 2 X . Lemma 3 [14] The operators Eα (s) and Eα,α (s) are strongly continuous for s > 0. Also, if 0 < α < 1 and 0 ≤ β < 2 and 0 ≤ s1 < s2 ≤ b, then there exists ϑ > 0 with αβ [Eα (s2 ) − Eα (s1 )]X  H˜ β ≤ ϑ(s2 − s1 ) 2 X  and [Eα,α (s2 ) − Eα,α (s1 )]X  H˜ β ≤ αβ ϑ(s2 − s1 ) 2 X . Lemma 4 [14] Let Sα (s) = s α−1 Eα,α (s), for 0 < α < 1 and 0 ≤ β < 2, there exists (2−β)α−2 a positive constant ϑ with Sα (s)X  H˜ β ≤ ϑs 2 X  and [Sα (s2 ) − Sα (s1 )]X  H˜ β 2−(2−β)α ≤ ϑ(s2 − s1 ) 2 X , for any 0 ≤ s1 < s2 ≤ b. Further, let us recall the concepts of HMNC. Throughout this section, let the nonempty subsets of H which is bounded be BH and χ be a HMNC on H. Definition 2 [17] HMNC is defined as χ (ϒ) = inf{ > 0 : ϒ has a finite -net} for all ϒ ∈ BH . For any b > 0 HMNC on C(I, Rn ) is given by, χb (ϒ) =

1 max y(s) − y(τ )Rn . lim sup 2 δ→0 y∈ϒ s,τ ∈I, |s−τ | 0, β ≥ 0} s∈I

endowed with the uniform convergence topology. For proving the solvability of (3), assume the following: (A1) Let ϑ F > 0 be a real number then the bounded bilinear operator F satisfies: (i) EF(z)2 ≤ ϑ F Ez2 and EF(z 1 ) − F(z 2 )2 ≤ ϑ F E(z 1 2 + z 2 2 ) z 1 − z 2 2 . (ii) For any countable bounded set ϒ ⊂ H which is equicontinuous there exists a positive constant ζ F : R+ → R+ with χ (F(s, ϒ)) ≤ ζ F (s)χ (ϒ), for s ∈ I. (A2) The mapping g satisfies the following: (i) For every s ∈ I, g(s, ·) is continuous and for every z ∈ H, g(·, z) is strongly measurable. (ii) There exists a function l g : I → R+ which is continuous and a non decreasing continuous function θg : R+ → R+ with Eg(z(s))2 ≤ l g (s)θg (Ez(s)2 ),

for a.e s ∈ I.

(iii) For any countable bounded set ϒ ⊂ H which is equicontinuous there exists positive constant ζg : R+ → R+ with

Optimal Mild Solutions of Time-Fractional Stochastic …

χ (g(ϒ)) ≤ ζg (s)χ (ϒ), (A3) The deterministic mapping  satisfies

s 0

99

for s ∈ I.

(τ )2L0 dτ < ∞. For every s ∈ I, 2

there exists a positive constant l with E(τ )2L0 ≤ l uniformly in I. 2

Theorem 2 Let all the hypotheses (A1)–(A3) be fulfilled, then there exists a mild solution on C(I, H˜ β ) for (3) with (2 − β)α − 1 > 0 provided that 4ϑ 2 b(2−β)α [ζ F L1 + ζg L1 ] < 1. (2 − β)α − 1

(6)

Proof The system (3) is transformed into a fixed point problem. Let the operator  : B k → C(I, H˜ β ) is given as s (z)(s) =Eα (s)a +

(s − τ )α−1 Eα,α (s − τ )[F(z(τ )) + g(z(τ ))]dτ

0

s +

(s − τ )α−1 Eα,α (s − τ )(τ )d Z H (τ ), s ∈ I.

0

The continuous functions F,  and g are integrable on I with aid of Bochner theorem [18]. Therefore, the set (z) is well defined on B k . Claim that  has a fixed point, which in turn is the mild solution for the system (3). Now, the proof is divided into four steps for proving the hypothesis. Step 1:  maps bounded sets into bounded sets in B k . Let k > 0 and z ∈ B k , then Ez(s)2 ≤ k. We have to show that for every z ∈ B k there exists a positive constant k1 with E(z)(s)2 ≤ k1 . In the view of Lemma 1, s 2  E(z)(s)2 ≤ 4 EEα (s)a2 + E (s − τ )α−1 Eα,α (s − τ )F(z(τ ))dτ 0

s 2 + E (s − τ )α−1 Eα,α (s − τ )g(z(τ ))dτ 0

s 2 + E (s − τ )α−1 Eα,α (s − τ )(τ )d Z H (τ ) 0

s  ≤ 4 ϑ 2 b−αβ Ea2 + bϑ 2 (s − τ )(2−β)α−2 ϑ F Ez(τ )2 dτ 0

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s + bϑ

2

(s − τ )(2−β)α−2 l g (τ )θg Ez(τ )2 dτ + ϑ 2 CH s 2H−1

0

s ×

(s − τ )(2−β)α−2 E(τ )2L0 dτ



2

0

4ϑ 2 b(2−β)α ϑ F (k) + θg (k)l g L1 ≤ 4ϑ 2 b−αβ Ea2 + (2 − β)α − 1  + CH b2H−2 l := k1 . Thus, {(z)(s) : z(s) ∈ B k } is bounded. k Step 2: (z)(s) is continuous on B k . Let {z n }∞ n=1 be a sequence in B and k lim z n = z, z ∈ B . Consider

n→∞

s E(z n )(s) − (z)(s) ≤ 2b

ESα (s − τ )2H˜ β EF(z n (τ )) − F(z(τ ))2 dτ

2

0

s + 2b

ESα (s − τ )2H˜ β Eg(z n (τ )) − g(z(τ ))2 dτ 0

s ≤2bϑ ϑ F 2

(s − τ )(2−β)α−2 E[z n (τ )2 + z(τ )2 ]z n − z2 dτ + 2bϑ 2

0

s ×

(s − τ )(2−β)α−2 Eg(z n (τ )) − g(z(τ ))2 dτ

0

−→ 0 as n → ∞. Hence, (z)(s) is continuous on B k . Step 3:  is equicontinuous. Claim that  maps bounded sets B k into an equicontinuous sets of functions. Let z ∈ B k , s1 , s2 ∈ I and 0 ≤ s1 < s2 ≤ b. Consider E(z)(s2 ) − (z)(s1 )2 ≤ 3E[Eα (s2 ) − Eα (s1 )]a2 s1 + 3E [(s2 − τ )α−1 Eα,α (s2 − τ ) − (s1 − τ )α−1 Eα,α (s1 − τ )] 0

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101

s1  2 × F(z(τ ))dτ + g(z(τ ))dτ + (τ )d Z H (τ ) + 3E [(s2 − τ )α−1

s2

 2 × Eα,α (s2 − τ )] F(z(τ ))dτ + g(z(τ ))dτ + (τ )d Z H (τ ) s1

αβ

≤3ϑ (s2 − s1 ) Ea + 9s1 ϑ 2

2

(s2 − s1 )2−(2−β)α

2 0

s1 

2H−1 2 2 2 × EF(z(τ )) + Eg(z(τ )) dτ + 9ϑ CH s1 (s2 − s1 )2−(2−β)α 0

s2 × E(τ )2L0 dτ + 9ϑ 2 (s2 − s1 ) 2

(s2 − s1 )(2−β)α−2 EF(z(τ ))2

s1

s2  2 2 2H−1 + Eg(z(τ )) dτ + 9ϑ CH (s2 − s1 ) (s2 − s1 )(2−β)α−2 E(τ )2L0 dτ 2

s1

9s1 ϑ 2 (s2 − s1 )3−(2−β)α [ϑ F (k) + θg (k)l g L1 + CH (2 − β)α − 3 9ϑ 2 (s2 − s1 )(2−β)α [ϑ F (k) + θg (k)l g L1 + CH (s2 − s1 )2H−2 l ] × s12H−2 l ] + (2 − β)α − 1 → 0 as s2 − s1 → 0.

≤3ϑ 2 (s2 − s1 )αβ Ea2 −

The above expression holds with aid of Lebesgue’s dominated convergence theorem [19]. Hence,  is continuous from the right in (0, b]. Likewise,  is continuous from the left in (0, b]. Therefore,  is equicontinuous. Step 4: The operator  is condensing. Let M = co(B k ), where co represents the closure of convex hull. It is easier to examine that  : M → M and M ⊂ C(I, H˜ β ). Claim that  : M → M is a condensing operator. For any W ⊂ M, a countable set W0 = {z n } ⊂ W exists by Lemma 5, with χ ((W)) ≤ 2χ ((W0 )). Hence, W0 ⊂ W is equicontinuous by the equicontinuity of W. Hence by assumptions (A1), (A2) and Lemma 6, s Eχ ((W0 )(s)) ≤ 2ϑ b 2

2

(s − τ )(2−β)α−2 Eχ (F(τ, W0 (τ )))2 dτ + 2ϑ 2 b

0

s × 0

≤

(s − τ )(2−β)α−2 Eχ (g(τ, W0 (τ )))2 dτ

2ϑ 2 b(2−β)α  ζ F L1 + ζg L1 Eχ (W)2 . (2 − β)α − 1

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Since (W0 ) ⊂ W is equicontinuous and bounded, by Lemma 7, we have χ ((W0 )) = max χ ((W0 ))(s). Then, s∈I

Eχ ((W))2 ≤

4ϑ 2 b(2−β)α  ζ F L1 + ζg L1 Eχ (W)2 . (2 − β)α − 1

In the view of (6),  : M → M is a k−set contractive. Therefore,  has a fixed point which is the mild solution of (3) by Theorem 1.

5 Optimal Mild Solutions The OMSs for the time-fractional stochastic NSE is given in this section. Let  be the set of all bounded mild solutions z(s) to the system (3) over I and is given by μ(z) ˆ = sup Ez(s)2 . Further, assume that  = ∅. s∈I

Definition 4 [11] Let z ∗ (s) be a bounded mild solution of (3). It becomes an OMS ˆ of (3) when μ(z ˆ ∗ ) ≡ μˆ ∗ = inf μ(z). z∈

Lemma 8 [11] Let Y be a uniformly convex Banach space and D be a nonempty closed subset of Y which is also convex. Let w ∈ / D, then there will be a unique qˆ0 ∈ D with w − qˆ0  = inf w − q. ˆ q∈D ˆ

Theorem 3 Assume the nontrivial functions F, g and  are convex in H, and Theorem 2 fulfills, then there exists an OMS to the system (3). Proof To claim the existence of OMS to the system (3), it is sufficient to show that  is a convex and closed set since the trivial solution 0 ∈ / . After verifying the conditions on Lemma 8, the OMS can be obtained. First, claim that  is convex. Let z 1 (s) and z 2 (s) be two distinct mild solutions belonging to  and λ ∈ [0, 1] be a real number such that z(s) = λz 1 (s) + (1 − λ)z 2 (s), s ∈ I. For every s ∈ I, by (A1)−(A3) , z(s) is continuous. s z(s) =Eα (s)a +

 (s − τ )α−1 Eα,α (s − τ ) [F(λz 1 (τ ) + (1 − λ)z 2 (τ ))]dτ

0

+ [g(λ(z 1 (τ )) + (1 − λ)(z 2 (τ )))]dτ + (τ )d Z H (τ ) s = Eα (s)a +

(s − τ )

α−1

s

Eα,α (s − τ )[F(z(τ )) + g(z(τ ))]dτ +

0

× Eα,α (s − τ )(τ )d Z H (τ ).

0

(s − τ )α−1

Optimal Mild Solutions of Time-Fractional Stochastic …

103

Thus, z(s) becomes the mild solution of (3). Here, z(s) is bounded over I since ˆ 1 ) + (1 − λ)μ(z ˆ 2 ) < ∞. μ(z) ˆ = sup Ez(s)2 ≤ λμ(z s∈I

Thus, z(s) ∈ . Next, claim that  is closed. Let z n (s) be a sequence in  with lim z n (s) = n→∞

z(s), s ∈ I. Consider s z n (s) = Eα (s)a +

(s − τ )α−1 Eα,α (s − τ )[F(z n (τ )) + g(z n (τ ))]dτ

0

s +

(s − τ )α−1 Eα,α (s − τ )(τ )d Z H (τ ).

0

For s ∈ I, applying limits to the above equation, we get s z(s) = Eα (s)a +

(s − τ )α−1 Eα,α (s − τ )[F(z(τ )) + g(z(τ ))]dτ

0

s +

(s − τ )α−1 Eα,α (s − τ )(τ )d Z H (τ ).

0

Then z(s) is the mild solution of (3). Next, claim that z(s) is bounded over I. Let z(s) = [z(s) − z n (s)] + z n (s) for every s ∈ I. Consider,  s Ez(s)2 ≤ 2 2b ESα (s − τ )2 EF(z n (τ )) − F(z(τ ))2 dτ 0

s + 2b

ESα (s − τ )2 Eg(z n (τ )) − g(z(τ ))2 dτ + Ez n (s)2



0

s ≤ 4bϑ 2 ϑ F

(s − τ )(2−β)α−2 E[z n (τ )2 + z(τ )2 ]z n − z2 dτ + 4bϑ 2

0

s ×

(s − τ )(2−β)α−2 Eg(z n (τ )) − g(z(τ ))2 dτ + 2Ez n (s)2 .

0

By Lebesgue’s dominated convergence theorem [19] and choosing n large enough, for every  > 0, Ez(s)2 ≤  + 2Ez n (s)2 , s ∈ I. We have

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μ(z) ˆ ≤  + 2μ(z ˆ n ) < ∞. Hence, z ∈  which concludes the proof.

6 Conclusion The solvability of time-fractional stochastic NSE governed by the Rosenblatt process has been obtained in this manuscript. The sufficient conditions for the existence of mild solution for the proposed system (1) are obtained by relaxing the stronger conditions like Lipschitz continuous and compactness of the operator using HMNC and fixed point theorem. Further, the existence of OMS for (1) has been established. The proposed system can be used to model long memory processes, and the obtained results in this manuscript generalize the existing solvability results in the literature [7] due to the presence of the Rosenblatt process. In future, the regularity properties of time-fractional stochastic NSEs can be studied based on the methodology used in this paper.

References 1. Vanhorn, W.: The Stokes Equations. Akademie Verlag, Berlin (1994) 2. Zhou, Y., Peng, L.: On the time-fractional Navier-Stokes equations. Comput. Math. Appl. 73(6), 874–891 (2017) 3. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier (2006) 4. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993) 5. Zhou, Y., Peng, L.: Weak solutions of the time-fractional Navier-Stokes equations and optimal control. Comput. Math. Appl. 73, 1016–1027 (2017) 6. Chen, P., Li, Y.: Nonlocal Cauchy problem for fractional stochastic evolution equations in Hilbert spaces. Collect. Math. 66, 63–76 (2015) 7. Xu, L., Shen, T., Yang, X., Liang, J.: Analysis of time fractional and space nonlocal stochastic incompressible Navier-Stokes equation driven by white noise. Comput. Math. Appl. 78(5), 1669–1680 (2019) 8. Fernando, B.P.W., Rudiger, B., Sritharan, S.S.: Mild solution of stochastic Navier-Stokes equation with jump noise in L p -spaces. Math. Nachr. 288(14), 1615–1621 (2015) 9. Tudor, C.A.: Analysis of the Rosenblatt process. ESAIM: Probab. Stat. 12, 230–257 (2008) 10. Shen, G., Ren, Y.: Neutral stochastic partial differential equations with delay driven by Rosenblatt process in a Hilbert space. J. Korean Stat. Soc. 44(1), 123–133 (2015) 11. N’Guerekata, G.: On weak-almost periodic mild solutions of some linear abstract differential equations. Amer. Inst. Math. Sci. 2003, 672–677 (2003) 12. Debbouche, A., El-borai, M.M.: Weak almost periodic and optimal mild solutions of fractional evolution equations. Electron. J. Differ. Equ. 2009, 1–8 (2009) 13. Yan, Z., Han, L.: Optimal mild solutions for a class of nonlocal multi-valued stochastic delay differential equations. J. Optim. Theory Appl. 181(3), 1053–1075 (2019) 14. Zou, G.A., Lv, G., Wu, J.L.: Stochastic Navier-Stokes equations with Caputo derivative driven by fractional noises. J. Math. Anal. Appl. 461(1), 595–609 (2018)

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15. Durga, N., Muthukumar, P.: Optimal control of Sobolev-type stochastic Hilfer fractional noninstantaneous impulsive differential inclusion involving Poisson jumps and Clarke subdifferential. IET Control Theory Appl. 14(6), 887–899 (2020) 16. Saravanakumar, S., Balasubramaniam, P.: On impulsive Hilfer fractional stochastic differential system driven by Rosenblatt process. Stoch. Anal. Appl. 37(6), 955–976 (2019) 17. Luong, V.T.: Decay mild solutions for two-term time fractional differential equations in Banach spaces. J. Fixed Point Theory Appl. 18, 417–432 (2016) 18. Deimling, K.: Nonlinear Functional Analysis. Springer, New York (1985) 19. Barra, G.D.: Measure Theory and Integration. Woodhead Publishing, Cambridge (2003)

Shape Preserving Hermite Interpolation Reproducing Ellipse Shubhashree Bebarta and Mahendra Kumar Jena

Abstract In this paper, trigonometric splines are used for Hermite interpolation of the data consist of functional values and their derivatives. If the derivative data is given, then we reparameterize the data suitably. Otherwise, we find them using a necessary and sufficient condition. We show that our Hermite interpolant preserves the monotonicity. We introduce the notion of H -convex and show that convexity of H -convex data is preserved by our Hermite interpolant. The biggest advantage of our method is that it reproduces ellipse. We also provide some numerical examples to validate our theory. Keywords Hermite interpolation · Quadratic trigonometric spline · Shape preservation · Ellipse

1 Introduction In recent times, it is possible to visualize a real-world phenomenon as a mathematical model in a computer. Given a few initial data of an object, it is possible nowadays to recreate the whole object mathematically. Two specific techniques are used for this: Approximation [1] and Interpolation [1]. Both techniques have their own advantages and disadvantages. Approximation techniques are unable to recreate the original object. At the same time, interpolation techniques recreate the original object [2]. Two standard interpolation techniques are used: Lagrange interpolation and Hermite interpolation. Unfortunately, the interpolants by these methods produce wiggles, oscillations, and loops although originally they are absent in the object [3]. In this matter, spline interpolations [3, 4] provide some help. However, they do not preserve the shapes: monotonicity and convexity. Thus, Hermite interpolation techniques [5] that preserves the shapes are most welcome. Sometimes objects have elliptical shapes [6]. Various shape preserving techniques use the approximation of S. Bebarta · M. K. Jena (B) Veer Surendra Sai University of Technology, Burla, Sambalpur 768018, Odisha, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_10

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these shapes [7]. Hence, a shape preserving Hermite interpolation that reproduces ellipses will be an advantage in the world of geometric design. This paper introduces a novel shape preserving the Hermite interpolation method that reproduces ellipse. The main features of the proposed method are: • Find a necessary and sufficient condition for the Hermite interpolation by a quadratic trigonometric spline. • The Hermite interpolant preserves the monotonicity of the original data. Moreover, the convexity of the original data is also preserved provided the data is H -convex. • Reproduces constants and ellipses. The paper is organized into seven sections. Section 1 is the introduction. In Sect. 2, quadratic trigonometric splines are defined. In this section, the Hermite spline interpolant is also found. Monotonicity of the Hermite interpolant is studied in Sect. 3. We describe the Hermite interpolant H (x) preserves the convexity in Sect. 4. In Sect. 5, some numerical examples are given. Comparison is studied in Sect. 6. The conclusion is given in Sect. 7.

2 Quadratic Trigonometric Spline and Hermite Interpolation L-splines are introduced by [8]. Trigonometric splines are special kind of L-splines. Given m = 2p let {v1 , . . . , vm } = {C(ξ/2), S(ξ/2), . . . , C[((p − 1)/2)ξ ], S[((p − 1)/2)ξ ], } Similarly, if m = 2p + 1, let {u1 , . . . , um } = {1, C(ξ ), S(ξ ), . . . , C(pξ ), S(pξ )} and define the m-dimensional space Hm by  Hm =

span{v1 , . . . , vm }, m = 2p span{u1 , . . . , um }, m = 2p + 1.

(1)

In the above as well as in the sequel, we take S(ξ ) = sin(ξ ) and C(ξ ) = cos(ξ ). The space S(Hm ; M; ) of trigonometric spline of order m with knots ξ1 , . . . , ξk of multiplicities m1 , . . . , mk is defined as follows: Suppose  = {a = ξ0 < ξ1 < · · · < ξk+1 = b} is a partition of the interval [a, b] and suppose M = (m1 , . . . , mk ) is a vector of integers with 1 ≤ mj ≤ m, j = 1, 2, . . . , k. Then

Shape Preserving Hermite Interpolation Reproducing Ellipse

⎫ ⎧ ⎪ ⎪ ⎪ H : there exist H0 , . . . , Hk ∈ Hm with H (ξ ) ⎪ ⎬ ⎨ = Hj (ξ ) for ξ ∈ Ij = [ξj , ξj+1 ), j = . S(Hm ; M; ) = 0, 1, . . . , k, and Dj−1 Hj−1 (ξj ) = Dj−1 Hj (ξj ), ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ j = 1, 2, . . . , m − mj , j = 1, 2, . . . , k

109

(2)

We emphasize on quadratic trigonometric splines, that is, m = 3 and M = (1, 1, . . . , 1). A particular kind of spline H is obtained by Hermite interpolation. In particular, we take 

Hj (ξj ) = fj and Hj (ξj+1 ) = fj+1 d H (ξ ) = dj and ddξ Hj (ξj+1 ) = dj+1 , dξ j j

(3)

where fj = f (ξj ) for some unknown function fj and dj is an approximation to f  (ξj ), j = 0, 1, . . . , k + 1. The problem (3) is known as the Hermite interpolation problem for the data set (4) D := {(ξj , fj , dj ) : j = 0, 1, . . . , k + 1}. The interpolant H is called as a Hermite spline [5]. This H exists provided D satisfies an important condition. It is enough to show this condition for Hj . Lemma 1 The necessary and sufficient condition for Hj to satisfy (3) is 

dj + dj+1 ξj+1 − ξj . = cot fj+1 − fj 2

(5)

Proof Let Hj (ξ ) = a0 + a1 C(ξ ) + a2 S(ξ ), where a0 , a1 and a2 are unknown real numbers to be determined. The conditions Hj (ξj ) = dj and Hj (ξj+1 ) = dj+1 imply in matrix form:

  

−S(ξj ) C(ξj ) dj a1 = . (6) a2 dj+1 −S(ξj+1 ) C(ξj+1 ) This equation has a unique solution if and only if S(ξj+1 − ξj ) = 0. In this case a1 =

dj C(ξj+1 ) − dj+1 C(ξj ) S(ξj+1 − ξj )

(7)

a2 =

dj S(ξj+1 ) − dj+1 S(ξj ) . S(ξj+1 − ξj )

(8)

and

The conditions Hj (ξj ) = fj and Hj (ξj+1 ) = fj+1 implies fj+1 − fj = a1 (C(ξj+1 ) − C(ξj )) + a2 (S(ξj+1 ) − S(ξj )). Substituting the values of a1 and a2 in the above equation, we get

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fj+1 − fj =

(C(ξj+1 ) − C(ξj ))(dj C(ξj+1 ) − dj+1 C(ξj )) S(ξj+1 − ξj ) (S(ξj+1 ) − S(ξj ))(dj S(ξj+1 ) − dj+1 S(ξj )) + S(ξj+1 − ξj )

Simplifying, we get

dj − dj C(ξj − ξj+1 ) − dj+1 C(ξj − ξj+1 ) + dj+1 S(ξj+1 − ξj )   ξ −ξ 

2(dj + dj+1 )S 2 j+12 j ξj+1 − ξj = = (dj + dj+1 ) tan . S(ξj+1 − ξj ) 2

fj+1 − fj =

1

 In the proof of above lemma, we have taken Hj (ξ ) = a0 + a1 C(ξ ) + a2 S(ξ ). It is found that a1 and a2 are given by (7) and (8), respectively. It is easy to find that a0 = fj+1 − a1 C(ξj+1 ) − a2 S(ξj+1 ). Taking all these together, we have −dj C(ξj+1 ) + dj+1 C(ξj ) −dj S(ξj+1 ) + dj+1 S(ξj ) C(ξj+1 ) + S(ξj+1 ) S(ξj+1 − ξj ) S(ξj+1 − ξj ) dj C(ξj+1 ) − dj+1 C(ξj ) dj S(ξj+1 ) − dj+1 S(ξj ) + C(ξ ) + S(ξ ). S(ξj+1 − ξj ) S(ξj+1 − ξj )

Hj (ξ ) = fj+1 +

Taking coefficients of dj+1 and dj together, we get −dj + dj+1 C(ξj+1 − ξj ) dj C(ξj+1 − ξ ) − dj+1 C(ξ − ξj ) + S(ξj+1 − ξj ) S(ξj+1 − ξj ) dj (C(ξj+1 − ξ ) − 1) + dj+1 (C(ξj+1 − ξj ) − C(ξ − ξj )) = fj+1 + S(ξj+1 − ξj ) = fj+1 + dj S(ξ − ξj ) dj (C(ξj+1 − ξj )C(ξ − ξj ) − 1) + dj+1 (C(ξj+1 − ξj ) − C(ξ − ξj )) + S(ξj+1 − ξj )   ξ −ξ C(ξj+1 − ξj )(−2 S 2 2 j dj + dj + dj+1 ) = fj+1 + dj S(ξ − ξj ) + S(ξj+1 − ξj )   2 ξ −ξj dj+1 ) (dj + dj+1 − 2 S 2 − S(ξj+1 − ξj )

 ξ − ξj dj+1 − dj C(ξj+1 − ξj ) = fj+1 + dj S(ξ − ξj ) + 2 S 2 2 S(ξj+1 − ξj )

 ξj+1 − ξj − tan (dj + dj+1 ). 2

Hj (ξ ) = fj+1 +

Shape Preserving Hermite Interpolation Reproducing Ellipse

111

After substitution of (5) into the last equation, we get the following lemma. Lemma 2 The Hermite interpolant Hj has the following form:

Hj (ξ ) = fj + dj S(ξ − ξj ) + 2 S

2

ξ − ξj 2



dj+1 − dj C(ξj+1 − ξj ) . S(ξj+1 − ξj )

(9)

In the sequel, we use the above form of Hj (ξ ).

3 Monotonicity Definition 1 A function g : I → R is said to be monotonically increasing in I if g  (ξ ) ≥ 0 ∀ ξ ∈ I . Similarly, if g  (ξ ) ≤ 0, ∀ ξ ∈ I , the function is monotonically decreasing in I . We are always dealing with discrete data. Thus, we provide an equivalent definition of monotonicity for discrete data.

Definition 2 The data D := { ξj , fj , dj , j = 0, 1, . . . , k + 1} is said to be monotonically increasing if fj ≤ fj+1 , dj ≥ 0 for ξ0 < ξ1 < · · · < ξk+1 .

Definition 3 The data D := { ξj , fj , dj , j = 0, 1, . . . , k + 1} is said to be monotonically decreasing if fj ≥ fj+1 , dj ≤ 0 for ξ0 < ξ1 < · · · < ξk+1 . Theorem 1 Let Ij = [ξj , ξj+1 ] and assume that 0 < ξj+1 − ξj < π . Let D is monotonically increasing. Then the Hermite spline Hj (ξ ) is monotonically increasing in Ij if and only if (5) is satisfied. For similar reason, Hj (ξ ) is monotonically decreasing in Ij if the data D is monotonically decreasing. Proof Let I = [ξj , ξj+1 ] and Hj be the quadratic trigonometric polynomial which solves the Hermite interpolation Problem (3). Thus, without loss of generality, we show the monotonicity of Hj in I . We observe that dj+1 − C(ξj+1 − ξj )dj S(ξ − ξj ) S(ξj+1 − ξj ) S(ξj+1 − ξ ) S(ξ − ξj ) + dj+1 = dj S(ξj+1 − ξj ) S(ξj+1 − ξj )

Hj (ξ ) = dj C(ξ − ξj ) +

(10)

Since 0 < ξj+1 − ξj < π and ξ ∈ [ξj , ξj+1 ], RHS of (10) is the circular barycentric combination [9] of dj and dj+1 . Let D is monotonically increasing, that is, dj ≥ 0 and dj+1 ≥ 0. Then Hj (ξ ) ≥ 0 is a property of circular barycentric combination. Hence,

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Hj (ξ ) is monotonically increasing in Ij . Next, let us consider that D is monotonically decreasing, that is, dj ≤ 0 and dj+1 ≤ 0. For this data  Hj (ξ ) ≤ max{dj , dj+1 }

The part

 ξ +ξ

j+1 −ξ 2  ξ −ξ  j+1 j C 2

C

j

S(ξj+1 − ξ ) S(ξj+1 − ξj )

+

S(ξ − ξj ) S(ξj+1 − ξj )

 = max{dj , dj+1 }

C

 ξ +ξ j

C

ξ

 −ξ  . −ξ

j+1

2 j+1

j

2



is always positive and max{dj , dj+1 } ≤ 0. Hence, H  (ξ ) < 0

for ξ ∈ [ξj , ξj+1 ]. Therefore, H is monotonically decreasing in Ij .



4 Convexity and Concavity Definition 4 (Convex function) A function g : I → R is said to be convex in I if g  (ξ ) ≥ 0, ∀ ξ ∈ I and if g  (ξ ) < 0, ∀ ξ ∈ I , the function is concave in I . Analogously, a discrete data set D is said to be convex according to the following definition. Definition 5 (Convex Data and Concave data) Let the Hermite data f −fj D = {(ξj , fj , dj ), j = 0, 1, . . . , k + 1} be given to us. Let us define fj = ξj+1 , j+1 −ξj j = 0, 1, . . . , k. Then the Hermite data D is convex if d0 < f0 < d1 < f1 < d2 < · · · < dk < fk < dk+1 . Similarly, the Hermite data D is concave if d0 > f0 > d1 > f1 > d2 > · · · > dk > fk > dk+1 . We now define a H -convex and H -concave data set. Definition 6 (H -convex and H-concave Data) Let the Hermite data D = {(ξj , fj , dj ), f −fj j = 0, 1, . . . , k + 1} be given to us. Let us define t fj = S(ξj+1 , j = 0, 1, . . . , k. j+1 −ξj ) t Then the Hermite data D is H -convex if  fj < dj+1 , j = 0, 1, . . . , k. Similarly, the Hermite data D is H -concave if t fj < dj , j = 0, 1, . . . , k. First, we consider the shape of monotonically increasing data, that is, dj ≥ 0 ∀ j = 0, 1, . . . , k + 1. Theorem 2 Suppose, the convex Hermite data D = {(ξj , fj , dj ), j = 0, 1, . . . , k + 1} is convex (or concave). Then the Hermite interpolant H (ξ ) preserves the shape of D provided it is H -convex (or H -concave). Proof Differentiating (10), we get Hj (ξ ) = and

1 S(ξj+1 − ξj )

{−dj C(ξj+1 − ξ ) + dj+1 C(ξ − ξj )},

(11)

Shape Preserving Hermite Interpolation Reproducing Ellipse

Hj (ξ ) =

1 S(ξj+1 − ξj )

113

{−dj S(ξj+1 − ξ ) − dj+1 S(ξ − ξj )},

(12)

This implies Hj (ξ ) < 0. That means Hj (ξ ) is a decreasing function. Since ξ ∈ [ξj , ξj+1 ] its minimum occurs at ξ = ξj+1 . That is Hj (ξ ) > Hj (ξj+1 ) ∀ ξ ∈ [ξj , ξj+1 ]. Now, Hj (ξj+1 ) =  Since cot

ξj+1 −ξj 2



Hj (ξj+1 )

=

dj +dj+1 , fj+1 −fj

−dj + dj+1 C(ξj+1 − ξj ) . S(ξj+1 − ξj )

we have ξ

−ξ

dj+1 − (fj+1 − fj ) cot( j+12 j ) + dj+1 cot(ξj+1 − ξj ) =− S(ξj+1 − ξj ) 



fj+1 − fj ξj+1 − ξj dj+1 − . = cot 2 S(ξj+1 − ξj )

−f

j Thus, if S(ξj+1 < dj+1 , then Hj (ξj+1 ) > 0. Consequently, Hj (ξ ) > 0 ∀ξ ∈ I . j+1 −ξj ) Hence, H (ξ ) is convex in [ξ0 , ξk+1 ]. Similarly, maximum of Hj (ξ ) occurs at ξ = ξj .

f

That is, Hj (ξ ) < Hj (ξj ), for all ξ ∈ [ξj , ξj+1 ]. Note that Hj (ξj ) =   ξ −ξ d +d Since cot j+12 j = fjj+1 −fj+1j , we have  Hj (ξj ) =

cot

ξj+1 −ξj 2



dj+1 −dj C(ξj+1 −ξj ) . S(ξj+1 −ξj )

(fj+1 − fj ) − dj − dj C(ξj+1 − ξj )

S(ξj+1 − ξj )  

fj+1 − fj ξj+1 − ξj − dj . = cot 2 S(ξj+1 − ξj )

Hence

fj+1 −fj S(ξj+1 −ξj )

− dj < 0 implies Hj (ξj ) < 0 and Hj (ξ ) < 0 for all ξ ∈ [ξj , ξj+1 ].

Consequently, H (ξ ) is concave in [ξ0 , ξk+1 ] if

fj+1 −fj S(ξj+1 −ξj )

− dj < 0, j = 0, 1, . . . , k.

Now, we consider the shape of monotonically decreasing data, that is, dj ≤ 0 ∀j = 0, 1, . . . , k + 1. Theorem 3 Suppose, the convex Hermite data D = {(ξj , fj , dj ), j = 0, 1, . . . , k + 1} is convex and monotonically decreasing. Then the Hermite interpolant H (ξ ) preserves the convexity of D. If D is concave, then H (ξ ) preserves its concavity provided D is H -concave. Proof We have seen that Hj (ξ ) is a solution of (D3 + D)g ≡ 0. Hence, Hj (ξ ) = −ξ )

S(ξ −ξ )

j+1 + dj+1 S(ξj+1 −ξj j ) . Hence, Hj (ξ ) = −Hj (ξ ). But from (10) Hj (ξ ) = dj S(ξj+1 −ξj )

S(ξ

114

S. Bebarta and M. K. Jena S(ξ −ξ )

−ξ )

j+1 −dj S(ξj+1 − dj+1 S(ξj+1 −ξj j ) which is always positive. Hence, Hj (ξ ) is an increasing −ξj ) function in [ξj , ξj+1 ]. We get, Hj (ξ ) > Hj (ξj ) for all ξ ∈ [ξj , ξj+1 ]. This implies

S(ξ

Hj (ξ ) > Hj (ξj ) =

dj+1 − dj dj+1 − C(ξj+1 − ξj )dj > > 0. S(ξj+1 − ξj ) S(ξj+1 − ξj )

Therefore, Hj is convex in I = [ξj , ξj+1 ]. To prove concavity note that, we have shown that Hj (ξ ) is an increasing function in [ξj , ξj+1 ]. Hence Hj (ξ ) ≤ Hj (ξj+1 ) for all   ξ −ξ f −fj ) = Hj (ξj+1 ). Since D is concave, ξ ∈ [ξj , ξj+1 ]. But cot j+12 j (dj+1 − S(ξj+1 j+1 −ξj )   ξ −ξ f −fj dj+1 < dj and hence Hj (ξ ) ≤ Hj (ξj+1 ) < cot j+12 j (dj − S(ξj+1 ) < 0 if D is j+1 −ξj ) H -concave. This completes the proof. 

5 Numerical Examples In this section, we consider several examples and justify that the Hermite interpolation that is introduced here is a shape preserving interpolation. Moreover, it preserves an ellipse (Table 1). Example 1 We have the following vector data. The parametrization is found using (5). After this, we get the Hermite data D1 , which is shown in Table 2. The table says that f −fj and first four data are convex and last four data are concave. Let C1 (j) = dj − S(ξj+1 j+1 −ξj ) f

−f

j C2 (j) = dj+1 − S(ξj+1 . Table 3 shows that C2 (j), j = 1, 2, 3 are positive. Hence, the j+1 −ξj ) corresponding data is H -convex. Thus, the Hermite interpolant of D1 in [1, 3.4042] is convex. Similarly, C1 (j), j = 4, 5, 6 is positive. Hence, the corresponding data is H -concave. Thus, the Hermite interpolant of D1 in [3.4042, 5.8084] is concave. This shape preserving property is shown in Fig. 1.

Example 2 In this example, data (functional value and derivative) are sampled from the function f (ξ ) = tan(ξ − 1.5) + 8 using the seven sampling points 0.1, 0.5, 1.0,

Table 1 A vector data without parametrization fj 1 2 3 4 dj

0

π −2

Table 2 Hermite data D1 ξj 1.0000 2.4388 fj dj

1 0

2 π −2

5

6

7

π −1

π

π −1

π −2

0

3.0301

3.4042

3.7783

4.3696

5.8084

3 π −1

4 π

5 π −1

6 π −2

7 0

Shape Preserving Hermite Interpolation Reproducing Ellipse Table 3 H -convexity and H -concavity test j 1 2 3 C1 (j) C2 (j)

−1.0088 0.1328

−0.6523 0.3477

−0.5946 0.4054

115

4

5

6

0.4054 −0.5946

0.3477 −0.6523

0.1328 −1.0088

7 6

y-axis

5 4 3 2 1

1

1.5

2

2.5

3

3.5

x-axis

4

4.5

5

5.5

6

Fig. 1 Hermite interpolant of D1 Table 4 Hermite data D2 ξj 0.1000 0.3220 fj dj

2.2021 34.6155

6.4426 3.4255

0.7437

1.2104

1.6772

2.0989

2.2221

7.4537 1.2984

8.0000 1.0000

8.5463 1.2984

9.5574 3.4255

22.1014 199.8500

1.5, 2.0, 2.5, 3.0. We find parameters ξj ; j = 1, 2, . . . , 7 for this sampled data using (5). The parameterized data is given in Table 4. The Hermite interpolant is shown in Fig. 2. Clearly, it is shape preserving. In the end, we give an important example which says that our Hermite interpolation reproduces a constant and an ellipse. Example 3 (Reproduction of an ellipse) Let us consider the following data: D3 = {(ξj , 1, 0), j = 0, 1, . . . , 10} D4 = {(ξj , S(ξj ), C(ξj )), j = 0, 1, . . . , 10} D5 = {(ξj , C(ξj ), −S(ξj )), j = 0, 1, . . . , 10}. Let H 1, H 2, and H 3 are the Hermite interpolants to D3 , D4 , and D5 , respectively. With the help of Mathematica, it is found that H 1(ξ ) = 1, H 2(ξ ) = S(ξ ) and H 3(ξ ) = C(ξ ). Thus, the interpolant reproduces the space span{1, C(ξ ), S(ξ )}. In particular,

116

S. Bebarta and M. K. Jena 25

y-axis

20 15 10 5 0

0

0.5

1

x-axis

1.5

2

2.5

Fig. 2 Hermite interpolant of D2 Table 5 X-component data set F of ( 3x )2 + ( 2y )2 = 1 ξj

0

1

2

3

4

fj

0

2.5244

2.7279

0.4234

−2.2704 −2.9326 −2.8768 −2.1166 −0.8382 −0.2493

0

dfj

3

1.6209

−1.2484 −2.97

−1.9609 −0.6324 0.8510

Table 6 Y-component data set G of 3

x 2 3

4.5

+

4

y 2 2

5

5.5

6

6.2

2π

2.1260

2.8805

2.9896

3

5.5

6

6.2

2π

1.4173

1.9203

1.9931

2

1.4111

0.5588

0.1662

0

=1

tj

0

1

2

4.5

gj

2

1.0806

−0.8323 −1.9800 −1.3073 −0.4216 0.5673

d gj

0

−1.6829 −1.8186 −0.2822 1.5136

1.9551

5 1.9178

it reproduces an ellipse. For example, consider the following data which is taken from the ellipse ( 3x )2 + ( 2y )2 = 1 (Table 5). From our observation, it is obvious that HF (ξ ) = 3C(ξ ) and HG (ξ ) = 2S(ξ ). Hence,  



HF (ξ ) 2 HG (ξ ) 2 + = 1. 3 2 Therefore, (HF (ξ ), HG (ξ )) is the original ellipse. This is illustrated by Fig. 3.

6 Comparison In this section, we compare quadratic trigonometric spline (QTS) with quadratic polynomial spline (QPS). The comparison is made graphically.

Shape Preserving Hermite Interpolation Reproducing Ellipse

117 Hermite interpolant

Original data 2

2

1 y−axis

y−axis

1 0

0

−1

−1

−2 −3

−2 −3

−2

−1

0 x−axis

1

2

3

−2

−1

2

2

1

1

0

0

−1

−1

−2 −3

−2

−1

0

0 x−axis

1

2

3

1

2

3

Comparison

Original Ellipse

1

2

−2 −3

3

−2

−1

0

Fig. 3 Ellipse

Hermite interpolant by QPS [5]: The Hermite interpolant of the QPS, defined in the interval [ξj , ξj+1 ], j = 0, 1, . . . , n, is derived as follows: Hj (ξ ) = fj + dj (ξ − ξj ) +

(dj+1 − dj )(ξ − ξj )2 2(ξj+1 − ξj )

(13)

Hermite interpolant by QTS: The Hermite interpolant of the QTS, defined in the interval [ξj , ξj+1 ], j = 0, 1, . . . , n is derived as follows: Hj (ξ ) = fj + dj S(ξ − ξj ) + 2 S 2 (

ξ − ξj dj+1 − dj C(ξj+1 − ξj ) ) . 2 S(ξj+1 − ξj )

(14)

The QTS interpolant and QPS interpolant differ from each other slightly. In Fig. 4a, QTS interpolant of data (Table 4) is plotted, while in Fig. 4b QPS interpolant of same fj and dj data but with different parametrization is given. It is clear from the figures, shapes are almost the same. QTS interpolant of data (Table 2) is plotted, while in QPS interpolant of same fj and dj data but with different parametrization is given. The trigonometric Hermite scheme reproduces an ellipse; however, the polynomial Hermite scheme does not reproduce ellipse which is evident from Fig. 6 (Fig. 5; Table 6).

118

S. Bebarta and M. K. Jena b

a 25

20

20

15

15 y−axis

y−axis

25

10

10

5

5

0

0

2.5

2

1.5

1

0.5

0

2.5

2

1.5

1

0.5

0

x−axis

x−axis

Fig. 4 a Plot of QPS; b plot of QTS 7

QPS QTS

6

y−axis

5 4 3 2 1

7

6

5

4 x−axis

3

2

1

Fig. 5 Comparison between QPS interpolant and QTS interpolant Hermite interpolant

Original data 2

2

1 y−axis

y−axis

1 0 −1 −2 −3

0 −1 −2

−2

−1

0 x−axis

1

2

3

−3 −3

−2

1

0

−1

2

3

4

2

3

4

x−axis

Original Ellipse

Comparison Ellipse

2

2 1

1

0 0 −1 −1 −2 −3

−2

−2

−1

0

1

2

3

−3 −3

−2

−1

Fig. 6 Interpolation of ellipse data by QPS. Ellipse is not reproduced

0

1

Shape Preserving Hermite Interpolation Reproducing Ellipse

119

7 Concluding Remark A new Hermite interpolation method has been introduced. The interpolant is a C 1 quadratic trigonometric spline. It preserves both monotonicity and convexity. Moreover, it reproduces ellipse. The advantage of the method is that it requires no new knot insertion, and there are no extra parameters involved. The interpolants are nice looking and pleasant to the eye.

References 1. Phillips, G.M.: Interpolation and Approximation by Polynomials, vol. 14. Springer (2003) 2. Farin, G.: Curves and Surfaces for Computer-Aided Geometric Design: A Practical Guide. Elsevier (2014) 3. Boor, D.: A Practical Guide to Splines (1978) 4. Schoenberg, I.J.: On trigonometric spline interpolation. J. Math. Mech. 795–825 (1964). http:// www.jstor.org/stable/24901234 5. Schumaker, L.I: On shape preserving quadratic spline interpolation. SIAM J. Numer. Anal. 20(4), 854–864 (1983). https://doi.org/10.1137/0720057 6. Conti, C., Romani, L., Unser, M.: Ellipse-preserving Hermite interpolation and subdivision. J. Math. Anal. Appl. 426(1), 211–227 (2015). https://doi.org/10.1016/j.jmaa.2015.01.017 7. Kocic, L.M., Milovanovic, G.V.: Shape preserving approximations by polynomials and splines. Comput. Math. Appl. 33(11), 59–97 (1997). https://doi.org/10.1016/S0898-1221(97)00087-4 8. Schumaker, L.: Spline Functions: Basic Theory. Cambridge University Press, Cambridge (2007) 9. Jena, M.K., Shunmugaraj, P., Das, P.C.: A subdivision algorithm for generalized BernsteinBezier curves. Comput. Aided Geometr. Des. 18(7), 673–698 (2001). https://doi.org/10.1016/ S0167-8396(01)00061-9

Mellin Transform of Bose–Einstein Integral Functions Akbari Jahan

Abstract The Mellin transform of Bose–Einstein integral functions has been discussed in the present work. By the use of Mellin transform, these functions can be expressed in terms of power series, which is well suited for numerical computation in different domains of Physics and Mathematics. The numerical values of the Bose– Einstein integral functions have been computed using Gauss–Laguerre quadrature, and their comparative analyses have also been reported. Keywords Mellin transform · Bose–Einstein integral functions · Gauss–Laguerre quadrature PACS numbers: 02.30.Gp · 02.30.Rz · 02.60.-x

1 Introduction An integral transform is an operation or a transformation that yields a new function from a given function in an integral form but depends on a different variable. In the following expression, v t (y) = f (x)k(y, x)dx (1) u

the function t (y) is called the integral transform of f (x) and k(y, x) is its kernel. Depending upon the kernel k(y, x) and its suitable limits, different types of integral transforms are found. Mellin transform, named after a Finnish mathematician Robert Hjalmar Mellin, is one such integral transform in which the kernel is x y−1 , and the limits of integration are 0 to infinity. In other words, the transformation represented by A. Jahan (B) Department of Physics, North Eastern Regional Institute of Science and Technology, Nirjuli 791109, Arunachal Pradesh, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_11

121

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A. Jahan

∞ M(y) =

f (x) x y−1 dx

(2)

0

is the Mellin transform of a function f (x). Mellin transform is related to Laplace transform and Fourier transform except for the change of variable x to exponential function. Whereas Laplace transform and Fourier transform were associated with solving physical problems, Mellin transform, on the other hand, was introduced in a mathematical context [1]. There are several applications of Mellin transform, where it has been proved to be more convenient and effective as compared to Laplace transform and Fourier transform. The importance of Mellin transform is found in the fact that the magnitudes of Mellin transform of two functions, that are different in scales, are equal.

2 Bose–Einstein Integral Functions Several authors [2–6], over the years, have worked on the analytical study of Bose– Einstein integral functions. Consider the parameters of the Bose–Einstein gas, viz. complex function σ and α, a function of total number of molecules at absolute temperature. The Bose–Einstein integral functions, in terms of Dirichlet series [7] for powers of small values of α, may be expressed as 1 F(σ, α) = (σ )

∞

x σ −1 dx −1

e x+α 0

(3)

where (σ ) is the Gamma function. In the present work, we use Mellin transform method to express the Bose–Einstein integral functions in terms of power series because such series are reasonably suitable for several numerical calculations in different domains of Physics and Mathematics. Thus, the Mellin transform of Bose–Einstein integral functions, F(σ, α) or Fσ (α) [8], may be written as ∞ F(σ, α)α y−1 dα

M(σ, y) = 0

=

∞  ∞ 0

m=1

m −σ e−mα α y−1 dα

(4)

Mellin Transform of Bose–Einstein Integral Functions

123

3 Bose–Einstein Integral Functions as Power Series Functions The relation of Mellin transform of the Bose–Einstein integral functions F(σ, α) and its power series functions as given in Eq. (4) can be proved easily as follows:  ∞ x σ −1 x σ −1 dx of Eq. (3), consider the term x+α . In the integral x+α e −1 e −1 0 We can write x σ −1 x σ −1 e−(x+α) = −1 1 − e−(x+α)

ex+α

= x σ −1 e−(x+α)

∞ 

e−m(x+α)

m=0

=

∞ 

e−m(x+α) x σ −1

(5)

m=1

Therefore,

∞ 0

∞

∞

 x σ −1 dx = x+α e −1 m=1

e−m(x+α) x σ −1 dx

(6)

0

Substituting mx = y in Eq. (6), we have m dx = dy. We then obtain ∞ 0

∞   y σ −1 dy  x σ −1 −y −mα dx = e e ex+α − 1 m m m=1 ∞

0

∞ =

e

−y σ −1

y

dy

∞ 0

m −σ e−mα

m=1

0

But

∞ 

e−y y σ −1 dy = (σ ). Therefore, ∞ 0

∞  x σ −1 dx = (σ ) m −σ e−mα ex+α − 1 m=1

This implies 1 (σ )

∞ 0

∞

 x σ −1 dx = m −σ e−mα ex+α − 1 m=1

(7)

124

A. Jahan

Comparing Eqs. (3) and (7), we find that 1 F(σ, α) = (σ )

∞ 0

∞

 x σ −1 m −σ e−mα dx = x+α e −1 m=1

(8)

Thus, Eq. (8) gives the desired Bose–Einstein integral functions F(σ, α) in terms of power series functions. It may be noted that the functions F(σ, α) are monotonically decreasing functions of α which, for large α, merge into the exponential function. Substituting σ = 1 in Eq. (8), we get F(1, α) = =

∞ 

m −1 e−mα

m=1 ∞ 

e−mα m m=1

Putting e−α = k, we have F(1, α) =

∞  km m m=1

= − ln(1 − k) Thus,

F(1, α) = − ln(1 − e−α )

(9)

It may be noticed that α in Eq. (9) is a positive quantity, and this can be proved in a simple manner as follows: Let loga p = q. Then, p = a q . Similarly, let ln p = q, i.e., loge p = q. Then, p = eq . We know that eq is a positive quantity, i.e., eq > 0. Hence, p must also 1 always be a positive quantity and e−q is also a positive quantity (as e−q = q ). For e any variable α (other than q), we then have the condition e−α < 1 eα > 1 α>0

(10)

Thus, α is always a positive quantity. If it is negative, then the exponential function becomes divergent.

Mellin Transform of Bose–Einstein Integral Functions

125

4 Relation Between the Mellin Transform of Bose–Einstein Integral Functions and the Riemann Zeta Function The Mellin transform of the Bose–Einstein integral functions as given in Eq. (4) is re-written here as M(σ, y) =

∞  ∞ 0

m −σ e−mα α y−1 dα

m=1

∞ ∞ −mα   mα  y−1 e = dα mσ m m=1

(11)

0

Substituting mα = x, we have m dα = dx, i.e., dα = dx/m. Therefore, M(σ, y) =

∞  ∞ 0

e−x mσ m=1



x y−1 m y−1



dx m

   ∞   1 1 1 e−x x y−1 dx = σ y−1 m m m m=1 ∞

(12)

0

Since

∞ 0

e−x x y−1 dx = (y) and simplifying Eq. (12), it can be written as M(σ, y) = (y)

∞  m=1

1 m σ +y

(13)

where the second term on the RHS of Eq. (13) is the Riemann zeta function. Thus, M(σ, y) = (y) ζ (y + σ )

(14)

Equation (14), therefore, gives an important relation between the Mellin transform of the Bose–Einstein integral functions and the Riemann zeta function. While deriving the infinite series of the functions, F1/2 (α), F3/2 (α), F5/2 (α), etc., we often use two special functions, viz. the Gamma function and the Riemann zeta function of both positive and negative rational numbers.

126

A. Jahan

5 Some Numerical Values of F1/2 (α), F3/2 (α) and F5/2 (α) Using Gauss–Laguerre Quadrature Substituting the value of α = 0.5 in the Bose–Einstein integral function F3/2 (α), Eq. (3) can be written as F3/2 (0.5) =

1 (3/2)

∞

x 3/2−1 dx −1

(15)

ex+0.5 0

Considering the Gauss–Laguerre quadrature [8, 10], we must have e−x f (x) =

x 3/2−1 ex+0.5 − 1

f (x) =

x 1/2 − e−x

That is,

(16)

e0.5

Let us take the number of points n = 6. Table 1 lists the values of the variable xi and the corresponding weights wi . The values of f (xi ) [given by Eq. (16)] and the products with their corresponding weights are also listed. From Table 1, we have wi f (xi ) ≈ 0.731992, that gives the integral value of Eq. (15). Substituting this value, we obtain the value of F3/2 (0.5) ≈ 0.826. The correct value ≈0.815 and, thus, the percentage error are ≈1.23%. Consider one more value of n for the same function. Table 2 lists the values of the variable xi and their corresponding weights wi for the node n = 10. Following the same procedure as above, we obtain the value of F3/2 (0.5) ≈ 0.817 and the percentage error is ≈0.245%. We thus observe that the percentage error gets reduced if we go for higher nodes.

Table 1 List of variables and weights for node n = 6 xi wi f (xi ) 0.222846 1.188932 2.992736 5.775143 9.837467 15.982873

(−1) 4.589646 (−1) 4.170008 (−1) 1.133733 (−2) 1.039919 (−4) 2.610172 (−7) 8.985479

0.556364 0.811190 1.008219 1.460334 1.092429 2.424823

wi f (xi ) 0.255351 0.338266 0.122691 0.015186 0.000496567 0.000002179 wi f (xi ) ≈ 0.731992

Mellin Transform of Bose–Einstein Integral Functions

127

Table 2 List of variables and weights for node n = 10 xi wi f (xi ) 0.137793 0.729454 1.808342 3.401433 5.552496 8.330152 11.843785 16.279257 21.996585 29.820697

(−1) 3.084411 (−1) 4.011199 (−1) 2.180682 (−2) 6.208745 (−3) 9.501516 (−4) 7.530083 (−5) 2.825923 (−7) 4.249313 (−9) 1.839564 (−13) 9.911827

wi f (xi )

0.477469 0.732143 0.905677 1.141700 1.432582 1.750825 2.087372 2.447203 2.844660 3.317711

0.147271 0.293677 0.197499 0.070885 0.013611 1.318385 × 10−3 5.898753 × 10−5 1.039893 × 10−6 5.232934 × 10−9 3.288458 × 10−12 wi f (xi ) ≈ 0.724321

Table 3 Comparative analysis of F1/2 (α) α

F1/2 (α) Correct value

F1/2 (α) Observed value

% error

Remarks

0.1 0.5 1.0 2.0

4.161 1.141 0.507 0.150

2.106 0.832 0.390 0.119

−49.39 −27.08 −23.07 −20.67

Very poor Convergence

Table 4 Comparative analysis of F3/2 (α) α

F3/2 (α) Correct value

F3/2 (α) Observed value

% error

Remarks

0.1 0.5 1.0 2.0

1.635 0.815 0.426 0.142

1.678 0.825 0.433 0.143

2.63 1.226 1.643 0.704

Fair Convergence

To sum up, the comparison between the correct values and the observed values of the Bose–Einstein integral functions F1/2 (α), F3/2 (α) and F5/2 (α) is shown in Tables 3, 4 and 5.

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A. Jahan

Table 5 Comparative analysis of F5/2 (α) α

F5/2 (α) Correct value

F5/2 (α) Observed value

% error

Remarks

0.1 0.5 1.0 2.0

1.147 0.688 0.398 0.138

1.149 0.691 0.424 0.139

0.174 0.436 0.532 0.007

Good Convergence

6 Conclusion The Mellin transforms of Bose–Einstein integral functions have been thoroughly studied. The numerical values of the Bose–Einstein integral functions have been obtained using Gauss–Laguerre quadrature, and their comparative study between the correct values and the observed values have also been reported in detail. From the evaluation of the Bose–Einstein integral functions for different values of σ and α, it can be inferred that there is very poor convergence for the function F1/2 (α) (very high percentage error). On the other hand, the function F5/2 (α) shows good convergence for higher nodes. Acknowledgements The author gratefully acknowledges late Prof. P. R. Subramanian for his valuable suggestions.

References 1. Bertrand, J., Bertrand, P., Ovarlez, J.: The Mellin Transform: The Transforms and Applications Handbook, 2nd edn. CRC Press, Boca Raton (2000)  −1

d. Appl. 2. Dingle, R.B.: The Bose-Einstein integrals B p (η) = ( p!)−1 0∞  p e−η − 1 Sci. Res. 6, 240 (1957) 3. Haber, H.E., Weldon, H.A.: On the relativistic Bose–Einstein integrals. J. Math. Phys. 23, 1852 (1982) 4. Babolian, E., Arzhang Hajikandi, A.: Numerical computation of the Riemann zeta function and prime counting function by using Gauss-Hermite and Gauss-Laguerre quadratures. Int. J. Comput. Math. 87, 3420 (2010) 5. London, F.: On the Bose-Einstein condensation. Phys. Rev. 54, 947 (1938) 6. London, F.: The state of liquid helium near absolute zero. J. Phys. Chem. 43, 49 (1939) 7. Kreyszig, E.: Advanced Engineering Mathematics, 10th edn. Wiley, New York (2010) 8. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover, New York (1972) 9. Robinson, J.E.: Note on the Bose–Einstein integral functions. Phys. Rev. 83, 678 (1951) 10. Arfken, G.B., Weber, H.J., Harris, F.E.: Mathematical Methods for Physicists, 7th edn. Academic, New York (2012)

A Note on Solution of Linear Partial Differential Equations with Variable Coefficients Formed by an Algebraic Function Using Sumudu Transform with Sm Operator Sandip M. Sonawane and S. B. Kiwne Abstract A different approach to solve initial value problems form by an algebraic functions (homogeneous and non-homogeneous functions) using Sumudu transform is given in this article. Two-dimensional Sm operator with two variable Sumudu transform is well defined and its existence is proved, and the duality relations with other transforms and Sumudu transform with Sm operator are proved in the given article. Keywords Laplace transform · Sumudu transform · Sm operator · pde with variable coefficients

1 Introduction Laplace transform, Fourier transform are good tools for the solution of differential equations. Applications of Fourier transform, Laplace transform are given by Debnath [1]. Engineering applications with examples are given by Vaishista and Gupta [2], Bali and Goyal [3]. Debnath [4] gave applications of two-dimensional Laplace transform for functional, integral, and partial differential equations. The concept of Sumudu transform was first introduced by Watugula [5, 6] developed by weerakoon [7] and applied by Asiru [8–10] to the discrete dynamic system, integral equation of convolution type. Two-dimensional Sumudu transform developed by Tchuenche and Mbare [11]. Kilicman and Gadain [12, 13] give applications of Sumudu transforms. Kiwne and Sonawane [14] solved partial differential equations using Sumudu transform with Hermite polynomial. The solution of partial differential equations like u

∂ ∂ z [u, w] + w z [u, w] = n z [u, w], ∂u ∂w

(1)

S. M. Sonawane (B) SRES’s Sanjivani College, Kopargaon, Maharashtra, India S. B. Kiwne Department of Mathematics, Deogiri Science College, Aurangabad, Maharashtra, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_12

129

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S. M. Sonawane and S. B. Kiwne

here, z [u , w] is homogeneous function of degree n (for more detail see Bali and Goyal [3]) not solved by any integral transform. Kiwne and Sonawane [15] solved Euler-Cauchy’s differential equations with some cases but not able to solved Eq. (1). Kiwne and Sonawane [16, 17] used Double Kamal and Double Aboodh transform to solve partial differential equations but did not get the solution of Eq. (1). In the present article, author introduced Sm operator and solved linear partial differential equations with variable coefficients using Euler’s theorem of homogeneous functions and initial value problems that contains Eq. (1). The duality relations given by Taha et al. [18] and Sonawane and Kiwne [19] are used to proved theorems in this article. In the first section, definitions and second section theorems with applications are given. Let us start with definition of two-dimensional Sumudu transform, Definition 1 (Sumudu transform [11]) Let z(u,w), be exponentially ordered function then the Sumudu transform of two variables is,

1 S [z(u, w)] = v1 v2

∞ ∞ e 0

  − vu + vw 1

2

z(u, w)dudw = Z m (v1 , v2 )

(2)

0

Definition 2 Let z = z(x1 , x2 , ...xm ), be function having continuous first order partial derivative then the Sm operator is defined as, Sm (z) =

m  i=1

xi

∂z , m>1 ∂ xi

(3)

It can be written in vector notation. ˆ Let n = 3 we get, S3 = r¯ .∇z, here r¯ = x1 iˆ + x2 jˆ + x3 k. Definition 3 (Sumudu transform with S2 operator) Let z(u, w), be exponentially ordered function having continuous first order partial derivative then, 1 Ss [z(u, w)] = v1 v2

∞ ∞ e 0

0

  − vu + vw 1

2

  ∂ ∂ u +w z(u, w)dudw = Z S (v1 , v2 ) ∂u ∂w

(4) We called definition (3) as Ss transform in further calculations of this article. Let us check the existence of definition (3)

Sumudu Transform with Sm Operator

131

2 Theorems on Sumudu Transform with Sm Operator Theorem 1 Let, z(u, w) bean exponentially ordered function such that S[z(u, w)] =  u

+w

Z m (v1 , v2 ) , z(u, w) ≤ Me a1 a2 and z(u, w) have continuous first order partial derivative then Ss transform of z(u, w) exists for all v1 , v2 . Theorem 2 Let, Ss [z 1 (u, w)] = Z 1s (v1 , v2 ) and Ss [z 2 (u, w)] = Z 2s (v1 , v2 ) then Ss [α1 z 1 (u, w) + α2 z 2 (u, w)] = α1 Z 1s (v1 , v2 ) + α2 Z 2s (v1 , v2 ) Proof Let, Ss [α1 z 1 (u, w) + α2 z 2 (u, w)] 1 = v1 v2

∞ ∞ e 0

  − vu + vw 1

0

2

  ∂ ∂ u +w [α1 z 1 (u , w) + α2 z 2 (u , w)] dudw ∂u ∂w

α1 = v1 v2 α2 + v1 v2

∞ ∞ e 0



  − vu + vw



1

2

0

∞ ∞ e 0

  − vu + vw

0

1

2

∂ ∂ u +w ∂u ∂w

∂ ∂ u +w ∂u ∂w

 z 1 (u , w)dudw  z 2 (u , w)dudw

= α1 Z 1s (v1 , v2 ) + α2 Z 2s (v1 , v2 ) Theorem 3 Let z(u, w) is homogeneous function of degree n, and suppose that, SS [z(u , w)] = Z s (v1 , v2 ) S [z(u , w)] = Z m (v1 , v2 ) be the Sumudu transform of z(u , w) L [ z (u , w)] = Z (v1 , v2 ) be the Laplace transform of z(u , w) M [z(u , w)] = Z c (v1 , v2 ) be the Laplace-Carson (Mahgoub) transform of z(u , w) (v) K [z(u , w)] = Z k (v1 , v2 ) be the Kamal transform of z (u , w) [19] (vi) A [z(u , w)] = Z A (v1 , v2 ) be the Aboodh transform of z(u , w) [17], then

(i) (ii) (iii) (iv)

1. Z s (v1 , v2 ) = n Z m (v1 , v2 )  2. Z s (v1 , v2 ) = v1nv2 Z v11 , v12   3. Z s (v1 , v2 ) = n Z c v11 , v12 4. U S (v1 , v2 ) = v1nv2 Z k (v1 , v2 )   5. U S (v1 , v2 ) = v2nv2 Z A v11 , v12 1 2

Proof Let z(u, w) is homogeneous function of degree n, then by using Euler’s theorem one can get u

∂ ∂ z(u, w) + w z(u, w) = n z(u, w), ∂u ∂w

(5)

132

S. M. Sonawane and S. B. Kiwne

Let, Z s (v1 , v2 ) = Ss [z(u, w)] =

1 v1 v2

∞ ∞ e 0

  − vu + vw 1

 u

2

0

 ∂ ∂ +w [z(u, w)] dudw ∂u ∂w

using Eq. (5), one can get Z s (v1 , v2 ) =

n v1 v2

∞ ∞ e 0

  − vu + vw 1

2

z(u, w)dudw

0

= n Z m (v1 , v2 ) On similar steps, one can prove the remaining equalities using Eq. (5), and the duality results given by Taha et al. [18], Sonawane and Kiwne [15, 19]. Theorem 4 Let, SS [z(u, w)] = Z s (v1 , v2 ) and S [z(u, w)] = Z m (v1 , v2 ), 1. if z(u, w) is homogeneous function of degree n, then   Ss e(au+bw) z(u, w) = S (n + au + bw)e(au+bw) z(u, w) 2. If z(u, w) is non-homogeneous algebraic function then

 v1 1 v2 Zs , (1 − av1 )(1 − bv2 ) 1 − av1 1 − bv2  + S (au + bw)e(au+bw) z(u, w)

 Ss e(au+bw) z(u, w) =

Proof 1. Using (5) one will get, SS e(au+bw) z(u, w)

=

1 v1 v2

∞ ∞ e 0

  − vu + vw 1

2

 (au+bw) e nz(u, w) + e(au+bw) (au + bw)z(u, w) dudw

0

 = S (n + au + bw)e(au+bw) z(u, w) 2. If z(u, w) is non-homogeneous algebraic function then by definition,

Sumudu Transform with Sm Operator



Ss e(au+bw) z(u, w) =

=

1 v1 v2

1 v1 v2 ∞∞

133

∞∞ e

  − vu + vw 1

u

2



∂ (au+bw) ∂ +w e z(u, w) dudw ∂u ∂w

0 0     −u v1 −a −w v1 −b

e

1

(u)

2

0 0

 ∂ ∂ + (w) [z(u , w)] dudw ∂u ∂w

    ∞∞ 1 −u v1 −a −w v1 −b 1 2 e [(au + bw)z(u, w)] dudw v1 v2 0 0



v1 v2 1 Zs + S (au + bw)e(au+bw) z(u, w) , = (1 − av1 )(1 − bv2 ) 1 − av1 1 − bv2

+

Theorem 5 Suppose that z(u) and z(w) be functions having continuous first order derivative and (i) S[z(u)] = Z m (v1 ), (ii) S[z(w)] = Z m (v2 ), (iii)S[uz(u)] = Z 1m (v1 ), (iv)S[wz(w)] = Z 1m (v2 ) then 1. Ss [z(u)] = v11 Z 1m (v1 ) − Z m (v1 ) 2. Ss [z(w)] = v12 Z 1m (v2 ) − Z m (v2 ) 3. Ss [z(u)z(w)] =





1 1 v1 Z 1m (v1 ) − Z m (v1 ) Z m (v2 ) + Z m (v1 ) v2 Z 1m (v2 ) − Z m (v2 ) 



∞∞ − u +w ∂ Proof Let, Ss [z(u)] = v11v2 0 0 e v1 v2 u ∂u z(u)dudw   ∞ ∞ 1 1 = 0 − v2 0 ue−(u/v1 ) z(u)du − v1 0 e−(u/v1 ) z(u)du 1

= v11 Z 1m (v1 ) − Z m (v1 ) on similar steps, we get second and third result.

Theorem 6 Suppose that z(u) and z(w) be functions having continuous first order derivative and (i) S[z(u)] = Z m (v1 ), (ii) S[z(w)] = Z m (v2 ), (iii)S[uz(u)] = Z 1m (v1 ), (iv)S[wz(w)] = Z 1m (v2 ) then



Ss [z(au)z(bw)] = av1 Z 1m (av1 ) − Z m (av1 ) Z m (bv2 ) + Z m (av1 ) bv1 Z 1m (bv2 ) − Z m (bv2 ) 1 2

∞∞

  − vu + vw

∂ 1 2 [au ∂ z(au) Proof Let, Ss [z(au)z(bw)] = z(bw) + bw ∂w 0 0 e ∂u z(bw)z(au)]dudw using substitution ax = u and by = v, one can get  ∞ ∞ − avu + bvv 1 1 2 [u ∂ z(u) z(v) + v ∂ z(v) z(u)]dudv Ss [z(au)z(bw)] = abv1 v2 e ∂u ∂v 1 v1 v2

0 0

Using Theorem (5), one get required result.

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Theorem 7 Let, Ss [z(u, w)] = Z s (v1 , v2 ) and S [z(u, w)] = Z m (v1 , v2 ) then a b − v1 + v 1  1 2 S[(u + a1 )z u (u, w)] + Ss [z(u − a1 , w − b1 )H (u − a1 , w − b1 )] = e (w + b1 )S[z w (u, w)] , where H (u − a1 , w − b1 ) = 1, i f u > a1 , w > b1 and 0, i f u < a1 , w < b1 Proof Consider, Ss [z(u − a1 , w − b1 )H (u − a1 , w − b1 )] 1 = v1 v2

∞ ∞ e

  − vu + vw 1

2

a1 b1

  ∂ ∂ z(u − a1 , w − b1 )dudw u +w ∂u ∂w

Using substitution u − a1 = t1 and w − b1 = t2 and separating integrals , one can get 

1 − e = v1 v2 =e

  a b − v1 + v 1 1

2

a1 v1

b

+ v1

 ∞ ∞

e

2

0

0

  t t − v1 + v2 1

2

  ∂ ∂ (t1 + a1 ) z(t1 , t2 )dt1 dt2 + (t2 + b1 ) ∂t1 ∂t2

[S[(u + a1 )z u (u, w)] + (w + b1 )S[z w (u, w)]]

2.1 Examples of Sumudu Transform with Sm Operator  1. Ss [u n + w n ] = n(n + 1) v1n + v2n 2. Ss [u m w n ] = (m + n)(m + 1)(n + 1)v1m v2n

 av1 bv2 1 + 3. Ss e(au+bw) == (1−av1 )(1−bv ) 1−av 1−bv 2 1 2 Similarly one can write,

 av1 bv2 Ss e−(au+bw) = (1+av1−1 + )(1+bv2 ) 1+av1 1+bv2 4. Ss [cosh(au + bw)] = 5. Ss [sinh(au + bw)] = 6. 7.



av1 1−av1 av1 1 2(1−av1 )(1−bv2 ) 1−av1 1 2(1−av1 )(1−bv2 )

+ +

abv1 v2 (1−a 2 v12 )−2a 2 v12 Ss [cos(au + bw)] = (1+a 2 v 2 )2 (1+b2 v 2 ) 1 2 (1−a 2 v12 )−2abv1 v2 Ss [sin(au + bw)] = av1 (1+a 2 v2 )2 (1+b2 v2 ) 1 2



bv2 av1 1 1−bv2 − 2(1+av1 )(1+bv2 ) 1+av1

bv2 av1 1 1−bv2 + 2(1+av1 )(1+bv2 ) 1+av1

− +





v1 v2 (1−b2 v22 )−2bv22 (1+a 2 v12 )(1+b2 v22 )2 (1−b2 v 2 )−2abv v bv2 (1+a 2 v22)(1+b2 v12 )22 1 2

+ +

bv2 1+bv2

bv2 1+bv2

Sumudu Transform with Sm Operator

135

3 Sumudu Transform with Partial Differential Equations Example 1 Consider the equation u

∂z(u, w) ∂z(u, w) +w = 6u 5 w + 4uw 3 ∂u ∂w

 −

+w

u

(6)



By multiplying v11v2 e v1 v2 , both sides of equation (6) , integrate with u and w in the first quadrant of XOY-Plane. 1 v1 v2

∞ ∞ e 0

  − vu + vw 1

2

0

  ∂z(u, w) ∂z(u, w) u +w dudw = 6(6) p 5 q + 4(4) pq 3 ∂u ∂w

Ss [z(u, w)] = 6(6) p 5 q + 4(4) pq 3 Taking Inverse Ss -transform both side, using result 2 from Sect. 2, z(u, w) = u 5 w + uw 3 Example 2 Let us solve the equation u

∂ (uw)1/2 ∂ z(u, w) + w z(u, w) = − ∂u ∂w (u + w)2  −

(7)



u

+w

u

∂z(u , w) π 1 ∂z(u , w) +w dudw = − √ √ ∂u ∂w 2 ( v1 + v2 )2

By multiplying v11v2 e v1 v2 , both sides of Eq. (7), integrate with u and w in the first quadrant of XOY-Plane. 1 v1 v2

∞ ∞ e 0

0

  − vu + vw 1

2

Ss [z(u, w)] = −

z(u, w) = Using Theorem (3)and n = −1.

1 π √ √ 2 ( v1 + v2 )2 (uw)1/2 (u + w)2

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Example 3 Let us solve the equation u2

2 ∂2 ∂2 2 ∂ z(u, w) + 2uw z(u, w) = 2(u 2 + w 2 ) z(u, w) + w ∂u 2 ∂u∂w ∂w 2

(8)

We write Eq. (8) as,

 

∂ ∂ 2 ∂z(u, w) ∂z(u, w) u +w +w = 2(u 2 + w 2 ) (9) z(u , w) − u ∂u ∂w ∂u ∂w  −

u

+w



By multiplying v11v2 e v1 v2 , both sides of Eq. (9), integrate with u and w in the first quadrant of XOY-Plane.

Ss

∂ ∂ u +w ∂u ∂w

 z(u , w) − Ss [z(u, w)] = 4[v12 + v22 ]

Taking Inverse Sumudu transform

 ∂ 2 ∂ u z(u, w) − z(u, w) = [u 2 + w 2 ] +w ∂u ∂w n

(10)

Again repeat the same procedure as above for Eq. (10), one can get Ss [z (u , w)] − S[z(u, w)] = n4 [v12 + v22 ] Ss [z (u , w)] − n1 Ss [z (u , w)] = n4 [v12 + v22 ] z (u , w) = u 2 + w 2 , Using Theorem (3)and n = 2. Example 4 Let us solve the equation containing z(u, w, t) such that it is homogeneous in u, w of degree n,   ∂ ∂ ∂ z(u, w, t) + c u +w z (u, w, t) = 0 ∂t ∂u ∂w

(11)

with initial condition z(u, w,0) = u 2 w 2  −

u

+w

By multiplying v11v2 e v1 v2 , both sides of Eq. (11), integrate with u and w in the first quadrant of XOY-Plane. 1 v1 v2

c + v1 v2

∞ ∞ e 0

0

∞ ∞ e 0

  − vu + vw 1

0

  − vu + vw 1

2



2

∂ z(u, w, t)dudw ∂t

 ∂ ∂ u +w z(u, w, t)dudw = 0 ∂u ∂w

Sumudu Transform with Sm Operator

137

Z m (v1 , v2 , t) + cZ s (v1 , v2 , t) = 0 Z m (v1 , v2 , t) + ncZ m (v1 , v2 , t) = 0 Solving this linear equation, one can get Z m (v1 , v2 , t) = c1 e−nct using initial condition, one will get, Z m (v1 , v2 , t) = 4v12 v22 e−nct z(u, w, t) = (u 2 w 2 )e−nct here, n = 4 so the answer is ,z(u, w, t) = (u 2 w 2 )e−4ct d dt d dt

Example 5 Let us solve the equation containing z(u, w, t) such that it is homogeneous in u, w of degree n,   ∂ ∂ ∂2 z(u, w, t) + w z(u, w, t) =0 z(u, w, t) + c u ∂t 2 ∂u ∂w

(12)

with initial condition z(u, w,0) = φ(u, w), dtd z(u, w, 0) = 0  −

u

+w

By multiplying v11v2 e v1 v2 , both sides of Eq. (12), integrate with u and w in the first quadrant of XOY-Plane. 1 v1 v2

c + v1 v2

∞ ∞ e 0

0

∞ ∞ e 0

  − vu + vw 1

2

0

  − vu + vw 1

2

∂2 z(u, w, t)dudw ∂t 2

  ∂ ∂ u +w z(u, w, t)dudw = 0 ∂u ∂w

d2 Z m (v1 , v2 , t) + cZ s (v1 , v2 , t) = 0 dt 2 d2 Z m (v1 , v2 , t) + ncZ m (v1 , v2 , t) = 0 dt 2 Solving this linear equation, one can get √ √ Z m (v1 , v2 , t) = c1 cos nct + sin nct using initial condition, one can get, c1 = φm (v1 , v2 ) and c2 = 0 √ z(u, w, t) = φ(u, w)cos nct Example 6 Let us solve the equation containing z(u, w, t) such that it is homogeneous in u, w of degree n,   ∂ ∂2 2 2 ∂ 2 ∂ u z(u, w, t) + w z (u, w, t) = c + 2uw ∂t 2 ∂u 2 ∂u∂w ∂w 2

(13)

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with initial condition z(u, w,0) = uw and z(u, w, t) = 0 as t → ∞  −

u

+w

By multiplying v11v2 e v1 v2 , both sides of Eq. (13), integrate with u and w in the first quadrant of XOY-Plane. 1 v1 v2

=

c2 v1 v2

∞ ∞ e 0

0

  − vu + vw 1

2

∂2 z(u, w, t)dudw ∂t 2

 ∞∞  u w   ∂ ∂ − + 2 ∂ z(u, w, t) dudw = 0 z(u, w, t) + w e v1 v2 u 2 z(u, w, t) + 2uw ∂u∂w ∂u 2 ∂w 2 0 0

d2 dt 2

Z m (v1 , v2 , t) + n(n − 1)c2 Z m (v1 , v2 , t) = 0 √ √ Z m (v1 , v2 , t) = c1 ec n(n−1)t + c2 e−c n(n−1)t using initial condition, √ one will get, c1 = 0, c2 = v1 v2 z(u, w, t) = uw e−c√n(n−1)t here n = 2 so the solution is, z(u, w, t) = uw e−c 2t .

4 Conclusion In the present article, the initial-valued partial differential equations due to algebraic functions are solved. As we know that such equations may have more than one solutions, we may not able to find all solutions using the concept introduced in the given paper. Acknowledgements The authors are very much thankful to Sanjivani College of Engineering, Kopargaon, Savitribai Phule Pune University, Pune and Dr. Babasaheb Ambedkar Marathwada University, Aurangabad, for providing all facilities for this work.

References 1. Debnath, L.: Integral Transforms and Their Applications, 9-272, 2nd edn. Taylor and Francis Group, Chapman and Hall/CRC Press, Boca Raton (2015) 2. Vasishtha, A., Gupta, R.: Integral Transforms, pp. 1–173. Krishna Prakashan Mandir, Meerut (1985) 3. Bali, N., Goyal, M.: A Text Book of Engineering Mathematics, pp. 412–414. Laxmi Publications, New Delhi (2007) 4. Debnath, L.: The double laplace transform and their properties with applications to functional, integral and partial differential equations. Int. J. Appl. Comput. Math. 02, 223–241 (2016) 5. Watugala, G.K.: Sumudu transform: a new integral transform to solve differential equations and control engineering problems. Int. J. Math. Educ. Sci. Technol. 24, 35–43 (1993)

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6. Watugala, G.K.: The Sumudu transform for functions of two variables. Math. Engr. Indust. 8, 293–302 (2002) 7. Weerakoon, S.: Application of Sumudu transform to partial differential equations. Int. J. Math. Educ. Sci. Technol. 25, 277–283 (1994) 8. Asiru, M.A.: Sumudu transform and solution of integral equations of convolution type. Int. J. Math. Educ. Sci. Technol. 32, 906–910 (2001) 9. Asiru, M.A.: Further properties of the sumudu transform and its applications. Int. J. Math. Educ. Sci. Technol. 33, 441–449 (2002) 10. Asiru, M.A.: Application of the Sumudu transform to discrete dynamic systems. Int. J. Math. Educ. Sci. Technol. 34, 944–950 (2003) 11. Tchuenche, J., Mbare, N.: An application of the double Sumudu transform. Appl. Math. Sci. 01, 31–39 (2007) 12. Kilicman, A., Gadain, H.E.: An application of double Laplace transform and double Sumudu transform. Lobachevskii J. Math. 30, 214–223 (2009) 13. Kilicman, A., Gadain, H.E.: On the applications of laplace and Sumudu transforms. J. Franklin I. 347, 848–862 (2010) 14. Kiwne, S.B., Sonawane, S.M.: Application of Sumudu transform of two variable functions with hermite polynomial to partial differential equations. Adv. Math. Sci. J. 9, 107–118 (2020) 15. Kiwne, S.B., Sonawane, S.M.: Mahgoub transform fundamental properties and applications. Int. J. Comput. Math. Sci. 7, 500–511 (2018) 16. Kiwne, S.B., Sonawane, S.M.: Double Kamal transform theorems for solution of partial differential equations. In: Proceedings of IMBIC 13th International Conference IMBIC on MSAST, vol. 13, pp. 111–120 (2019) 17. Kiwne, S.B., Sonawane, S.M.: On the applications of Laplace and Aboodh transforms in engineering field. Int. Int. J. Pure Appl. Math. 15, 1079–1101 (2019) 18. Taha, H., Nuruddeen, I., Sedeeg, H.: Duality between Kamal and Mahgoub transform and some famous integral transforms. Br. J. App. Sci. Technol. 20, 1–8 (2017) 19. Sonawane, S.M., Kiwne, S.B.: Double Kamal transform: properties and applications. J. Appl. Sci. Comput. 7, 1727–1739 (2019)

Mathematical Modelling and Simulation in Various Disciplines

A Computational Approach for Disease Diagnosis Using Information Embedded in the Relationships Between MicroRNA and Their Target Messenger RNA Srirupa Dasgupta, Medhashree Ghosh, Abhinandan Khan, Goutam Saha, and Rajat Kumar Pal Abstract Cancer is one of the severe diseases, and it has been biologically observed that miRNAs have an especially critical role as the cause of several types of cancers. It has been observed clinically that microRNAs have a specific set of target messenger RNAs (mRNAs). In this work, we have studied how certain relationships between miRNA and their target mRNA can be responsible for cancers by experimenting with the samples of miRNA and mRNA for lung cancer. We have proposed an approach to model the miRNA–mRNA relationship through graphical representation for lung adenocarcinoma. It has been observed that one miRNA has multiple target mRNAs, and thus, several graphs have been produced because of their relational study. Further, we have used the mean-shift clustering approach to specify a particular similar set of graphs of this relationship. The biological characteristics of these graphs can represent the miRNA–mRNA relationships for lung cancer. This same study can be done for several types of cancers. Keywords mRNA · miRNA · Mean-shift clustering

S. Dasgupta Government College of Engineering and Leather Technology, LB-11, Sector-III, Saltlake, Kolkata 700106, India M. Ghosh Indian Institute of Technology Patna, Patna, Bihar 801106, India A. Khan (B) · R. K. Pal University of Calcutta, Acharya Prafulla Chandra Roy Shiksha Prangan, JD-2, Sector-III, Saltlake, Kolkata 700106, India G. Saha North-Eastern Hill University, Umshing Mawkynroh, Shillong, Meghalaya 793022, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_13

143

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1 Introduction MicroRNAs or miRNAs can be defined as small non-coding types of RNAs [8] that signify an important function in mRNA [9] quality degradation in posttranscriptional expressions of genes. Genes are the blueprints of all the activities of the living systems. Since, one gene controls the regulation of another gene; this post-transcriptional degradation will cause desynchronisation in their genetic activities. This may result in several diseases in higher animal species like human beings. It has been clinically proved that mRNAs carry genetic information for protein building in the human body. Proteins are the basis of life as they execute all the important functions of every cell in a human body. Proteins are produced by utilising the encoded information obtained from the mRNAs. It has been clinically identified that miRNAs have a specific set of target mRNAs. When one miRNA attacks its target mRNA, the mRNA gene expression is deregulated. Hence, the quality of protein is compromised and degraded. Thus, several diseases occur in the human body, and cancer is one among them. The samples of mRNAs are normally present in the cytoplasm of the cell or the DNA. The samples of miRNA can be collected through blood, urine, or saliva of any human body. In present days, cancer is one of the severe diseases. Cancer may be infected in the human body by deregulation of the gene expression during the protein synthesis procedure. Based on the increasing number of cancer patients, our motive of this work is to model the relational association between miRNAs and the corresponding target mRNAs for a specific biochemical condition, through graphical representation. As we know, single miRNA can attack several mRNAs, and it has been clinically observed that for a certain type of cancer, there will be several miRNAs responsible for targeting several mRNAs. Thus, several graphs have been produced as a result. Later on, we have applied the mean-shift clustering approach to finding out the set of similar graphs which can define the relationship of miRNA–mRNA for a specific type of cancer. In our case study, we have used lung cancer [5] data which is globally available in GDC portal [6].

2 Literature Survey One study regarding microRNA and target gene pairs for tumourigenesis and its development were done by observing the correlation of miRNA and mRNA expression profiles. The materials considered for this research purpose were of 11 types of cancers, namely bladder urothelial carcinoma, kidney chromophobe, hepatocellular liver carcinoma, kidney renal cell carcinoma, lung squamous cell cancer, lung adenocarcinoma, kidney renal-papillary cell carcinoma, invasive breast carcinoma, squamous cell carcinoma of the head and neck. For each type of cancers, firstly, the expressed miRNAs and the tumour genes were obtained. Then, we take the help of two scientifically obtained miRNA–mRNA target relational databases, miRTarBase,

A Computational Approach for Disease Diagnosis …

145

which contains information of more than fifty thousand miRNA–mRNA relations, and miRecords, which contains animal miRNA–mRNA information. These acquired critical target analysis of miRNA–mRNA pairs of genes. Both miRNA and mRNA samples with the expression values were collected, and the correlation analysis was performed named Pearson method to obtain the miRNA target genes for normal and tumour samples, respectively. As a result, they found 4743 numbers of critical miRNA target pairs for these above-mentioned types of cancers, and 4572 of them expressed fragile correlation in tumour than in standard. This analysis showed that the correlation between miRNAs and target gene pairs was massively decreased in tumour, and these significant sets they got in this study were concerned in cellular bond, propagation, and relocation [1]. Another study regarding miRNA–mRNA reciprocity which could be related to bone metabolism by experimenting correlations and inter-individual inconsistency in levels of miRNA and interactive relationship between a miRNA and its target mRNA levels in a large faction of human osteoblasts (HOBs) found in the course of orthopaedic surgery. There were 24 miRNAs whose differential expressions were analysed, and among them, we found nine miRNAs exhibiting differential expression among males and females. The following miRNAs, namely hsa-miR-30c2, hsa-miR29b, and hsa-miR-125, were identified as miRNAs, and their corresponding targets were also recognised for this study as significant modulators of bone metabolism. Next, they applied a cumulative survey of overall correlations of miRNA–mRNA interactive analysis, mRNA expression profiling. Differential expression experimentation and studies of bio-informatics to check the target mRNA genes for the miRNAs which have the prospective to control extracellular matrix creation and osteoblast differentiation. Scientific analysis by overexpression and dismantle of miRNAs proved that the miRNAs which are expressed differentially, namely, hsa-miR-30c2, hsa-miR-125b, and hsa-miR-29b, target genes very pertinent to metabolism of bone. These miRNAs orchestrate the actions of key controllers of extracellular matrix proteins and osteoblast differentiation by their concurrent activities on target pairs of gene and ways to regulate the outline of gene counts [2]. A network-based approach was proposed to find out the potential interaction of miRNA–mRNA in case of tumourigenesis to recognise the carcinoma-responsible miRNAs and to guess their target set of genes depending on the count levels of the genes. The technique was implemented with the help of the following data sets— acute myeloid leukaemia (AML), invasive breast carcinoma (BRCA), and kidney renal clear cell carcinoma (KIRC). This technique proposed an approach by applying the lasso algorithm with graphical interface to find the significant microRNAs and the interaction of miRNA and target mRNA in tumourigenesis depending on the gene expression profiles of miRNAs and the corresponding target mRNAs. A graphical network which is bi-partite by nature with the miRNAs as main nodes was generated to investigate the interactivity among the miRNAs and their target mRNAs. The experimental analysis explained that in the top twenty miRNAs organised by their intensities, 90.0% for AML, 70.0% for KIRC, and 70.0% for BRCA miRNAs were discovered to be connected with the cancers. Along with this, the results verified that

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the projected strategy could be useful for forecasting the miRNA–mRNA interactivities in tumourigenesis and recognising the carcinoma-responsible miRNAs as the possible drug analysis [3]. A systematic study was done in miRNA–mRNA expression levels to report limited findings on aged human atria. In this study, for aged human atria, they have observed the miRNA and mRNA interactive relational networks to identify the responsible miRNA genes for aged human atria. The data of twelve patients who have done aortic valve replacement, their miRNA sequence, and RNA sequence were studied for the research purpose. All the patients were classified into four groups depending upon their ages, and all of them were in sinus rhythm (SR). Differential expression (DE) computation was performed to find miRNAs and target mRNA genes responsible for the disease. The interactions of miRNA and mRNA were discovered by using the method of correlation (Pearson) and miRNA target analysis methodologies. As the results, they found seven miRNAs, among which four were upregulated, and three were downregulated. Moreover, 42 genes, among which 23 and 19 were upregulated and downregulated, respectively, had been expressed differentially in tissues of the right artery of the human body. Bio-informatic experiments revealed 114 sets of miRNA–mRNA interactivity on human AA and four different types of correlations. Though, this study discovered miRNA–mRNA interactivity networks for AA, yielding deep perception into the growth of human AA. Further experiments are required to examine the possible worth of these aforementioned miRNA–mRNA interactivities in human AA or any associated cardiovascular disease [4]. In this paper, we have described the literature survey, proposed methodology, along with the experimental results elaborately in the following sections.

3 Proposed Methodology We have collected the zipped folder for both the miRNA and mRNA data sets. We have downloaded the sample sheets for both of the data sets as there will be a common ID for every sample for both mRNA and miRNA data sets. There is a total of 585 cases of mRNA data set and 567 for miRNA data set. We have extracted the overlapping samples by studying the sample sheets for both the data sets (Tables 1, 2, 3 and 4). Table 1 mRNA input sheet sample

mRNA ID

D1

D2

ENSG00000270112.3

0

0

ENSG00000167578.15

4

3

ENSG00000273842.1

0

0

ENSG00000078237.5

5

6

ENSG00000146083.10

14

11

A Computational Approach for Disease Diagnosis … Table 2 Sample of miRNA output after DESeq

Table 3 Sample of miRNA output after DESeq

Table 4 miRNA and their corresponding target mRNA

147

miRNA

p-val

p-adj

hsa-mir-144

1.10E−09

1.43E−06

hsa-mir-4529

3.88E−08

2.52E−05

hsa-mir-143

3.38E−07

0.000146

hsa-mir-210

1.24E−06

0.000403

hsa-mir-577

1.64E−06

0.000424

miRNA

p-val

p-adj

hsa-mir-144

1.10E−09

1.43E−06

hsa-mir-4529

3.88E−08

2.52E−05

hsa-mir-143

3.38E−07

0.000146

hsa-mir-210

1.24E−06

0.000403

hsa-mir-577

1.64E−06

0.000424

miRNA ID

mRNA ID

hsa-miR-144

ENSG00000189056

hsa-miR-4529

ENSG00000063015

hsa-miR-4529

ENSG00000100505

hsa-miR-4529

ENSG00000113368

hsa-miR-210

ENSG00000128266

There will be a total of 532 files for each type of unique data sets. Next, we have mapped the data one to one as in for one ID in the sample sheet, and there will be one miRNA and one mRNA file. We have stored both the data of the particular ID in the same column of the final Excel sheet, which will be generated after the pre-processing of data. Moreover, only read counts for miRNAs and FPKM values for mRNAs have been taken into consideration. These particular values will be in the second columns for each sample. We have extracted those as per our requirements. We will get two Excel sheets as our input files for the next steps. The numbers of miRNAs are 1880, and the numbers of mRNA samples are 60,482. There are two types of samples, diseased and normal. We have marked those according to the sample sheet for each of the data sets. For every sample ID, the type is mentioned in the sample sheet. Either it will be ‘solid tissue normal’ or ‘primary tumour’ as normal and diseased sample correspondingly. We have marked them accordingly for the diseased sample we have labelled ‘d’, and for normal, we have marked ‘n’. A part of the input sheet of mRNA is shown in Table 1. The same process makes the same sheet for miRNA.

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Our next job was to perform the differential expressions for both the data to find out the deregulated mRNAs and miRNAs. Differential expressions can signify the change of reading counts or FPKM values in miRNA and mRNA correspondingly. It can detect which genes are deregulated. It has been clinically seen that for any disease, neither every gene are not affected equally, nor every miRNA are equally responsible. That is why, we have found the significantly deregulated genes and responsible set of miRNAs by calculating the differential expressions. The biocManager package ‘DESeq’ in R-studio is used for this purpose. ‘DESeq’ operation showed us how many miRNAs or mRNAs are deregulated (either upregulated or downregulated). It showed us the p-val for every miRNA and mRNA. We selected the p-val < 0.05 for miRNAs to find out the significant data. In the case of mRNAs, we considered it p-val < 0.01, for the sake of simplicity in miRNA–mRNA target prediction. Because of selecting the particular p-val, we have reduced the number of miRNAs to 30. Next, we have found the targets for these 30 miRNAs. As we already know, the number of mRNAs is 60482, that is a huge data to study at a time, that is why we have reduced the number of mRNAs to 133 by taking the p-val < 0.01 as these numbers are sufficient to make a significant decision about the relationship for any cancer data. For target searching for each miRNA, we need to check the online database MirDB [7], which contains the miRNA names and corresponding target mRNA gene names. As of now, we have mRNA gene IDs as the output of DESeq operation explained in the previous section. For every gene ID (e.g. ENSG00000242268), there is a unique gene name. These names can be found on the online database of mRNA, i.e. BioMart [6]. Now, for graph plotting, we need to check among those deregulated mRNAs which ones are targeted by deregulated miRNAs. We have created an Excel file to store the miRNAs and its’ corresponding targets. The following tables will show the list of some miRNAs with their related targets. As we have mentioned earlier, we have 30 miRNAs on which we are working, and we have reduced the set of mRNAs to 133, so we only get 125 graphs among these. There is a possibility of plotting for all mRNAs in the data set, for future experiments. Table 4 is showing some examples of miRNA and corresponding target mRNAs. For graph plotting purpose, we have used MS Excel. First of all, we need to note all 532 data points of a particular miRNA and copy a target mRNA’s all 532 data points. In our graph, miRNA is placed at X-axis, and mRNA is at Y-axis. We have used the normalised count for the data of miRNAs and mRNAs. Normalised counts were found during the ‘DESeq’ operation. As we know, every miRNA can target one or more mRNAs, so for 30 miRNAs, we have 125 graphs among this set of mRNAs which we have found as the output of DESeq operation. Next, for every graph, we have drawn the trend-line, which can be drawn by using Excel functions. As the graphs are scattered, we have considered polynomial trend-line of degree 2. The following graph will show the example of a miRNA–mRNA scattered plotting and its trend-line along with the equation.

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3.1 Graphical Representation The graph in Fig. 1 is the scattered plotting for miRNA has-mir-143 and its target mRNA ENSG00000198176, where the representation of the X-axis is miRNA, and the representation of the Y-axis is the mRNA data points. There is a total of 532 data points, as we have mentioned earlier as the number of samples is 532. We have secured the trend-line for every graph covering maximum data points. Every trend-line has a unique equation in the form of y = ax 2 + bx + c,

(1)

where a, b, and c are coefficients of the equation. We will store the values of a, b, and c for further phase with respect to the graph names. A table is shown to explain how to store the coefficients. This is just a part of the total set. A total number of graphs for our work are 125 as of now. Further, we can experiment on all the mRNAs to plot graphs. The three columns are the coefficients a, b, and c. Table 5 is showing the coefficient-storing approach. Next, we targeted to find out a similar set of graphs. For this purpose, we have used the mean-shift clustering method. We have used the above coefficient file as input. As the number of clusters was not specified in advance, the results showed us four clusters with 109, 13, 2, and 1 number of elements correspondingly. In the result, the cluster with 109 elements has the maximum number of graphs with similar characteristics, i.e. the contents of that cluster have identical characteristics 2000

Series1

1800

Poly. (Series1)

1600 1400

y = 2E-12x2 - 1E-05x + 28.14

1200 1000 800 600 400 200 0 0

1000000 2000000 3000000 4000000 5000000 6000000

Fig. 1 Example of one graphical representation of one miRNA and its target mRNA

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Table 5 Coefficient stored corresponding to the graph names Row No.

Graph name

a

b

0

hsa-mir-143vsENSG00000101746

2.00E−14

−2.00E−07

c

1

hsa-let-7f-1vsENSG00000072274

3.00E−09

0.00E+00

2

hsa-let-7f-1vsENSG00000101746

5.00E−11

−8.00E−06

0.312

3

hsa-let-7f-1vsENSG00000105511

−1.00E−10

−4.00E−05

9.614

4

hsa-let-7f-1vsENSG00000107295

1.00E−10

−1.00E−05

0.448

5

hsa-let-7f-1vsENSG00000122756

2.00E−10

−1.00E−05

0.421

6

hsa-let-7f-1vsENSG00000133958

7.00E−12

−1.00E−05

0.057

0.235 23.920

based on their trend-line equations. Hence, we have concluded that this particular cluster represents the relational interaction of miRNAs and their corresponding target mRNAs for lung cancer. In this case, the bandwidth has to be given manually by the user; we have used brute-force method to apply this bandwidth for our data. In this project, as far now, we have dealt with the problem of screening and filtering the raw data. Moreover, we have found out the differential expression profiles for each of the data sets. We have plotted the graphs depending on these differentially expressed genes for both miRNAs and mRNAs. As we have discussed earlier, one miRNA can target several mRNAs; as a result, a set of several graphs was produced as our result. Further, we have clustered the graphs with the same characteristics depending on the trend-lines. The particular cluster with maximum elements represents the relationship of miRNA–mRNA interaction for lung adenocarcinoma. Our resultant set of graphs is the following which represents the miRNA–mRNA relationship. Table 6 consists of the names of the graphs with their coefficients. We observed that a single cluster consists of 109 numbers of graphs. This clustering was based on the equation for each graph which we found by the trend-lines. This particular set of 109 graphs is our results which may further be used for biological experiments to find out the medical significance for the cancer research purpose. For accuracy purpose, we have compared the output of the clustering phase with another type of clustering approach, i.e. k-means clustering with four clusters. The results were the same as mean-shift clustering. This was for checking the accuracy of our results. At first, we did not know the number of clusters that is why we applied a density-based clustering approach to perform the analysis. Later on, we got to see the number of clusters was four for our instance. Therefore, we tried to make an output comparison between the two types of clustering approaches. In this relational study of microRNAs and their target messenger RNAs, we tried to find out some similarity among all the responsible miRNAs and the corresponding deregulated mRNAs. In this work, we have dealt with data collection and

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Table 6 Resultant graph names with their corresponding coefficients Graph names

a

b

hsa-mir-143 versus ENSG00000101746

2.00E−14

−2.00E−07

c

hsa-let-7f-1 versus ENSG00000101746

5.00E−11

−8.00E−06

0.312

has-let-7f-1 versus ENSG00000105511

−1.00E−10

−4.00E−05

9.614

hsa-let-7f-1 versus ENSG00000107295

1.00E−10

−1.00E−05

0.448

hsa-let-7f-1 versus ENSG00000122756

2.00E−10

−1.00E−05

0.421

hsa-let-7f-1 versus ENSG00000133958

7.00E−12

−1.00E−05

0.057

hsa-let-7f-1 versus ENSG00000145681

3.00E−11

−7.00E−06

0.425

hsa-let-7f-1 versus ENSG00000153956

2.00E−10

−2.00E−05

1.594

hsa-let-7f-1 versus ENSG00000162624

1.00E−11

−2.00E−06

1.108

hsa-let-7f-1 versus ENSG00000179603

2.00E−11

−2.00E−06

0.048

hsa-let-7f-1 versus ENSG00000197977

−4.00E−11

3.00E−06

0.43

hsa-let-7f-2 versus ENSG00000107295

1.00E−10

−1.00E−05

0.448

hsa-let-7f-2 versus ENSG00000152954

3.00E−12

−6.00E−07

0.034

hsa-let-7f-2 versus ENSG00000153956

2.00E−10

−2.00E−05

1.595

hsa-let-7f-2 versus ENSG00000162624

1.00E−11

−2.00E−06

0.11

hsa-let-7f-2 versus ENSG00000183145

−6.00E−11

−1.00E−06

1.008

hsa-let-7f-2 versus ENSG00000188580

−3.00E−12

−2.00E−06

0.279

hsa-let-7f-2 versus ENSG00000196074

8.00E−11

−2.00E−05

1.527

hsa-let-7f-2 versus ENSG00000197977

−4.00E−11

3.00E−06

0.432

hsa-mir-9-3 versus ENSG00000103460

−1.00E−10

1.00E−05

5.888

hsa-mir-9-3 versus ENSG00000120251

−5.00E−11

1.00E−05

-0.009

hsa-mir-9-3 versus ENSG00000145681

4.00E−11

−4.00E−06

0.305

hsa-mir-9-3 versus ENSG00000174469

−2.00E−10

5.00E−05

0.518

hsa-mir-9-3 versus ENSG00000196074

−2.00E−10

5.00E−05

0.902

hsa-mir-30a versus ENSG00000100505

−3.00E−12

2.00E−07

0.344

hsa-mir-30a versus ENSG00000120068

6.00E−11

−3.00E−05

3.715

hsa-mir-30a versus ENSG00000120251

2.00E−12

−1.00E−06

0.102

hsa-mir-30a versus ENSG00000141431

3.00E−12

−2.00E−06

0.219

hsa-mir-30a versus ENSG00000143195

2.00E−13

−4.00E−07

0.13

hsa-mir-30a versus ENSG00000156687

−5.00E−13

−4.00E−07

0.184

hsa-mir-30a versus ENSG00000162624

3.00E−12

−2.00E−06

0.175

hsa-mir-30a versus ENSG00000163888

1.00E−11

−8.00E−06

0.999

hsa-mir-30a versus ENSG00000175175

6.00E−12

−3.00E−06

0.376

hsa-mir-30a versus ENSG00000197977

3.00E−12

−2.00E−06

0.642

0.235

hsa-mir-31 versus ENSG00000131711

9.00E−08

2.00E−03

4.328

hsa-mir-31 versus ENSG000000167619

1.00E−07

0.00E+00

0.427 (continued)

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Table 6 (continued) Graph names

a

b

hsa-mir-31 versus ENSG000000169427

5.00E−08

0.00E+00

c 0.171

hsa-mir-31 versus ENSG000000181634

1.00E−06

−4.00E−03

3.655

hsa-mir-143 versus ENSG00000100505

3.00E−14

−2.00E−07

0.408

hsa-mir-143 versus ENSG00000169427

2.00E−14

−2.00E−07

0.216

hsa-mir-144 versus ENSG00000189056

−3.00E−10

−3.00E−06

0.381

hsa-mir-147b versus ENSG00000023171

0.00E+00

4.20E−02

1.052

hsa-mir-147b versus ENSG00000063015

3.00E−05

−7.00E−03

0.284

hsa-mir-147b versus ENSG00000100505

1.00E−05

−2.00E−03

0.342

hsa-mir-147b versus ENSG00000101746

8.00E−06

−2.00E−03

0.203

hsa-mir-147b versus ENSG00000120251

4.00E−06

0.00E+00

0.037

hsa-mir-147b versus ENSG00000131711

1.00E−05

−2.00E−03

4.603

hsa-mir-147b versus ENSG000000133958

4.00E−06

−1.00E−03

0.047

hsa-mir-147b versus ENSG000000153531

0.00E+00

−3.00E−02

3.201

hsa-mir-147b versus ENSG000000154928

5.00E−05

−9.00E−03

0.514

hsa-mir-147b versus ENSG000000156687

−2.00E−06

8.00E−03

0.023

hsa-mir-147b versus ENSG000000244122

−3.00E−06

0.00E+00

0.042

hsa-mir-147b versus ENSG0000001881634

0.00E+00

−5.50E−02

3.907

hsa-mir-190a versus ENSG00000113739

0.00E+00

−7.00E−02

4.073

hsa-mir-190a versus ENSG00000131711

−4.00E−05

−2.40E−02

4.932

hsa-mir-190a versus ENSG00000156687

−4.00E−05

2.00E−03

0.12

hsa-mir-190a versus ENSG00000169427

5.00E−05

−6.00E−03

0.227

hsa-mir-195 versus ENSG00000103460

−1.00E−07

−2.00E−03

6.438

hsa-mir-195 versus ENSG00000107295

1.00E−06

0.00E+00

0.397

hsa-mir-195 versus ENSG00000143195

−2.00E−07

0.00E+00

0.062

hsa-mir-195 versus ENSG00000153956

8.00E−09

0.00E+00

1.234

hsa-mir-195 versus ENSG00000156687

9.00E−07

−1.00E−03

0.395

hsa-mir-195 versus ENSG00000175175

5.00E−07

0.00E+00

0.337

hsa-mir-195 versus ENSG00000184601

2.00E−08

−4.00E−05

0.019

hsa-mir-195 versus ENSG00000189056

7.00E−07

−1.00E−03

0.533

hsa-mir-195 versus ENSG00000261115

8.00E−07

−1.00E−03

0.598

hsa-mir-196a-1 versus ENSG00000063015

3.00E−08

0.00E+00

0.245

hsa-mir-196a-1 versus ENSG00000101746

−5.00E−09

2.00E−05

hsa-mir-196a-1 versus ENSG00000139352

−4.00E−06

2.40E−02

hsa-mir-196a-1 versus ENSG00000156687

−1.00E−08

6.00E−05

0.134

hsa-mir-196a-1versus ENSG00000174469

9.00E−09

0.00E+00

0.741

hsa-mir-196a-1 versus ENSG00000180318

−4.00E−08

0.00E+00

0.174 10.93

0.178 (continued)

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Table 6 (continued) Graph names

a

b

hsa-mir-196a-1 versus ENSG00000205097

2.00E−09

−2.00E−05

c 0.016

hsa-mir-206 versus ENSG00000103460

5.00E−05

−6.30E−02

6.017

hsa-mir-206 versus ENSG00000113739

1.00E−05

−1.70E−02

3.378

hsa-mir-206 versus ENSG00000141431

2.00E−06

−2.00E−03

0.121

hsa-mir-206 versus ENSG00000154928

2.00E−06

−1.00E−03

0.43

hsa-mir-206 versus ENSG00000205502

−3.00E−05

3.40E−02

2.904

hsa-mir-206 versus ENSG000001774469

9.00E−06

−8.00E−03

0.712

hsa-mir-6744 versus ENSG00000177181

0.00E+00

−4.30E−02

1.728

hsa-mir-210 versus ENSG00000101746

6.00E−09

−6.00E−05

0.199

hsa-mir-210 versus ENSG00000128266

2.00E−09

−3.00E−05

1.013

hsa-mir-210 versus ENSG00000141431

−1.00E−09

4.00E−05

hsa-mir-210 versus ENSG00000163873

−1.00E−09

3.00E−05

0.049

hsa-mir-210 versus ENSG00000167771

−7.00E−09

0.00E+00

0.904

hsa-mir-210 versus ENSG00000197977

6.00E−09

−6.00E−05

0.464

hsa-mir-210 versus ENSG00000261115

3.00E−10

2.00E−05

0.245

hsa-mir-218-1 versus ENSG00000156687

2.00E−06

−1.00E−03

0.295

hsa-mir-218-1 versus ENSG00000261115

1.00E−06

−1.00E−03

0.479

hsa-mir-218-2 versus ENSG0000153956

−4.00E−06

4.00E−03

0.833

hsa-mir-218-2 versus ENSG0000244122

−2.00E−07

9.00E−05

0.038

hsa-mir-301b versus ENSG00000141431

2.00E−05

1.70E−02

-0.083

hsa-mir-301b versus ENSG00000156687

−2.00E−05

3.00E−03

0.114

hsa-mir-577 versus ENSG00000005108

2.00E−06

0.00E+00

0.701

hsa-mir-577 versus ENSG00000100604

−6.00E−05

1.11E−01

-0.998

hsa-mir-577 versus ENSG00000103460

−3.00E−05

2.00E−02

5.378

hsa-mir-577 versus ENSG00000107295

−3.00E−06

3.00E−03

0.23

hsa-mir-577 versus ENSG00000123080

−3.00E−05

2.20E−02

3.992

hsa-mir-577 versus ENSG00000152954

5.00E−07

1.00E−03

-0.017

hsa-mir-577 versus ENSG00000156687

−3.00E−06

2.00E−03

0.079

hsa-mir-577 versus ENSG00000167619

−4.00E−06

4.00E−03

0.285

hsa-mir-577 versus ENSG00000175175

−4.00E−06

5.00E−03

0.055

hsa-mir-577 versus ENSG00000177181

−1.00E−05

1.10E−02

1.342

hsa-mir-577 versus ENSG00000196074

−6.00E−06

6.00E−03

0.888

hsa-mir-4529 versus ENSG00000156687

0.00E+00

2.10E−02

0.135

hsa-mir-4529 versus ENSG000001134443

1.30E−02

−2.57E−01

3.593

hsa-mir-4529 versus ENSG00000100505

0.00E+00

−4.40E−02

0.349

hsa-mir-4652 versus ENSG00000258986

4.00E−05

−1.00E−03

0.162

hsa-mir-4678 versus ENSG00000154415

−1.40E−02

5.80E−02

-0.01

0.011 (continued)

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Table 6 (continued) Graph names

a

b

hsa-mir-4678 versus ENSG00000189056

3.90E−02

−1.98E−01

c 0.403

pre-processing. We found the inputs for the next phase, which is performing the ‘DESeq’ operation. Next, we calculated normalised count and differential counts of the data to find out which are the significantly deregulated genes. Further, we have found 30 miRNAs and corresponding 125 target mRNAs to plot the relational graphs. In this graphical representation, we have formulated the polynomial equations for each of the graphs. This equation has given us the coefficients for every graph, and depending on those, we have used the mean-shift clustering method to find out a similar group of graphs. We found a cluster of 109 graphs out of 125, so we can conclude that this cluster consists of the graphs which can represent the relational characteristics of miRNA and their target mRNA for lung cancer. This cluster can be further biologically experimented to find out more significance. There may be some scope of improvement in this research work. This same procedure can be followed for more types of cancer data set to find out the common characteristics among various kinds of cancers. An algorithm can also be devised to find out the perfect bandwidth for mean-shift clustering method so that instead of using a brute-force method to assign bandwidth, we can assign it properly. Moreover, we have reduced the size of the data from 60,000 to 133 mRNAs for the sake of clarity in our work. For future research purpose, we can include more mRNA data from the same data set. However, this research will help the modern medical science to pursue cancer treatment and drug design effectively.

References 1. Li, X., Yu, X., He, Y., Meng, Y., Liang, J., Huang, L., Du, H., Wang, X., Liu, W.: Integrated Analysis of MicroRNA (miRNA) and mRNA Profiles Reveals Reduced Correlation between MicroRNA and Target Gene in Cancer. Biomed Research International (2018) 2. Laxman, N., Rubin, C.-J., Mallmin, H., Nilsson, O., Pastinen, T., Grundberg, E., Kindmark, A.: Global miRNA expression and correlation with mRNA levels in primary human bone cells. RNA J. (2015) 3. Xue, J., Xie, F., Xu, J., Liu, Y., Liang, Y., Wen, Z., Li, M.: A new network-based strategy for predicting the potential miRNA–mRNA interactions in tumorigenesis. Int. J. Genomics (2017) 4. Yao, Y., Jiang, C., Wang, F., Yan, H., Long, D., Zhao, J., Wang, J., Zhang, C., Li, Y., Tian, X., Wang, Q.K., Wu, G., Zhang, Z.: Integrative analysis of miRNA and mRNA expression profiles associated with human atrial aging. Front. Psychol. (2019) 5. GDC Homepage. https://portal.gdc.cancer.gov/. Accessed 21 Dec 2019 6. BioMart Homepage. https://asia.ensembl.org/info/data/biomart/index.html. Accessed 22 Feb 2020 7. MirDB Homepage. http://www.mirdb.org/. Accessed 2 Mar 2020 8. Micro RNA Wikipedia. https://en.wikipedia.org/wiki/MicroRNA. Accessed 22 Dec 2019

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9. Messenger RNA Wikipedia. https://en.wikipedia.org/wiki/MessengerRNA. Accessed 22 Dec 2019 10. Investopedia. https://www.investopedia.com/terms/m/mode. Accessed 28 Feb 2020 11. Geeks for Geeks Mean-shift Clustering. https://www.geeksforgeeks.org/ml-mean-shift-cluste ring/. Accessed 16 Mar 2020

Computational Reconstruction of Gene Regulatory Networks Using Half-Systems Incorporating False Positive Reduction Techniques Prianka Dey, Abhinandan Khan, Goutam Saha, and Rajat Kumar Pal

Abstract All biological activities performed by living organisms are controlled by gene-gene interactions of a regulatory nature, and these regulatory relationships are often characterised by a gene regulatory network. Remodelling of such networks from temporal expression profiles of genes is an incredibly intriguing task, and a biologically accurate reconstruction is yet to be achieved. Also, existing computational methods predict a significant number of false positives. Here, we have posed a technique for the reduction of false positives by bringing together two metaheuristic techniques. A hybrid swarm intelligence technique has been introduced here, which is a composite of artificial bee colony optimisation and dragonfly algorithm. We have used half-systems for modelling the network dynamics. Experimentation has been performed using the proposed methodology on the 8-gene SOS DNA Repair network of Escherichia coli and a DREAM3 Challenge network comprising 10 genes. The acquired results show that the posed methodology can reduce the false positives of the predicted network to a considerable extent. Keywords Artificial bee colony · Dragonfly algorithm · False positives · Gene regulatory networks · Half-system · Netwok inference

1 Introduction The functional and structural unit of every living organism is a cell. Every biological process is driven by the interactions between cells. Deoxyribonucleic acid or DNA, a double helix structure, is the controller of all cells, and DNA consists of some P. Dey Future Institute of Engineering and Management, Sonarpur Station Road, Kolkata 700150, India A. Khan (B) · R. K. Pal University of Calcutta, Acharya Prafulla Chandra Roy Shiksha Prangan, JD-2, Sector-III, Saltlake, Kolkata 700106, India G. Saha North-Eastern Hill University, Shillong 793022, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_14

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small utility units called genes. For the proper and efficient functioning of a cell, proteins are the key players. Protein production is entirely reliant on the genes and the interactions between them. Protein production is usually accomplished in two steps: transcription and translation as has been highlighted in Molecular Biology [3] by the Central Dogma. Generally, gene produces a protein, and the same protein, or along with other proteins (complex), is responsible for controlling other genes’ protein production capability. It is clear that, there is a complex interaction amongst the genes, and these interactions can be easily implemented using a simple fully connected graph or through a network. This network is termed as a gene regulatory network (GRN). The amount of protein produced by a gene is the measure of its expression level, and this expression level is the estimation of the regulation of any gene. The regulation can be two types; either activation or inhibition. For the first case, expression of target genes are being driven by a group of genes or by a single gene, while in the latter case, the target gene expression is restrained or inhibited. To encompass the elementary cellular activities of living organism, we need to explain the inter-regulation of the genes. Though the Central Dogma proves the existence of such gene regulatory networks, it fails to provide any information on the nature and variety of such networks. This induces researchers worldwide to develop a tool or methodology for the proper elucidation of the gene expression levels and the relative genetic interactions precisely. However, the biological implementation is inconvenient. Thus, we have proposed a computational method for extracting the underlying dynamics of genetic regulation from temporal expression data. This process of reconstruction of genetic interactions from the time-series datasets is known as reverse engineering, network interface, or network identification. For this reverse engineering endeavour, researchers have used various methodologies, namely, Bayesian networks [13, 21], Boolean networks [5], recurrent neural networks [6, 17, 20], S-systems [18, 19], etc. Here, in this paper, we have implemented a recent strategy, namely, half-system (HS) [8], interpreting the GRNs using time-series expression data. Subsequently, here two nature-inspired metaheuristic algorithms have been used for the purpose of HS model parameter training. One is the well-established artificial bee colony optimisation (ABC) [4], while the other is the proposed hybridisation of ABC and dragonfly algorithm (DA) [12], termed as dragonfly algorithm inspired artificial bee colony (DAABC) optimisation. We have implemented our proposed approach on the 8-gene SOS DNA Repair network of E. coli [15] and a 10-gene network selected from the DREAM3 Challenges [10, 11, 14]. The rest of the paper has been arranged in the following manner. Section 2 introduces the fundamental scientific concepts behind the reconstruction of GRNs, emphasising on half-system, ABC, and DAABC. Section 3 presents a detailed explanation of the proposed methodology. The experimental outcomes have been presented and discussed in Sect. 4. Section 5 concludes the paper.

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2 Preliminaries Researchers have used various models and methods for inferring GRNs from timeseries expression datasets which we have discussed below, in brief, before delving into the details of HS and the optimisation algorithms. Bayesian network (BaN) [13, 21] is a probabilistic graphical model that can combine graph theory and probability. Using a directed acyclic graph (DAG) BaNs can be expressed easily; G(X, E), where xi ∈ X are random variables that represent expression levels of gene, and the edges (E) in the graph imitate the dependency among the genes. DAGs prevent BaNs from modelling GRNs with self-loops. Another approach is Boolean network (BoN) [5], one of the simplest ways to represent the dynamic nature of gene regulatory networks and the regulatory interactions. BoNs are however, incapacitated to approch incomplete gene expression data and require higher computational time. Chai et al. [2] and Kiani et al. [9] provide an overall picture of the various methodologies used for the reconstruction of GRNs. Recurrent neural network (RNN) is another neural network based approach that is quite robust and has been used by many contemporary researchers [6, 17, 20]. In the RNN model, the expression level of the i-th gene, i.e., xi , can be derived for a certain time instant, t + t, with the help of the levels of expression of the remaining genes (in the network) at the previous time instant, t, as follows: t xi (t + t) = · f τi

 N 

 wi,k xk + βi

k=1

where f (x) =

  t · xi (t) , + 1− τi

(1)

1 , 1 + e−x

βi denotes the reaction delay parameter, τi represents a kinetic constant, and N is the total number of genes in the network. The weights wi,k denote the nature and strength of a regulation that gene k exercises on gene i.

2.1 Half-Systems One of the most well-known models used for the reconstruction of GRNs is S-system (SS) [18], as following: xi (t + t) = (αi · t)

N  k=1

[xk (t)]gi,k − (βi · t)

N 

[xk (t)]h i,k + xi (t) ,

(2)

k=1

where N denotes the number of genes in a network and xi denotes the expression level of the i-th gene. The nature and type of regulation of gene i by gene k is

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represented by the exponential parameters, g j,k and h j,k . The first term in Eq. 2 symbolises synthesis the product of gene expression, while the second term denotes its degradation. (αi , βi ) denote non-negative rate constants. SS is a biologically specious model due to its power-law characteristics, but the number of parameters that is necessary is 2N (N + 1), where the number of genes in the network is denoted by N . This is almost double that of the number of parameters required in RNN (Eq. 1). This makes SS a computationally expensive model. Another dilemma is when both g j,k and h j,k are predicted to be of the same sign, which indicates dual regulation of a gene. To solve some of this issues, HS [8] was introduced with the motivation of combining the robustness of RNN with the biological plausibility of SS. The mathematical definition of HS has been reproduced from [8] as follows: xi (t + t) = (αi · t) ·

N 

[xk (t)]gi,k + (1 − μi · t) · xi (t) ,

(3)

k=1

where αi defines the rate constant, μi is a kinetic constant. Therefore, only N (N + 2) parameters are to be trained, instead of the 2N (N + 1) parameters as in SS. Here, gi,k > 0 denotes that gene k activates gene j, and g j,k < 0 specifies that gene k restrains gene j. Thus, the computational time requirements become comparable to RNN. Additionally, the problem of dual regulations is also solved. Moreover, being a power-law formalisation it retains the biological relevance of SS.

2.2 Artificial Bee Colony Optimisation Karaboga [4] developed the artificial bee colony (ABC) optimisation basing it upon the intelligent foraging behaviour of honeybees. Artificial bees fly in a problem space interacting with each other socially to find the food source with the highest amount of nectar (i.e., the minimum fitness). There are three types of bees in a colony: employed, onlooker, and scout bees. The colony is first divided equally into employed bees and onlooker bees. When a better food source cannot be found by an employed bee, i.e., when the bee gets stuck in a local minimum after a number of iterations, it becomes a scout bee and reinitialises its position to begin a new search process. The location of the food sources is randomly selected by the employed bees initially, according to the following:  xi = ximin + rand (0, 1) · ximax − ximin ,

(4)

where ximax and ximin are the lower and upper limits, respectively, of xi (i = 1, 2, 3, . . . , P, where P is the total number of solutions (food sources)). Next, these employed bees reach the selected food source and again search for a neighbouring food source according to the following:

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xinew = xiold + φi · (xi − xk ) ,

(5)

where k is a randomly selected food source, such that k = i, and φi is the random number within [1, −1]. The employed bees then evaluate the fitness of these food sources and share the information with the onlooker bees. An onlooker bee then selects a food source on the basis of a probability that, in turn, depends on the fitness of the food source. The mathematical formulation of this selection technique has been presented below: fiti , (6) pi =  P fiti i=1

where fiti is the fitness of solution i, and pi is the probability factor based on which onlooker bees choose the position of the next food source. The onlooker bees further perform local search around the selected food source according to Eq. 5. If it is identified that a discovered food source cannot be improved further, i.e., it gets stuck in a local minimum, the particular food source is discarded. The corresponding bee becomes a scout bee and it reinitialises a food source according Eq. 4. However, by the above-mentioned process, the biological relevance of the reconstructed GRN has not been achieved completely till now. The main problem of this reconstruction process is that the number of false positives (F P) predicted is more compared to the true positives (T P). This motivates us to propose a new hybrid technique, DAABC, which we have explained in the next section.

2.3 Dragonfly Inspired Artificial Bee Colony Optimisation Dragonfly algorithm (DA) [12] is a recent algorithm that is robust in terms of both exploration and exploitation. Dragonflies show two types of foraging behaviours: (i) static or steady action that leads to exploration and (ii) dynamic or vigorous action that helps in exploitation [12]. In contrast, ABC is robust in terms of its exploration capability only and lacks in exploitation. In this work, we have attempted to incorporate the robustness of DA into ABC to overcome the limitations of ABC regarding exploitation. The separation (S), cohesion (C), and alignment (A) parameters of DA have been considered here. To make the search process of employed bees more robust we have proposed Eq. 7: xinew = rand (0, 1) · xiold + s · Si + a · Ai + c · Ci ,

(7)

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where Si = −

P  

xiold − x old , j

j=1

Ai = − Ci =

P Si 1   old xi − x old = , and j P j=1 P

P 1  old x − xiold P j=1 j

Next, for the identification of onlooker bee position, to make exploitation more efficient, we have considered the best and worst fitness values as the food source position (F) and enemy position (E), respectively, of DA. The criteria for updating the onlooker bee is based on the probability value of the information gained. Here, this condition has been replaced by the best and worst positions of the fitness value. Hence, the search becomes equally concentrated towards the apparent best and worst solution, thus reducing the chances of being in local minima altogether and leading to efficient exploitation. The onlooker bee positions have been calculated according to the following proposed equation: xinew = rand (0, 1) · xiold + e · E i + f · Fi

(8)

In Eqs. 7 and 8, {a, c, e, f, s} are preset constants = 0.5.

3 Proposed Methodology In this part, we have discussed our proposed methodology to infer GRNs from temporal expression data. According to researchers [1], on an average, in a network any gene can be regulated by four to eight genes at most. Here, we have considered two networks, one is a 8-gene (in vivo) network and the other is a 10-gene network (in silico). Therefore, the maximum number of possible regulators in a genetic network has been assumed to be four  in this work. Based on this assumption, the search space for a GRN is reduced to N4 from 2 N , where N signifies the number of genes in a GRN. Furthermore, as all possible combinations of regulator are being considered, the biological validity of the resulting GRN is preserved. Next, let us discuss about the parameters that need to be learned for the HS model. If there are N genes, N (N + 2) parameters are to be trained, i.e. αi , gi j , and μi in Eq. 3. However, for large n, this becomes a significantly, computationally expensive problem. Thus, we have considered a decoupled approach [6, 8], where we have considered each gene separately. Therefore, we get N sub-problems, each having a dimension of (N + 2). Equation 9 defines the fitness function used in this work:

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2 1 

x˜i (t) − xi (t) , T t=1 T

MSE =

(9)

here x˜i (t) represents the predicted gene expression value at time-point t, while xi (t) denotes the original value of expression of gene i. Here, we have depicted GRNs as a directed graph G(E, V ). E signifies the edges (regulatory interactions amongst genes) and V defines the vertices (genes), of the graph. For computations, we have represented the graph as: G = [gi,k ] N ×N , N denoting the number of genes in the corresponding genetic network. The parameter gi,k can either have a value of 1 or 0, signifying the presence or absence of edges directed towards node i from node k, respectively. We have applied ABC and DAABC to optimise the parameters as claimed by the objective function given by Eq. 9. Combining the results obtained using both ABC and ABC-DA leads to the pruning in the number of F Ps in the final predicted GRN. This false positive reduction technique has been motivated by the authors [7]. In this approach, we have generated two final networks, one using ABC and the other using DAABC. Subsequently, we have combined both these networks, and generated the final structure using the edges common to both the networks only. Next, we have executed a selection scheme based on validity score, psi, j , specified to all edges for the reconstructed networks as, psi, j =

M 1  wi, j M 1

(10)

where wi j ∈ W , obtained in each simulation as the output; the total number of simulations being M, and psi, j ∈ [0, 1]. After the evaluation of psi, j , ∀i, j, we have constructed the resultant network according to the following condition: 1, if psi j ≥ μ, gi j = (11) 0, otherwise. Here, μ is the threshold corresponding to psi j , which governs if a specific edge is involved in the final reconstructed network or not. For the evaluation of the approach proposed in this work, the obtained GRN G ob has been compared with the original one, G or . And, an obtained edge have been categorised as either a true positive (T P), or a false positive (F P), or a true negative (T N ), or a false negative (F N ), as follows: (i) T P: giorj = gi j = 1, (ii) F P: giorj = 0 & gi j = 1, (iii) T N : giorj = gi j = 0, and (iv) F N : giorj = 1 & gi j = 0. We have used the following statistical metrics for the evaluation of the obtained results. • Sensitivity (Sn ): It defines the fraction of correctly predicted edges that is present in the original  network. • Specificity S p : It defines the fraction of the non-existent edges present in the original network that has been correctly predicted. • False Positive Rate (FPR): It denotes the number of non-existent edges which are not present in the original network but wrongly present in the predicted network.

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• Positive Predictive Value (PPV ): It denotes the number of correct edges in the original network which are interacting. • Accuracy (ACC): The fraction of of all interactions in the original network that have been truly predicted is given by this term. • F-score (F1 ): It is the harmonic mean of (Sn ) and (PPV ). The mathematical representations of the above parameters are as below: Sn = Sp = FPR = P PV = ACC = F1 =

TP T P + FN TN FP + T N FP = 1 − Sp FP + T N TP T P + FP TP +TN T P + FP + T N + FN 2T P 2T P + F P + F N

(12) (13) (14) (15) (16) (17)

4 Experimental Results and Discussions Here, we have first employed HS, ABC, and the proposed DAABC on a 8-gene network from experiment (in vivo) datasets. The network is the familiar E. coli SOS DNA Repair network [15]. Figure 1 shows the original network. This network consists of 8 genes: r uv A, uvr D, uvr A, lex A, uvr Y , r ec A, pol B, and umu DC, respectively. There are nine regulatory relationships amongst these eight genes. Table 1 presents the corresponding results. It is clearly visible from Table 1 that the proposed methodrecA

lexA

uvrD

uvrA

uvrY

umuDC

ruvA

polB

Fig. 1 The original structure of the E. coli SOS DNA Repair network [15]. The arrowheads represent activation, and the open circles denote inhibition

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Table 1 Experimental results for the 8-gene E. coli SOS DNA Repair network [15]. Method TP FP Sn FPR Sp P PV ACC Dataset 01 ABC (1) 8 21 DAABC (2) 5 12 (1) + (2) 4 6 Dataset 02 ABC (1) 7 21 DAABC (2) 4 12 4 6 (1) + (2) Dataset 03 ABC (1) 8 22 DAABC (2) 5 11 4 3 (1) + (2) Dataset 04 ABC (1) 3 24 14 DAABC (2) 6 (1) + (2) 2 12

G1

F1

0.889 0.556 0.444

0.382 0.218 0.109

0.618 0.782 0.891

0.276 0.294 0.400

0.656 0.750 0.828

0.421 0.385 0.421

0.778 0.444 0.444

0.382 0.218 0.109

0.618 0.782 0.891

0.250 0.250 0.400

0.641 0.734 0.828

0.378 0.320 0.421

0.889 0.556 0.444

0.400 0.200 0.055

0.600 0.800 0.945

0.267 0.313 0.571

0.641 0.766 0.875

0.410 0.400 0.500

0.333 0.667 0.222

0.436 0.255 0.218

0.564 0.745 0.782

0.111 0.300 0.143

0.531 0.734 0.703

0.167 0.414 0.174

G6

G10

G4

G3

G7

G9

G5

G8

G2

Fig. 2 The original network structure of the 10-gene DREAM3 [10, 11, 14] Challenge network. The arrowheads represent activation, and the open circles denote inhibition

ology has succeeded in reducing the number of F Ps significant. This is indicated by the markedly increased specificity S p and accuracy (ACC) values in almost all the cases. Next, we have applied the same procedure to a 10-gene DREAM3 [10, 11, 14] in silico Challenge network. In this case also, four datasets have been generated (using GNW [16]) to maintain consistency with the previous 8-gene case. This network has 11 interactions. Figure 2 shows the original structure of the network and Table 2

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Table 2 Experimental results for the 10-gene DREAM3 Challenge network [10, 11, 14]. Method TP FP Sn FPR Sp P PV ACC F1 Dataset 01 ABC (1) 6 32 17 DAABC (2) 5 (1) + (2) 4 11 Dataset 02 ABC (1) 6 31 DAABC (2) 3 13 2 3 (1) + (2) Dataset 03 ABC (1) 5 31 DAABC (2) 3 12 1 4 (1) + (2) Dataset 04 ABC (1) 4 33 6 DAABC (2) 2 (1) + (2) 2 2

0.545 0.455 0.364

0.360 0.191 0.124

0.640 0.809 0.876

0.158 0.227 0.267

0.630 0.770 0.820

0.245 0.303 0.308

0.545 0.273 0.182

0.348 0.146 0.034

0.652 0.854 0.966

0.162 0.188 0.400

0.640 0.790 0.880

0.250 0.222 0.250

0.455 0.273 0.091

0.348 0.135 0.045

0.652 0.865 0.955

0.139 0.200 0.200

0.630 0.800 0.860

0.213 0.231 0.125

0.364 0.182 0.182

0.371 0.067 0.022

0.629 0.933 0.978

0.108 0.250 0.500

0.600 0.850 0.890

0.167 0.211 0.267

presents the corresponding results. The GRN reconstruction formalism proposed in this work has succeeded in reducing the number of predicted F Ps to a significant extent in this case also, which can be observed clearly from the improved S p and ACC scores.

5 Conclusion Here, in this work, we have investigated the reconstruction of GRNs from time-series gene expression data. The same has been accomplished using half-system (to model the underlying dynamics present in the expression data) and ABC and the proposed DAABC (to train the half-system model parameters). The obtained results indicate that the specificity (S p ), accuracy (ACC), and F1 values of the inferred networks improve, if we combine the results obtained by the two metaheuristic techniques. We have implemented our proposed methodology on a in vivo benchmark, i.e. the 8-gene E. coli SOS DNA Repair network [15], as well as a DREAM3 [10, 11, 14] Challenge network comprising 10 genes. The technique proposed in this work has utilised an exhaustive searching strategy to extract the biologically probable GRN architectures. The search space has been reduced based on available biological information. The initial results of the proposed methodology are promising. The implementation of the same for larger networks needs to be investigated further. Additional biological information may also be provided to prevent the loss of obtained T Ps, as a result of the combination of the different structures obtained from various swarm intelligence techniques.

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References 1. Bolouri, H., Davidson, E.H.: Modeling transcriptional regulatory networks. BioEssays 24(12), 1118–1129 (2002) 2. Chai, L.E., Loh, S.K., Low, S.T., Mohamad, M.S., Deris, S., Zakaria, Z.: A review on the computational approaches for gene regulatory network construction. Comput. Biol. Med. 48, 55–65 (2014) 3. Crick, F.: Central dogma of molecular biology. Nature 227(5258), 561–563 (1970) 4. Karaboga, D.: An idea based on honey bee swarm for numerical optimization. Technical report, Technical Report-TR06, Erciyes University, Engineering Faculty, Computer Engineering Department (2005) 5. Kauffman, S.A.: Metabolic stability and epigenesis in randomly constructed genetic nets. J. Theoret. Biol. 22(3), 437–467 (1969) 6. Kentzoglanakis, K., Poole, M.: A swarm intelligence framework for reconstructing gene networks: searching for biologically plausible architectures. IEEE/ACM Trans. Comput. Biol. Bioinform. 9(2), 358–371 (2011) 7. Khan, A., Saha, G., Pal, R.K.: An approach for reduction of false predictions in reverse engineering of gene regulatory networks. J. Theoret. Biol. 445, 9–30 (2018) 8. Khan, A., Saha, G., Pal, R.K.: Modified half-system based method for reverse engineering of gene regulatory networks. IEEE/ACM Trans. Comput. Biol. Bioinform. 17(4), 1303–1316 (2020) 9. Kiani, N.A., Zenil, H., Olczak, J., Tegnér, J.: Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars Cell Dev. Biol. 51(Suppl. C), 44–52 (2016). http://www.sciencedirect.com/science/article/pii/ S108495211630012X 10. Marbach, D., Prill, R.J., Schaffter, T., Mattiussi, C., Floreano, D., Stolovitzky, G.: Revealing strengths and weaknesses of methods for gene network inference. Proc. Natl. Acad. Sci. 107(14), 6286–6291 (2010) 11. Marbach, D., Schaffter, T., Mattiussi, C., Floreano, D.: Generating realistic in silico gene networks for performance assessment of reverse engineering methods. J. Comput. Biol. 16(2), 229–239 (2009) 12. Mirjalili, S.: Dragonfly algorithm: a new meta-heuristic optimization technique for solving single-objective, discrete, and multi-objective problems. Neural Comput. Appl. 27(4), 1053– 1073 (2016) 13. Perrin, B.E., Ralaivola, L., Mazurie, A., Bottani, S., Mallet, J., d’Alche Buc, F.: Gene networks inference using dynamic Bayesian networks. Bioinformatics 19(suppl_2), ii138–ii148 (2003) 14. Prill, R.J., et al.: Towards a rigorous assessment of systems biology models: the DREAM3 challenges. PLOS One 5(2), e9202 1–18 (2010). https://doi.org/10.1371/journal.pone.0009202 15. Ronen, M., Rosenberg, R., Shraiman, B.I., Alon, U.: Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proc. Natl. Acad. Sci. 99(16), 10555–10560 (2002) 16. Schaffter, T., Marbach, D., Floreano, D.: Genenetweaver: in silico benchmark generation and performance profiling of network inference methods. Bioinformatics 27(16), 2263–2270 (2011) 17. Vohradsky, J.: Neural model of the genetic network. J. Biol. Chem. 276(39), 36168–36173 (2001) 18. Voit, E.O.: Computational Analysis of Biochemical Systems: A Practical Guide for Biochemists and Molecular Biologists. Cambridge University Press (2000) 19. Wang, H., Qian, L., Dougherty, E.: Inference of gene regulatory networks using s-system: a unified approach. IET Syst. Biol. 4(2), 145–156 (2010) 20. Xu, R., Wunsch II, D., Frank, R.: Inference of genetic regulatory networks with recurrent neural network models using particle swarm optimization. IEEE/ACM Trans. Comput. Biol. Bioinform. 4(4), 681–692 (2007) 21. Zou, M., Conzen, S.D.: A new dynamic Bayesian network (DBN) approach for identifying gene regulatory networks from time course microarray data. Bioinformatics 21(1), 71–79 (2005)

A Study with Magnetic Field on Stenosed Artery of Blood Flow Sarfraz Ahmed and Biju Kumar Dutta

Abstract The present study and its mathematical modelling were done to decide the effect of the magnetic field on blood moving through a pivotally asymmetric but radially symmetric atherosclerotic conduit. Herschel–Bulkley fluid model condition has been used to non-Newtonian character of blood flow in the presence of applied magnetic effects. The mathematical model is exposed to graphical and numerical analysis. It was revealed that within the sight of applied magnetic field, blood did not definitely change the stream designs, yet caused an apparent decline in the shear stresses and a marginally lower protection from stream. This hypothetical demonstration to cardiovascular infections is considered in our study. Keywords Herschel–Bulkley fluid flow · Stenosis · Non-dimensional flow resistance

1 Introduction Stenosis is responsible for many cardiovascular diseases. A group of scientist examined the blood vessels in a stenosed vein and mechanical conduct of the veins. It is now well-known that the localized in a blood vessel at one or more site of cardiovascular system is generally referred to a stenosis. Atherosclerosis stenosis is a kind of vascular diseases which is of frequent occurrence, particularly in mammalian arteries [1–4]. The coagulations can shape as emboli and block the littler vessels of the stenosis veins to the cardiovascular framework considered [5–8]. A significant decent number of scientist logically thinks that relating to blood course through stenosis supply routes have been completed [9, 10]. Herein, the use of magneto-hydrodynamics is used in physiological problem to build up a mathematical model for the assessment of the circulation system inside to examine the appealing impact through the stenosis vein fragments. Herschel–Bulkley condition is used to the non-Newtonian character of

S. Ahmed (B) · B. K. Dutta Department of Mathematics, The Assam Kaziranga University, Jorhat, Assam, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_15

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circulatory framework inside seeing applied magnetic effects. Nowadays, the cardiovascular afflictions are the world top explanation behind death, causing 33% of all passing all around [11–15]. The application of MHD standard in prescription and engineering is creating interest nowadays for authority [16–19]. A group of the researcher analysed the circulation system in a stenosed vein and mechanical lead of the veins, and the major explanation is the advancement of injuries called arthrosclerosis in the gigantic medium evaluated passageways in the human blood course. The rheology of blood of human being in the vascular homeostatic biomechanical segment that constitutes the cardiovascular system was examined [20–22]. A huge fair number of analytic assessments identifying with blood course through stenosis conductors have been done [23] to examine the effect of vein fixing on the stream typical for blood. This is to some extent influenced by various examinations for uncertain stream between equivalent plates [24–26]. In their paper [27–32], the authors and makers consider blood to be a Newtonian fluid and developed a mathematical model for looking into the lead of viscous-adaptable property of circulatory system in a stenosis game plan obviously. Considering present assessment, blood is considered to be non-Newtonian fluid and cast a numerical model in accordance to Herschel–Bulkley condition under the effect of the transverse alluring field [33–39]. The computational result for the assortment of skin grinding with centre point detachment in the area of the stenosis under the consequences of magnetic field is accessible in the graphic mode. The abstract and sums change in skin scouring of the stream restriction and the volumetric flow rate at different period of the nature of the stenosis inside to see transverse magnetic field have also been represented [40, 41]. In the recent paper [23], the author studied variation of skin friction along the vessel length for different values of n and τ H . In present, we study the variation of flow resistance, i.e. yield stress with sized of stenosis at 60 and 70%.

2 Mathematical Formulation The laminar flow of an incompressible fluid, blood in a steady manner is satisfied by Hershel–Bulkley equation through a stenosis artery in the presence of an applied magnetic field. The governing equation of the blood flow is taken [13]. Displayed equations are centred and set on a separate line. −

dp 1 d(r τ ) B2 = + σ u. dz r dr ρ

(1)

Here, τ display the shear stress and p the pressure of the blood at any point (Fig. 1). The formulation may be put as −

du = f (t) = f (x) = dz

1

− τ H ), τ ≥ τ H k (τ 0, τ ≤ τ H

(2)

A Study with Magnetic Field on Stenosed Artery …

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Fig. 1 Stenosis in arterial segment

In which, τ H is the yield shear stress; u be the axial velocity; k is a parameters which represent non-Newtonian effects. The above Eqs. (1) and (2) can be solved by using following boundary conditions: u = 0 at r = R(z) no slip condition. u = u s at r = R(z) where u s be the axial slip velocity. τ is finite at r = 0 regularity condition. Integrating above equation using above boundary condition, we get τ =−

B 2r σ u r dp − . 2 dz 2

(3)

B 2r σ u R dp − . 2 dz 2

(4)

The skin friction τ R is given by τR = −

Now, integrating (2) and using Eq. (4), then velocity function becomes u = us +

 n+1  r R(z) τR − τH (τR − τH )n+1 − k(n + 1)τR R

(5)

The mathematical formulation of the stenosed artery is written as [38]    δ m 2 ε2 z 2 R(z) = R0 1 − exp − R0 R02 Here, R0 R(z) δ

radius of normal artery. stenosed artery. maximum height of the stenosis.

(6)

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parametric constant ε and relative length (ε) of the stenosis and is defined as ε = R0 /L 0 . From (6), we obtained R(z) 2 = 1 − ae−bz , R0

(7)

where = δ/R0 , b = m 2 ε2 /R0 , and a, b are the parameter in the non-dimensional form.

3 Solution of the Problem The volumetric flow of rate Q can be calculated, and we get    2 τH τH (n+1) 1+ 1− k(n + 3) τR n + 1 τR  2 τH 2 + . (n + 1)(n + 3) τR

Q = π R(z)2 u S +

When

τH τR

n R(z)3 π τ R(z)

(8)

 1, Eq. (8) reduces to the form Q=

  n  n+3 π R3 τR − τH . k(n + 3) n+2

(9)

Using Eq. (3) in Eq. (5) yields dp = − dz



Qk(n + 2)2n π R n+3

 n1

 +

n+3 n+2

2

τR + B 2 σ u. R

(10)

Utilizing the condition that p = P1 at z = −L and p = P2 at z = L, integrating Eq. (10), we get  P1 − P2 =

Qk(n + 3)2 π R n+3 

+

n+3 n+2

2

n n

1

⎡

⎤

dz ⎥ ⎢ ⎣ L − L 0 + ∫   3 +1 ⎦ n L0

−L

R R0

 L 0 dz L0 τH L − L 0 + ∫ R + B 2 σ u ∫ dz. R0 −L R −L 0

(11)

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  2 , and the flow resistances λN in Flow resistances is defined by λ = P1 −P Q absence of constrictions is ⎤ ⎡ 1  1−n  n 2 4(n + 3) ⎣ k Q(n + 3) τH B σ u R0 ⎦ . (12) λN = + + Q R0 n + 2 2(n + 3) π R02 The flow resistances in dimensionless form is given by  1 L0 + λ=1− L L

(k Q(n+3)1−n π R03

 n1

τH σ u R0 I 3 I1 + n+2 I2 + B2(n+3) .  1 (k Q(n+3)1−n n 2τH B 2 σ u R0 + n+2 + 2(n+3) π R3 2

(13)

0

where L0

I1 = ∫  0

L0 L0 dz dz 1+ n3 , I2 = ∫ 1 − ae−bz 2  , I3 = ∫ dz. 0 0 R

(14)

R0

Substituting the expression for

R , R0

then integral I1 and I2 reduces

L0 1 I1 =   n3 +1 +   n3 +1 √ √ −bL 0 (4+2 3) −bL 0 (4−2 3) 12 12 1 − ae 2 1 − ae L0 1 + . I2 =  √ √ −bL 0 (4+2 3) −bL 0 (4−2 3) 12 12 1 − ae 2 1 − ae

(15)

(16)

From Eqs. (4) and (10), we obtained skin friction as 1

τR = −

B 2 σ u R0 Qk(n + 3)2n n R dp − + 2 dz 2 π R3



 n+3 τH . n+2

(17)

The skin friction in the absences of any constriction (R0 = R(z)) 1

Qk(n + 3)2n n τN = + π R03



 n+3 τH . n+2

A non-dimensional articulation for the skin grating might be placed as

(18)

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  1  n+3  R  n1 1 [Qk(n + 3)] n + τH π R03 n + n+2 R0 τR τ= =  3 . 1  3  n  n+3  τN 1 R n n [Qk(n + 3)] + τH π R0 R0 n+2

(19)

4 Result and Discussion In this paper, it is attempted to formulate a mathematical analysis for the blood flow in an arterial segment by using proper boundary condition to the rheological parameter like radius of stenosis, skin friction, yield stress, and flow resistance. The result is forwarded graphically for different values of parameters. It is likewise ensuring that the volumetric flow rate of blood is least at the throat and maximum at beginning of the stenosis. Accordingly, this model obviously settled the way that a slip at a supply route divider will quicken the stream and retard the protection from stream. An expansion in size of stenosis allows protection from blood course through supply routes in various imperative organs of the blood which may cause different cardiovascular ailments. The calculated analytical outcome for skin contact is got from present investigation. Figures 2 and 3 explain how the shapes of the stenosis vary with b and different lengths (L 0 = 0.5, 1, 1.5, 2) and a = 0.1, m = 1. Noteworthy outcomes are seen in the variation of stenosis by changing the values of b (m = 1, 2, 3, 4, 5, 6) and a = 0.1, L 0 = 2. It is perceived that the extensiveness of the stenosis decreases with increment in the estimation of m. It shows for various values of n,τH , and k. Figures 4 and 5 clarify how the stream resistance increment with the move in the pathology size and any that considers the manner in which the liquid once-over decreases the stream limitation increments. Figures 6 and 7 explain how the volumetric stream decay with a rise in yield pressure related volumetric stream rate diminishes with a fluid rundown n increase.

5 Conclusion This problem dismembered the relentless movement of blood through stenosed conductor with bended viewing non-Newtonian fluid (blood) as Herschel–Bulkley fluid model. The prime findings of the current numerical examination are as per the following: 1.

2.

The flow resistance increases with the increases of stenosis size and more so in light of the fact that the fluid record decreases with the increase of the stream resistance. The eventual outcome of non-Newtonian liquid on circulatory framework is the yield pressure inversely proportional to the stenosis length.

A Study with Magnetic Field on Stenosed Artery …

Fig. 2 Different shapes of stenosis (L 0 = 0.5, 1, 1.5, 2)

Fig. 3 Different shapes of stenosis (m = 1, 2, 3, 4, 5, 6)

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Fig. 4 Flow resistance variation with stenosis height (τ H = 0.05, k = 5)

Fig. 5 Flow resistance variation with stenosis height (τ H = 0.1, k = 6)

A Study with Magnetic Field on Stenosed Artery …

Fig. 6 Rate flow variation for yield stress τ H (n = 1, k = 5 at 60 and 70%)

Fig. 7 Rate flow variation for different n(τ H = 0.01, k = 8 at 80 and 90%)

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The stream diminishes with the ascent of the yield pressure and increase of stenosis thickness.

From the above conclusion, the author feels that much of this analysis is critical to review the scientific demonstration of blood stream in slim veins with gentle stenosis.

References 1. Sud, V., Sekhon, G.: Blood flow through the human arterial system in the presence of a steady magnetic field. Phys. Med. Biol. 34(7), 795–805 (1989) 2. Suri, P., Pushpa, R.: Blood flow in a branched arteries. Indian J. Pure Appl. Math. 12, 907–918 (1981) 3. Rathod, V., Parveen, S.: Math. Educ. 31(3), 121–133 (1997) 4. Fung, Y.: Mechanical Properties of Living Tissue, Chap 3, Biomechanics. Springer, New York (1981) 5. Temur’Yants, N., Shejhtkin, A., Kamynima, I., Naileich, V.: Effect of weak varying magnetic fields on the infradianrhythmics of the functional activity of rat leucocytes. Biophysics 41(4), 943–946 (1996) 6. Verma, S.: Mathematical model of unsteady blood flow through a narrow tapered vessel with stenosis. J. PAS 17, 96–108 (2011) 7. Merrill, E., Cokelet, G., Wells, A.: Non Newtonian rheology of human blood effect of fibrinogen deduced by subtraction. Circ. Res. 13(1), 48 (1963) 8. Blasius, H.: Math. U. Physik 58, 225 (1910) 9. Whitemore, R.: Rheology of Circulation. Oxford, Pergamon Press, New York (1968) 10. Smith, F.: Flow through constricted or dilated pipes and channels: Part 1. J. Mech. Appl. Math. 29(3), 343–365 (1976) 11. Prasad, K., Chanda, P.: Proc. Natl. Acad. Sci. 60(A), 317–326 (1990); Chandra, P.: J. Math. Phys. Sci. 431 (1983) 12. Misra, J., Patra, M., Misra, S.: A non-newtonian fluid model for blood flow through arteries under stenotic conditions. J. Biomech. 26(9), 1129–1141 (1993) 13. Bugliarello, G., Sevilla, J.: Velocity distribution and other characteristics of steady and pulsatile blood flow in fine glass tubes. Biorheolog 7, 85–107 (1970) 14. Srivastava, L.: Flow of couple stress fluid through stenotic blood vessels. J. Biomech 17, 479–485 (1985) 15. Popel, A., Giora, S., Joseph, O.: Advances in Bioheat and Mass Transfer, pp. 105–111. American Society of Mechanical Engineers, New York (1993) 16. Mukhopadhyay, S., De, P., Mandal, M., Layek, G.: Analysis of flow fields in a flexibletube with periodic constriction. Appl. Appl. Math. 6(1), 304–322 (2011) 17. Biswas, D., Dey, K.: Bingham plastic model of blood flow through an artery with slip. In: Proceedings of the National Conference on Recent Developments in Mathematics and Applications (2002) 18. Berman, A.: Laminar flow in an annulus with porous walls. J. Appl. Phys. 29, 71–75 (1958) 19. Charm, C., Kurland, G.: Blood Flow and Micro-Circulation. Biomedical Health Publication, A Wiley (1975) 20. Reiner, M., Scott-Blair, G.: The flow of the blood through narrow tubes. Nature 84, 354 (1959) 21. Radhakrishnamacharya, G., Chandra, P., Kaimal, M.: A hydrodynamical study of the flow in renal tubules. Bull. Math. Biol. 43(2), 151–163 (1981) 22. Korchevski, E., Marochnik, L.: Magnetohydynamic version of movement of blood. Biofizika 10, 371–373 (1965) 23. Ahmed, S., Dutta, B., Goswami, A.: Blood flow through arteries in presence of transverse magnetics field. J. Interdisc. Math. 22(7), 1083–1093 (2019)

A Study with Magnetic Field on Stenosed Artery … 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38.

39. 40. 41. 42. 43. 44. 45. 46. 47.

179

Barnothy, M.: Biological Effects of Magnetic field, vols. I & II, Plenum Press (1969) Belousova, L.: Biophysics 10, 404–405 (1965) Vardanyan, V.: Effect of magnetic field on blood flow. Biofizika 18(3), 491–496 (1973) Ramachandra, R., Desikachar, K.: Physiological-type flow in a circular pipe in the presence of a transverse magnetic field. J. Indian Inst. Sc 68, 247–260 (1988) Jain, R., Ronghe, A., Ali, S.: Appl. Sc. Periodical 1(4), 197–199 (1999) Orlov, L., Pochechoyea, G., Mokaye, W.: Biophysics 41(4), 959–962 (1996) Reddy, Y., Parveen, S.: Math. Educ. 31(3), 149–152 (1997) Schneck, D., Ostrach, S.: Pulsatile blood flow in a channel of small exponential divergence-I. The linear approximation for low mean Reynolds number. J. Fluids Eng. 97(3), 353–360 (1975) Kirwan, A.: Phys. Fluids 11, 1440 (1968) Barnothy, M.: Biological Effects of Magnetic Fields. Springer, Heidelberg (2013) Eirich, F.: Rheology Theory and Application, vol. l 4, Academic Press Inc., New York (1960) Soundalgaker, V.: Rev. Roum Phys. 15, 505 (1972) Singh, N., Shmokov, U.: Analysis of blood. Flow in narrow vessels through micropolar fluid. Model. Bionika 22, 55–57 (1988) Chaturani, P., Ranganathan, T.: Proceedings of International Conference of Mathematical Modeling in Science and Techonology (1988) Larimi, M., Ghanaat, A., Ramiar, A., Ranjbar, A.: Forced convection heat transfer in a channel under the influence of various non-uniform transverse magnetic field arrangements. Int. J. Mech. Sci. 118, 101–112 (2016) Sadiku, M., Kulkarni, S.: Principles of Electromagnetics. Oxford University Press (2015) Casson, N.: Rheology of Disperse Systems. In: Mill (ed.) pp. 84–102. Pergamon Press (1959) Chow, J., Soda, K.: Laminar flow in tubes with constriction. Phys. Fluids 5, 1700–1706 (1972) Young, D.: Fluid mechanics of arterial stenosis. J. Biomech. Engg. Trans. ASME 101, 157–175 (1979) Boyd, W.: A Textbook of Pathology: Structure and Function in Disease, Lippincott Williams and Wilkins (1963) Womersley, J.: Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known. J. Physiol. 127(3), 553–563 (1955) Cox, R.: Comparison of linearized wave propagation models for arterial blood flow analysis. J. Biomech. 2(3), 251 (1969) Fry, D.: Responses of the Arterial Wall to Certain Physical Factors, 12, pp. 93–125. Initiating Factors, Ciba Foundation Symposium (1973) Ling, S.: Modelling the nonlinear behaviour of arteries. In: American Institute Aeronautics and Astronautics paper No. 70–789. 3rd Fluid and Plasma Dynamics Conference, Los Angeles, California (1970)

Estimation of Reproduction Number of COVID-19 for the Northeastern States of India Using SIR Model Prabhdeep Singh, Arindam Sharma, Sandeep Sharma, and Pankaj Narula

Abstract Coronavirus disease (COVID-19) has been the cause of over a million deaths across the globe. The pandemic has affected the social, economic, and psychological facets of human life. India currently ranks second in the total number of cases in the world. There is an emergent need to understand the severity of the prevalence of the disease in India. In the present work, an SIR model in conjunction with daily case count has been implemented to analyze the transmission dynamics of COVID19 across the eight northeastern states of India. The parameters associated with this model, namely the infection, recovery, and death rates, have been estimated for the northeastern region of India. The infection rate is found to be in the range from 0.18 to 0.49, which is observed to be the least in Sikkim and the highest in Mizoram. The basic reproduction number for COVID-19 is found to vary between 1.1 and 1.3. Keywords COVID-19 · India · Mathematical modeling · Basic reproduction number

1 Introduction Coronavirus disease (COVID-19) is an infectious respiratory and viral disease caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). The COVID19 virus spreads through the respiratory droplets produced by an infected person while breathing, talking, coughing, and sneezing [1–4]. Later on, it has also been discovered that these droplets can sustain in the air for a long span of time [5–7]. P. Singh · A. Sharma Department of Computer Science and Engineering, Thapar Institute of Engineering and Technology, Patiala, Punjab 147004, India S. Sharma Department of Mathematics, DIT University, Dehradun, Uttarakhand 248009, India P. Narula (B) School of Mathematics, Thapar Institute of Engineering and Technology, Patiala, Punjab 147004, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_16

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Further, the spread of COVID-19 is also possible through touching contaminated objects and then touching the nose, mouth, or eyes [8–10]. First trace of COVID-19 was found in Wuhan (China) on December 1st, 2019 [11]. In the same month, the virus was also detected in the wastewater of Milan and Turin [12]. The disease spreads rapidly across Thailand and Japan before the end of January 2020. As of the end of October, there have been over 60 million reported cases of COVID-19 throughout the world [13]. Particularly in India, the first case of COVID-19 was reported on January 30th, 2020 [14]. Currently, with over nine million infections, India leads in the number of confirmed cases in Asia and ranks second globally [13]. Five southern states (Maharashtra, Karnataka, Andhra Pradesh, Tamil Nadu, Kerala) account for more than 50% of the total confirmed cases of the subcontinent [15]. The large number of COVID-19 cases has overburdened the health system in these states, as one out of five patients needs to be hospitalized [16]. This unprecedented situation has adversely affected the mental health and efficiency of health care professionals [17]. The prevalence of COVID-19 in the northeastern states of India is at a gradual pace. The region witnessed its first COVID-19 case on March 24th, 2020, almost two months after the first case in India. The delayed spread of COVID-19 in the northeastern states is due to its complex terrain and low connectivity with the other parts of the country. The public health infrastructure is not robust in this region as compared to the other states of the subcontinent [18]. Therefore, quantification of parameters associated with COVID-19 for this region demands immediate attention. This may assist the health care authorities in the optimal distribution of resources. The dynamics of infectious diseases can be modeled mathematically using deterministic and probabilistic techniques [19, 20]. A common deterministic approach is to divide the population into different compartments, namely susceptible, infected, and removed. Appropriate transfer rates such as infection, recovery, and death rates are included in order to model the movement of population from one compartment to another. The basic reproduction number (R0 ) is another significant parameter of infectious disease modeling which provides a threshold criterion for disease eradication. This is defined as the expected number of new infections generated by a single infected individual in the entire susceptible population. R0 > 1 is an indicative of disease outbreak, whereas R0 < 1 represents that the disease dies out. R0 = 1 underlines the endemic nature of the infection. R0 of COVID-19 has been estimated for several countries across the globe. Quite recently, the values of R0 for Sri Lanka, Nigeria, India, and Russia have been reported in the range from 0.93 to 1.29 [21, 22]. On the other hand, for the USA, China, and France, the R0 is observed to lie between 1.61 and 2.72 [21]. The estimates of R0 remained relatively higher for Italy (3.1) and Brazil (5.25) [23, 24]. To our best knowledge, R0 has not been reported for the northeastern states of India. Compartmental modeling has been widely implemented to model the transmission dynamics of COVID-19 [25–28]. In the case of COVID-19, it has been observed that a fundamental SIR model outperformed complex models such as the SEIR model [29]. Therefore, in the present endeavor, an SIR model has been utilized to analyze the severity of COVID-19 for the northeastern region of India. The parameters pertaining

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183

to the model such as infection rate, recovery rate, death rate, and diagnosis rate have been estimated. The Levenberg–Marquardt method in conjunction with Akaike information criteria (AIC) has been used to obtain the best fit of the daily COVID-19 case count.

2 Data and Methods The daily data of confirmed COVID-19 cases for eight northeastern states of India, namely Arunachal Pradesh, Assam, Manipur, Meghalaya, Mizoram, Nagaland, Sikkim, and Tripura have been retrieved from the archive of https://www.cov id19india.org/ for the present analysis [30]. The temporal span of this data ranges from 03/24/2020 to 10/25/2020. The number of active cases has been computed by subtracting the recovered and deceased cases from confirmed cases. In Fig. 1a, the political boundary of India has been depicted, whereas the distribution of total number of confirmed COVID-19 cases at a log scale has been shown in Fig. 1b. During the formulation of an SIR model, the total population (N(t)) is divided into three mutually exclusive compartments, namely susceptible (S(t)), infected (I(t)), and removed (R(t)). The governing equations of these compartments are as following: dS = −β S I dt

Fig. 1 Distribution of COVID-19 cases in northeastern region

(1)

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dI = β S I − (γ + d)I dt

(2)

dR =γI dt

(3)

The movement between these compartments depends on infection rate (β), recovery rate (γ ), and death rate (d). The number of susceptible people who are infected by an infected individual per unit time is proportional to the product of S and I with proportional coefficient β and, therefore, is equal to βSI. The number of people who get cured from COVID-19 from the compartment of infected individuals per unit time is equal to γI. The number of people dying due to the infection per unit time is equal to dI. The basic reproduction number (R0 ) for this model is defined as R0 =

β γ +d

(4)

It is possible that an infected individual is not a part of the reported cases. This can be attributed to various reasons such as asymptomatic infection, under-reporting, and false testing. Such cases can be potential superspreaders and can play a vital role in the transmission dynamics of COVID-19. It is, therefore, important to estimate the total infections in the population in order to accurately determine the spread of the disease. The ratio of the reported cases and the total infections is called the diagnosis rate (ρ) which has also been estimated along with infection rate, recovery rate, and death rate. The diagnosis rate provides a better estimation of the total number of infected individuals.

3 Results The value of I(0) has been estimated along with the infection rate (β), recovery rate (γ), and death rate (d) for four different ranges of diagnosis rates (ρ): (0.01, 0.1), (0.1, 0.2), (0.2, 0.5), and (0.5, 1). These ranges have been adopted from Weston et al. [29]. For each range of ρ, the remaining parameters have been estimated using the Levenberg–Marquardt method. Optimal choice of ρ has been made on the basis of AIC. The death rate due to COVID-19 has been converted to the infection fatality rate (IFR) using the relation IFR = 100 ×

d γ

(5)

The estimated values of infection rate, recovery rate, diagnosis rate, infection fatality rate along with basic reproduction number have been furnished in Table 1.

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Table 1 Estimated parameters along with basic reproduction number State

Infection rate (β)

Recovery rate (γ )

Diagnosis rate (ρ)

IFR (%)

R0

Arunachal Pradesh

0.20

0.16

0.09

1.34

1.24

Assam

0.20

0.15

0.04

3.46

1.25

Manipur

0.20

0.17

0.50

2.72

1.11

Meghalaya

0.18

0.14

0.02

0.02

1.30

Mizoram

0.49

0.43

0.07

1.19

1.13

Nagaland

0.20

0.16

0.06

2.89

1.17

Sikkim

0.18

0.15

0.03

0.36

1.27

Tripura

0.41

0.33

0.10

1.80

1.22

The infection rate is found to be the lowest in Meghalaya and the highest in Mizoram. The recovery rate is observed to lie between 0.14 and 0.43. The IFR of COVID-19 varies from 0 to 3.5% in the northeastern region of the subcontinent. The values of R0 are observed to lie between 1.1 and 1.3. Several studies have reported the values of R0 of COVID-19 for India under different scenarios. Hilton et al. have reported the values of R0 for India as 1.73 and 2.78 without age bias and with age bias, respectively [31]. Seema et al. have categorized the pandemic time interval into pre- and post-lockdown periods. The values of R0 have been estimated to be 2.6 and 1.57 for these periods, respectively [32]. Tridip et al. have used a deterministic approach and provided the early estimate of R0 during the period from mid- of March to the first week of May. The value of R0 in this study is determined to be approximately 2 for India [33]. Marwan Al-Raeei has carried out a deterministic analysis of a compartmental model and reported the value of R0 to be 1.26 for India [21]. This study considered the COVID-19 data up to July 2020. Marimuthu et al. have performed the exponential growth method and time-dependent technique to estimate the R0 (1.38) for the subcontinent [34]. These values are in close match to the value of R0 for northeastern states reported in the present study. In Fig. 2a, b, and d, the model has efficiently captured the peak of the COVID-19 cases for Arunachal Pradesh, Assam, and Meghalaya, respectively. It can also be seen that the pandemic is approximately halfway through its life cycle. The model projects that disease will die out in Arunachal Pradesh and Meghalaya around the 68th week from the onset, whereas in Assam, the disease is projected to die out in the 77th week. In Mizoram and Tripura, the predicted curve is found to be more leptokurtic in nature as shown in Fig. 2e and h, respectively. The shape of the curves suggests that states have already witnessed the peak of COVID-19. The infection is expected to die out in the 47th week for both these states. Even though the daily case data of Mizoram bears few irregularities such as multiple valleys and abrupt spikes, the SIR model explains the data adequately as shown in Fig. 2e. In the case of Manipur, the predicted curve exhibits the slow pace of infection in the state as compared to other states as shown in Fig. 2c. The peak of infection is projected in

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Fig. 2 Reported versus projected cases for northeastern states

the 44th week from the beginning of infection. For Nagaland and Sikkim, the SIR model could not capture the disease pattern efficiently as depicted in Fig. 2f and g. This is due to the presence of multiple sharp changes occurring at small intervals in the reported cases.

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4 Conclusion An SIR model has been used to study the ongoing COVID-19 pandemic scenario in eight northeastern states of India. The real data have been used to estimate the key model parameters. The estimated parameters were subsequently used to estimate the value of the basic reproduction numbers for all eight states. The present analysis suggests that an SIR model, despite its simple structure, can accurately describe the COVID-19 patterns in the northeastern states of India. The work provides many useful insights regarding the severity and projections of COVID-19 in the region. The main findings can be summarized as follows: 1.

2.

3. 4.

The infection rate varies from 0.18 to 0.49, whereas the infection fatality rate ranges from 0 to 3.5%. The basic reproduction number of COVID-19 is found to be the highest in Meghalaya (1.3) and the lowest in Manipur (1.1). Arunachal Pradesh, Assam, and Meghalaya are passing through the peak phase, whereas cases in Tripura and Mizoram are reaching nadir. The peak in COVID19 cases has not yet arrived in Manipur. The model also forecasts the end of the infections for the region. The major limitation of the study is that the model could not explain the data for Nagaland and Sikkim due to the multiple peaks observed over a short span of time.

References 1. World Health Organisation COVID-19 Q&A. https://www.who.int/news-room/q-a-detail/cor onavirus-disease-covid-19-how-is-it-transmitted. Accessed 28 Nov 2020 2. Chan, J.F.-W., et al.: A familial cluster of pneumonia associated with the 2019 novel coronavirus indicating person-to-person transmission: a study of a family cluster. Lancet 395(10223), 14–23 (2020). https://doi.org/10.1016/S0140-6736(20)30154-9 3. Ghinai, I., McPherson, T.D., Hunter, J.C., Kirking, H.L., Christiansen, D., Joshi, K., et al.: First known person-to-person transmission of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) in the USA. Lancet 395(10230), 1137–1144 (2020). https://doi.org/10.1016/ S0140-6736(20)30607-3 4. Huang, C., Wang, Y., Li, X., Ren, L., Zhao, J., Hu, Y., et al.: Clinical features of patients infected with 2019 novel coronavirus in Wuhan China. Lancet 395(10223), 497–506 (2020). https://doi.org/10.1016/S0140-6736(20)30183-5 5. Stadnytskyi, V., Bax, C.E., Bax, A., Anfinrud, P.: The airborne lifetime of small speech droplets and their potential importance in SARS-CoV-2 transmission. Proc. Natl. Acad. Sci. USA 117(22), 11875–11877 (2020). https://doi.org/10.1073/pnas.2006874117 6. Somsen, G.A., van Rijn, C., Kooij, S., Bem, R.A., Bonn, D.: Small droplet aerosols in poorly ventilated spaces and SARS-CoV-2 transmission. Lancet Respir. Med. 8(7), 658–659 (2020). https://doi.org/10.1016/S2213-2600(20)30245-9 7. Fears, A.C., et al.: Persistence of severe acute respiratory syndrome coronavirus 2 in aerosol suspensions. Emerg. Infect. Dis. 26(9), 2168–2171 (2020). https://doi.org/10.3201/eid2609. 201806

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8. Ong, S.W.X., Tan, Y.K., Chia, P.Y., Lee, T.H., Ng, O.T., Wong, M.S.Y.: Air, surface environmental, and personal protective equipment contamination by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) from a symptomatic patient. JAMA 323(16), 1610–1612 (2020). https://doi.org/10.1001/jama.2020.3227 9. Chia, P.Y., Coleman, K.K., Tan, Y.K.: Detection of air and surface contamination by SARSCoV-2 in hospital rooms of infected patients. Nat. Commun. 11(1), 2800 (2020). https://doi. org/10.1038/s41467-020-16670-2 10. Guo, Z., Wang, Z., Zhang, S., Li, X., Li, L., Li, C.: Aerosol and surface distribution of severe acute respiratory syndrome coronavirus 2 in hospital wards, Wuhan China. Emerg. Infect. Dis. 26(7), 1583–1591 (2020). https://doi.org/10.3201/eid2607.200885 11. Wu, Y.C., Chen, C.S., Chan, Y.J.: The outbreak of COVID-19: an overview. J. Chin. Med. Assoc. 83(3), 217–220 (2020). https://doi.org/10.1097/JCMA.0000000000000270 12. La Rosa, G., et al.: SARS-CoV-2 has been circulating in northern Italy since December 2019: evidence from environmental monitoring. Sci. Total Environ. 750 (2021). https://doi.org/10. 1016/j.scitotenv.2020.141711 13. World Health Organisation Coronavirus Disease (COVID-19) Dashboard. https://covid19.who. int/table. Accessed 28 Nov 2020 14. Ministry of Health and Family Welfare Press Release. https://pib.gov.in/PressReleaseIframe Page.aspx?PRID=1601095. Accessed 28 Nov 2020 15. Ministry of Health and Family Welfare COVID-19 Statewise Status. https://www.mohfw. gov.in/. Accessed 28 Nov 2020 16. World Health Organisation COVID-19 Media Statement. https://www.who.int/indonesia/news/ detail/08-03-2020-knowing-the-risk-for-covid-19. Accessed 28 Nov 2020 17. Suryavanshi, N., Kadam, A., Dhumal, G., et al.: Mental health and quality of life among healthcare professionals during the COVID-19 pandemic in India. Brain Behav. 10(01837) (2020). https://doi.org/10.1002/brb3.1837 18. Lyngdoh, L.M.: Inter-state variations in rural healthcare infrastructure in north-east India. NEHU J. 8(2) (2015) 19. Hethcote, H.W.: The mathematics of infectious diseases. SIAM Rev. 42(4), 599–653 (2000). https://doi.org/10.1137/S0036144500371907 20. Keeling, M.J., Danon, L.: Mathematical modelling of infectious diseases. Br. Med. Bull. 92(1), 33–42 (2009). https://doi.org/10.1093/bmb/ldp038 21. Al-Raeei, M.: The basic reproduction number of the new coronavirus pandemic with mortality for India, the Syrian Arab Republic, the United States, Yemen, China, France, Nigeria and Russia with different rate of cases. Clin. Epidemiol. Glob. Health (2020). https://doi.org/10. 1016/j.cegh.2020.08.005 22. Dharmaratne, S., Sudaraka, S., Abeyagunawardena, I., et al.: Estimation of the basic reproduction number (R0 ) for the novel coronavirus disease in Sri Lanka. Virol. J. 17(144) (2020). https://doi.org/10.1186/s12985-020-01411-0 23. D’Arienzo, M., Coniglio, A.: Assessment of the SARS-CoV-2 basic reproduction number, R0 , based on the early phase of COVID-19 outbreak in Italy. Biosafety Health 2(2), 57–59 (2020). https://doi.org/10.1016/j.bsheal.2020.03.004 24. Nuno, C.: Modeling the early evolution of the COVID-19 in Brazil: results from a Susceptible– Infectious–Quarantined–Recovered (SIQR) model. IJMPC 31(10), 1–7 (2020). https://doi.org/ 10.1142/S0129183120501351 25. Arino, J., Portet, S.: A simple model for COVID-19. Infect. Dis. Model. 5, 309–315 (2020). https://doi.org/10.1016/j.idm.2020.04.002 26. Chen, Y.-C., Lu, P.-E., Chang, C.-S., Liu, T.-H.: A time-dependent SIR model for COVID-19 with undetectable infected persons. IEEE Trans. Netw. Sci. Eng. (2020). https://doi.org/10. 1109/TNSE.2020.3024723 27. Cooper, I., Mondal, A., Antonopoulos, C.G.: A SIR model assumption for the spread of COVID19 in different communities. Chaos Solitons Fractals 139 (2020). https://doi.org/10.1016/j. chaos.2020.110057

Estimation of Reproduction Number of COVID-19 …

189

28. Zhao, Z.Y., Zhu, Y.Z., Xu, J.W., et al.: A five-compartment model of age-specific transmissibility of SARS-CoV-2. Infect. Dis. Poverty 9(117) (2020). https://doi.org/10.1186/s40249020-00735-x 29. Roda, W.C., Varughese, M.B., Han, D., Li, M.Y.: Why is it difficult to accurately predict the COVID-19 epidemic? Infect. Dis. Model. 5, 271–281 (2020). https://doi.org/10.1016/j.idm. 2020.03.001 30. COVID19-India Homepage. https://www.covid19india.org/. Accessed 29 Nov 2020 31. Hilton, J., Keeling, M.J.: Estimation of country-level basic reproductive ratios for novel Coronavirus (SARS-CoV-2/COVID-19) using synthetic contact matrices. PLOS Comput. Biol. 16(7) (2020). https://doi.org/10.1371/journal.pcbi.1008031 32. Patrikar, S.R., Kotwal, A., Bhatti, V.K., et al.: Incubation period and reproduction number for novel coronavirus 2019 (COVID-19) infections in India. Asia Pacific J. Public Health (2020). https://doi.org/10.1177/1010539520956427 33. Sardar, T., Nadim, S.S., Rana, S., Chattopadhyay, J.: Assessment of lockdown effect in some states and overall India: a predictive mathematical study on COVID-19 outbreak. Chaos Solitons Fractals 139 (2020). https://doi.org/10.1016/j.chaos.2020.110078 34. Marimuthu, S., Joy, M., Malavika, B., Nadaraj, A., Asirvatham, E.S., Jeyaseelan, L.: Modelling of reproduction number for COVID-19 in India and high incidence states. Clin. Epidemiol. Global Health (2020). https://doi.org/10.1016/j.cegh.2020.06.012

Novel Generalized Divergence Measure for Intuitionistic Fuzzy Sets and Its Applications in Medical Diagnosis and Pattern Recognition Adeeba Umar

and Ram Naresh Saraswat

Abstract The necessity of appropriate divergence measures is arising as they play a vital role in different kinds of problems which are related to dissimilarity, inference, and discrimination. Intuitionistic fuzzy sets (IFSs) are very useful to manage the unassured state of data. For the evaluation of relationships of IFSs, divergence measures of IFSs are necessary. The information of each set in the matrix is formulated by the introduced intuitionistic fuzzy divergence measure, where the matrix under fuzzy environment is applied to find the divergence between the two IFSs. The main motive of this paper is to introduce a new generalized measure with proof of its validity. The proposed divergence measure is applied to the problems of medical diagnosis and pattern recognition on real-world datasets to examine the effectiveness and practicality. Also, the newly developed method is compared with the extant methods which is demonstrated in an intuitionistic fuzzy environment. It is noticed that the newly developed divergence measure found better results in comparison with the other existing methods. Keywords Intuitionistic fuzzy set · Medical diagnosis · Divergence measure · Pattern recognition · Decision-making

A. Umar · R. N. Saraswat (B) Department of Mathematics and Statistics, Manipal University Jaipur, Jaipur, Rajasthan 303007, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_17

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1 Introduction Before fuzzy set (FS) [1] was introduced, the only method for measuring the uncertainty was probability. This theory brought the researchers with new method to work with uncertainty. Researchers began to think fuzziness differently. After this invention, many researchers gave many theories to generalize the notion of FS. Intuitionistic fuzzy set theory (IFST) [2] was the most renowned theory among other approaches which generalizes FST. In IFST, besides membership degree and nonmembership degree, one more parameter called intuitionistic index or degree of hesitancy was introduced. It is proved that intuitionistic fuzzy sets are very proficient as compared to fuzzy sets. A lot of researchers introduced information measures based on intuitionistic fuzzy sets, according to them. First, Burillo and Bustince [3] introduced intuitionistic fuzzy entropy. After that, many authors defined intuitionistic fuzzy entropies. Many researchers introduced new intuitionistic fuzzy entropies [4] in different regions like pattern recognition, multi-criteria decision-making [5–8]. Out of these applications, one application of IFSs has been given to find the divergence between two sets, which is being widely used by many researchers (De et al. [9]; Joshi and Kumar [10]). The different methods and applications using IFSs were presented by many researchers ([11–30]). The IFS [2] has become a useful tool to depict fuzziness. A new divergence measure is proposed here, seeking properties of exponential entropy which was given in [31]. The prime purpose of this manuscript is (1) to introduce a new intuitionistic fuzzy divergence measure as divergence/dissimilarity measures are very necessary and useful for many real-world situations. (2) To apply the new measure in medical investigation. (3) To apply the newly developed intuitionistic fuzzy measure in pattern recognition. This paper is organized as follows: The role of many researchers in this area and the main motive of the study are given in Sect. 1. Necessary definitions of FSs and IFSs are given in Sect. 2. In Sect. 3, a new intuitionistic fuzzy divergence measure is introduced with its validity proof. In Sect. 4, the application of novel intuitionistic fuzzy measure of divergence is given in medical diagnosis using practical examples. The application of newly developed intuitionistic fuzzy measure is given in pattern recognition through numerical illustrations in Sect. 5, to show the viability of the newly developed fuzzy divergence. Lastly, paper is concluded in Sect. 6.

2 Preliminaries First, some important concepts are reviewed connected to FST and IFST. Definition 2.1. Fuzzy set [1, 32]: Let X = (g1 , g2 , . . . ..gn ) be a finite universe of discourse. A fuzzy set E in X is defined as: E = {< gi , μ E (gi ) > |gi ∈ X }

(2.1)

Novel Generalized Divergence Measure for Intuitionistic Fuzzy Sets …

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where μ E (gi ) : X → [0, 1] is the membership function or membership degree of measure of belongingness of gi ∈ X to E. Fuzzy sets were characterized to IFSs as: Definition 2.2. Intuitionistic fuzzy set [2, 11] : An IFS E in X = (g1 , g2 , . . . ..gn ) is defined as: E = {< gi , μ E (gi ), υ E (gi ) > |gi ∈ X }

(2.2)

where μ E (gi ) : X → [0, 1] and υ E (gi ) : X → [0, 1] with the condition, 0 ≤ μ E (gi ) + υ E (gi ) ≤ 1. For gi ∈ X , μ E (gi ) and υ E (gi ), respectively, denote the membership function and the non-membership function of gi to E. Further, π E (gi ) = 1−μ E (gi )−υ E (gi ) denotes the hesitation degree or intuitionistic index of gi . Taking π E (gi ) = 0 implies υ E (gi ) = 1 − μ E (gi ) for all gi ∈ X , i.e., IFS becomes a FS. Therefore, FSs are particular cases of IFSs. Now, certain basic operations on IFSs are discussed: Definition 2.3 [11] : Let E, F ∈ I F S(X ) be the family of all IFSs in X, then: (i) (ii) (iii) (iv) (v)

E ⊆ F iff μ E (gi ) ≤ μ F (gi ) and υ E (gi ) ≤ υ F (gi ) for all gi ∈ X. E = F iff E ⊆ F and F ⊆ E. E c = {< gi , υ E (gi ), μ E (gi ) > |gi ∈ X } E ∪ F = {< gi , max(μ E (gi ), μ F (gi )), min(υ E (gi ), υ F (gi )) > |gi ∈ X } E ∩ F = {< gi , min(μ E (gi ), μ F (gi )), max(υ E (gi ), υ F (gi )) > |gi ∈ X }.

3 Novel Generalized Intuitionistic Fuzzy Divergence Measure This section proposes novel generalized intuitionistic fuzzy divergence measure and its properties with proofs. Li et al. [15] introduced a tool for converting IFSs into FSs.

3.1 Definition of Novel Generalized Intuitionistic Fuzzy Divergence Measure Consider E and F be two IFSs defined on X = (g1 , g2 . . . ..gn ). The dissimilarity measure between IFSs E and F is given by

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⎤  (μ (gi ))k log μ E (gi ) + (μ F (gi ))k log μ F (gi ) ⎥ ⎢ ⎥ ⎢   ⎥ ⎢ μ E (gi )+μ F (gi ) log ⎥ ⎢− 2 ⎥ ⎢ ⎥ ⎢  ⎥ ⎢ 1 k k +  υ g +υ g k−1 (υ E (gi )) log υ E (gi ) + (υ F (gi )) log υ F (gi ) ⎥ ⎢ n ⎥ E( i) F( i) 1 ⎢ 2 ⎥ ⎢    M(E, F) = ⎥ ⎢ (υ E (gi )) K +(υ F (gi ))k υ E (gi )+υ F (gi ) ⎥ ⎢−  n log  k−1 2 i=1 ⎢ ⎥ υ E (gi )+υ F (gi ) ⎥ ⎢ 2 ⎥  ⎢ 1 k k ⎥ ⎢+ + log π log π (g )) (g ) (π (g )) (g ) (π  E i E i F i F i k−1 ⎥ ⎢ π g +π g ⎥ ⎢  E ( i )2 F ( i )   ⎥ ⎢ (π (g )) K +(π (g ))k π E (gi )+π F (gi ) F i ⎦ ⎣− E i log  ⎡



1

 E μ E (gi )+μ F (gi ) k−1 2  (μ E (gi )) K +(μ F (gi ))k   μ E (gi )+μ F (gi ) k−1 2

π E (gi )+π F (gi ) k−1 2

2

(3.1) Now, here a question arises about the validity of the proposed divergence measure. For this, we satisfy the properties defined in [7]. Theorem The proposed novel generalized divergence measure satisfies the following properties: (i) (ii) (iii) (iv)

0 ≤ M(E, F) ≤ 1 M(E, F) = 0 if and only if E = F. M(E, F) = M(F, E) If D ⊆ E ⊆ F, D, E, F ∈ I F S(X ), then M(D, E) ≤ M(D, F) and M(E, F) ≤ M(D, F)

Proof (i) (ii) (iv)

Since (3.1) is a convex function, therefore, it gives us 0 ≤ M(E, F) ≤ 1. And, (iii) properties are obviously satisfied by M(E, F). For D ⊆ E ⊆ F, |D − E| ≤ |D − F| and |E − F| ≤ |D − F|

Therefore, M(D, E) ≤ M(D, F) and M(E, F) ≤ M(D, F).

4 Case Study of Pattern Recognition The main purpose of this section is to find a particular pattern by comparing their characteristics. The method is described as: Assume “a set of m patterns,” which is known ℘1 , ℘2 . . . . . . ℘m with classifications ℵ1 , ℵ2 . . . . . . ℵm . The patterns in the universe of discourse X = {g1 , g2 , . . . .gn } are represented by the IFSs, which is given by

 ℘i = < gi , μ℘i (gi ), υ℘i (gi ) > |gi ∈ X , Let an unknown pattern represented by IFSs as

i = 1, 2, 3, . . . .m

Novel Generalized Divergence Measure for Intuitionistic Fuzzy Sets … Table 1 Calculated values of intuitionistic fuzzy divergence measure

195

M(E, F)

℘1

℘2

℘3

K=1

0.0722

0.0344

0.0656

K=2

0.2014

0.0995

0.1824

K=3

0.4941

0.2868

0.4270

 i = < gi , μ i (gi ), υ i (gi ) > |gi ∈ X Our objective is to classify to one of the classes. To check the ability “of the proposed measure between” two IFSs, it is applied in real-world problems of pattern recognition. Here, the minimum divergence principle is used, which was suggested by Shore and Grey [17] as:   w ∗ = arg min ℘w , The minimum divergence measure is calculated, where the sample ( ) is to be assigned to the class (℘w ), so that the divergence measure between and ℘w is minimum. Example 1 Consider three known patterns ℘1 , ℘2 , and ℘3 having classifications ℵ1 , ℵ2 , and ℵ3 , respectively. These are given by the three “IFSs in the universe of discourse” X = {g1 , g2 , g3 } as follows: ℘1 = {(g1 , 0.2, 0.5), (g2 , 0.5, 0.4), (g3 , 0.2, 0.4)} ℘2 = {(g1 , 0.4, 0.3), (g2 , 0.6, 0.1), (g3 , 0.5, 0.2)} ℘3 = {(g1 , 0.1, 0.4), (g2 , 0.3, 0.5), (g3 , 0.7, 0.1)} Let be a pattern to be identified represented by the following IFS: = {(g1 , 0.4, 0.5), (g2 , 0.3, 0.2), (g3 , 0.6, 0.3)} Our purpose is to classify . Calculated numerical values of divergence measures are shown in Table 1.

4.1 Comparison with Other Divergence Measures Now, the performance of the measure (3.1) is compared with some other extant divergence measures presented in [5–7, 14, 16]. Here, [16] introduced the measure given by

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d L (E, F) =

 n   |μ E (gi ) − μ F (gi )| + |υ E (gi ) + υ F (gi )|

(4.1)

2

i=1

And, [7] introduced the distance between IFSs E and F using Hausdorff distance given by d H Y (E, F) =

H (I E (gi ), I F (gi )) n

(4.2)

where I E (gi ) and I F (gi ) are the subintervals on [0,1] given by I E (gi ) = [μ E (gi ), 1 − υ E (gi )] and I F (gi ) = [μ F (gi ), 1 − υ F (gi )], and the Hausdorff distance H (E, F) = max{|e1 − f 1 |, |e2 − f 2 |} is defined for two intervals E = [e1 , e2 ] and F = [ f 1 , f 2 ]. Vlachos and Sergiadis [5] introduced the divergence measure as: DV S (E, F) = I V S (E, F) + I V S (F, E) I V S (E, F) =

n 

 μ E (gi ) ln

i=1

μ E (gi ) 1 2 (μ E (gi ) + μ F (gi ))

+ υ E (gi ) ln

(4.3) 

υ E (gi ) 1 2 (υ E (gi ) + υ F (gi ))

where [6] introduced intuitionistic fuzzy measure between IFSs E and F as D Z J (E, F) = D(E, F) + D(F, E)

(4.4)

where D(E, F) =

n  i=1

 μ E (gi )+1−υ E (gi )

log2

2 + 1−μ E (gi2)+υ E (gi )

{μ E (gi )+1−υ E (gi )}/2 1 4 [{μ E (gi )+1−υ E (gi )}+{μ F (gi )+1−υ F (gi )}] {1−μ E (gi )+υ E (gi )}/2 2 1 [{1−μ E (gi )+υ E (gi )}+{1−μ F (gi )+υ F (gi )}] 4



log

Joshi et al. [14] suggested a new divergence measure which is given by ⎡

⎤ 1 η (α1 μ E (gi ) + α2 μ F (gi ))η + (α1 υ E (gi ) + α2 υ F (gi ))η ⎢ ⎥ ⎢ +(α1 γ E (gi ) + α2 γ F (gi ))η ⎥ ⎢ ⎥ n ⎢  1 ⎥  η η} η η ⎢ ⎥ μ μ + υ υ + α + α (α (α (g ) (g )) (g ) (g )) 1 F i 2 E i 1 F i 2 E i D J K (E, F) = ⎢+ ⎥; η > 0( = 1) η ⎢ ⎥ η(1 − η) +(α γ γ + α (g ) (g )) 1 F i 2 E i i=1 ⎢ ⎥ 1 ⎢ ⎥ η η η ⎣ −((μ E (gi )) + (υ E (gi )) + (γ E (gi )) ) η ⎦ 1

−((μ F (gi ))η + (υ F (gi ))η + (γ F (gi ))η ) η

(4.5) Comparison between new and existing divergence measures is given in Table 2.

Novel Generalized Divergence Measure for Intuitionistic Fuzzy Sets … Table 2 Comparison of different intuitionistic fuzzy divergence measures

197



℘1

℘2

℘3

M(℘i , )

0.0722

0.0344

0.0656

d L (℘i , )

0.5500

0.4000

0.5000

d H Y (℘i , )

0.2667

0.2000

0.2667

DV S (℘i , )

0.3838

0.1079

0.2246

D Z J (℘i , )

0.1064

0.0786

0.0888

D J K (℘i , )

0.0950

0.0453

0.0881

From the calculated values of different measures (4.1), (4.2), (4.3), (4.4), and (4.5) presented in the table, it can be noted that all measures of divergence adhere the classification of pattern to ℘2 . Therefore, the new intuitionistic fuzzy divergence measure is extensively good in context of pattern recognition (PR).

5 Case Study of Medical Diagnosis For curing a disease, its appropriate diagnosis is necessary. The doctor may not have time to go through the medical history of a patient as at all the times, it is not available; hence, the doctor suggests a diagnosis of a disease based on only symptoms. Therefore, there exists a firm relationship between disease and its diagnosis. But most often, different diseases show similar symptoms. Hence, there comes uncertainty. The IFST [2] is an emphatic tool to deal with vagueness or uncertainty. Many authors like [5, 9, 19–21], and [12] discussed different perspectives of FSs and their applications in medical investigation. In the given example, new divergence measure is used to diagnose a disease for a patient. Example 5.1 We have adapted the given example from De et al. [9]. We consider a set of patients O = {Jack (O1 ), Paul (O2 ), Lisa (O3 ), Susan (O4 )}, a set of diagnosis W = {viral fever (W1 ), malaria (W2 ), typhoid (W3 ), stomach problem (W4 ), chest problem (W5 )}, and a set of symptoms S = {temperature (S1 ), headache (S2 ), stomach pain (S3 ), cough (S4 ), chest pain (S5 )}. Table 3 represents the characteristics of symptoms Table 3 Characteristics of symptoms for diagnosis W1

W2

W3

W4

W5

S1

(0.4,0.0,0.6)

(0.7,0.0,0.3)

(0.3,0.3,0.4)

(0.1,0.7,0.2)

(0.1,0.8,0.1)

S2

(0.3,0.5,0.2)

(0.2,0.6,0.2)

(0.6,0.1,0.3)

(0.2,0.4,0.4)

(0.0,0.8,0.2)

S3

(0.1,0.7,0.2)

(0.0,0.9,0.1)

(0.2,0.7,0.1)

(0.8,0.0,0.2)

(0.2,0.8,0.0)

S4

(0.4,0.3,0.3)

(0.7,0.0,0.3)

(0.2,0.6,0.2)

(0.2,0.7,0.1)

(0.8,0.0,0.0)

S5

(0.1,0.7,0.2)

(0.1,0.8,0.1)

(0.1,0.9,0.0)

(0.2,0.7,0.1)

(0.8,0.1,0.1)

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Table 4 Characteristics of symptoms for patients S1

S2

S3

S4

S5

O1

(0.8,0.1,0.1)

(0.6,0.1,0.3)

(0.2,0.8,0.0)

(0.6,0.1,0.3)

(0.1,0.6,0.3)

O2

(0.0,0.8,0.2)

(0.4,0.4,0.2)

(0.6,0.1,0.3)

(0.1,0.7,0.2)

(0.1,0.8,0.1)

O3

(0.8,0.1,0.1)

(0.8,0.1,0.1)

(0.0,0.6,0.4)

(0.2,0.7,0.1)

(0.0,0.5,0.5)

O4

(0.6,0.1,0.3)

(0.5,0.4,0.1)

(0.3,0.4,0.3)

(0.7,0.2,0.1)

(0.3,0.4,0.3)

and diagnosis, and the characteristics of patients with the particular symptoms are demonstrated in Table 4. To propound a diagnosis, divergence measures between possible diagnosis and patients by symptoms are calculated. The diagnosis which occurs most of the number of times is the suggested diagnosis. For this, we use the minimum divergence principle introduced in [17]. Calculated values of new divergence measure (3.1) for k = 1, 2, and 3 are represented in Table 5. Table 5 Calculated diagnosis results of new measure For k = 1

W1

W2

w3

W4

W5

O1

0.0683

0.0684

0.0773

0.1873

0.2204

O2

0.1406

0.2335

0.0859

0.0250

0.1591

O3

0.0906

0.1384

0.0926

0.1972

0.2475

O4

0.0434

0.0781

0.0864

0.1239

0.1696

Result: O1 (W1 ), O2 (W4 ), O3 (W1 ), O4 (W1 ) For k = 2

W1

W2

W3

W4

W5

O1

0.0889

0.0831

0.1605

0.6276

0.7624

O2

0.2423

0.4534

0.1658

0.0103

0.3253

O3

0.1386

0.2348

0.1015

0.3558

0.4536

O4

0.0349

0.1104

0.1311

0.2384

0.3399

Result: O1 (W2 ), O2 (W4 ), O3 (W3 ), O4 (W1 ) For k = 3

W1

W2

W3

W4

W5

O1

0.4033

0.3776

0.4259

0.9362

1.0296

O2

0.6769

1.0461

0.4648

0.1789

0.7985

O3

0.4994

0.6959

0.4806

0.9432

1.1601

O4

0.2698

0.4184

0.4831

0.6523

0.8651

Result: O1 (W2 ), O2 (W4 ), O3 (W3 ), O4 (W1 )

Novel Generalized Divergence Measure for Intuitionistic Fuzzy Sets …

199

5.1 Comparison with Other Intuitionistic Fuzzy Divergence Measures Here, the implementation of newly derived divergence measures is compared “with other” divergence measures. Srivastava and Maheshwari [18] introduced a version of parametric divergence measure given by ⎡ ⎢ D S M (E, F) = 1 − log2 ⎢ ⎣

1+

1 n

n 

(min(μ E (gi ) + δ1 π E (gi )), (μ F (gi ) + δ1 π F μ F (gi )))

⎤

⎥ i=1 ⎥ ⎦ + min((υ E (gi ) + δ2 π E (gi )), (υ F (gi ) + δ2 π F μ F (gi ))) + min((π E (gi ) − (δ1 + δ2 )π E (gi )), (π F (gi ) − (δ1 + δ2 )π E (gi )))

(5.1)

The parametric divergence measure was introduced in [5] as DV S (E, F)  ⎞ ⎡  μ E (gi )+δ1 π E (gi ) n  (μ E (gi ) + δ1 π E (gi )) log (0.5)((μ E (gi )+δ 1 π E (gi ))+(μ F (gi )+δ2 π F (gi )))  ⎠ ⎣ = (υ E (gi )+(1−δ1 )π E (gi )) +(υ E (gi ) + (1 − δ1 )υ E (gi )) log (0.5)((υ E (gi )+(1−δ i=1 )π (g )+(υ (g )+(1−δ )π (g )))) 1 E i 2 F i F i

(5.2) The [8] proposed a new measure of divergence of IFSs given as ⎡

D H Y (E, F) = −

1 1−

α   ⎤ μ E (gi )+μ F (gi ) (μ E (gi ))α +(μ F (gi ))α − 2 ⎢  2 α  ⎥ ⎢ (υ E (gi ))α +(υ F (gi ))α ⎥ F (gi ) − ⎥; α ⎢ + υ E (gi )+υ 2 2 α⎣  α  ⎦ α α F (gi ) F (gi )) + π E (gi )+π − (π E (gi )) +(π 2 2

> 0(= 1) (5.3)

Then, [10] introduced an exponential Jensen intuitionistic fuzzy divergence measure, which is given by ⎤ (δ1 μ E (gi ) + δ2 μ F (gi ))e(1−(δ1 με (gi )+δ2 μ F (gi ))) ⎥ ⎢ +(δ1 υ E (gi ) + δ2 υ F (gi ))e(1−(δ1 υ E (gi )+δ2 υ F (gi )) n ⎥ 1 ⎢ ⎥ ⎢ π π (1−(δ (g )+δ (g ))) z 1 2 i F i D J K (E, F) = ⎥ ⎢ +(δ1 π E (gi ) + δ2 π F (gi ))e ⎥ ⎢  n i=1 ⎣ −δ μ (g )e(1−μ E (gi )) + υ (g )e(1−υ E (gi )) + π (g )e(1−π E (gi )) ⎦ 1 E i E i E i  (1−μ (g )) (1−υ (g )) (1−π (g )) F i F i F i −δ2 μ F (gi )e − υ F (gi )e − π F (gi )e ⎡

(5.4)

On comparing with different divergence measures given in [5, 8, 18], and [10], the results are given in Table 6. From the above analysis, it can be seen that the output of the measures (5.1)– (5.4) shows the variation in their results and also demonstrates the steadiness of the outcomes of the proposed method.

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Table 6 Comparison of different algorithms Vlachos and Sergiadis (2007)

Hung and Yang (2008)

Srivastava and Joshi and Maheshwari Kumar (2016) (2018)

Proposed measure

Correct disease

Jack

Viral fever or malaria

Viral fever or malaria

Viral fever or malaria

Malaria

Malaria

Malaria

Paul

Stomach problem

Stomach problem

Stomach problem

Stomach problem

Stomach problem

Stomach problem

Lisa

Typhoid

Typhoid

Typhoid

Typhoid

Typhoid

Typhoid

Susan

Viral fever

Viral fever

Viral fever

Viral fever

Viral fever

Viral fever

6 Conclusions In spite of the verity that many measures of divergence between IFSs have been carried into effect, there is still a possibility to develop better measures of divergence so as to find applications in different fields. In this study, a novel intuitionistic fuzzy measure is introduced with its validity proof. The application of the derived measure of divergence is exhibited in pattern recognition and medical diagnosis. Lastly, a comparison between the new and the extant methods presents the consistency of the outcomes of the newly developed method. Therefore, it can be concluded that the newly derived divergence measure is an appropriate measure of divergence for solving many real-life problems associated with pattern recognition and medical investigation. Further, the future research direction of this study is as follows: (i) Parametric generalized divergence measure can be developed. (ii) The newly developed divergence measure can be extended and applied under interval-valued intuitionistic fuzzy soft set (IVIFSS) environment. (iii) Various other practical problems can also be solved using the proposed divergence measure.

References 1. Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965) 2. Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20, 87–96 (1986) 3. Burillo, P., Bustince, H.: Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets. Fuzzy Sets Syst. 118, 306–316 (2001) 4. Montes, I., Pal, N.R., Montes, S.: Entropy measures for Atanassov intuitionistic fuzzy sets based on divergence. Soft. Comput. 22(15), 5051–5071 (2018) 5. Vlachos, I.K., Sergiadis, G.D.: Intuitionistic fuzzy information: application to pattern recognition. Pattern Recogn. Lett. 28(2), 197–206 (2007) 6. Zhang, Q., Jiang, S.: A note on information entropy measure for vague sets. Inf. Sci. 178, 4184–4191 (2008) 7. Hung, W.L., Yang, M.S.: Similarity measures of intuitionistic fuzzy sets based on Hausdorff distance. Pattern Recogn. Lett. 25, 1603–1611 (2004) 8. Hung, W.L., Yang, M.S.: On the j-divergence of intuitionistic fuzzy sets and its application to pattern recognition. Inf. Sci. 178(6), 1641–1650 (2008)

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9. De, S.K., Biswas, R., Roy, A.R.: An application of intuitionistic fuzzy sets in medical diagnosis. Fuzzy Sets Syst. 117(2), 209–213 (2001) 10. Joshi, R., Kumar, S.: Exponential Jensen intuitionistic fuzzy divergence measure with applications in medical investigation and pattern recognition. Soft Comput. 1–14 (2018) 11. Atanassov, K.T.: Intuitionistic fuzzy sets. Physica-Verlag, Heidelberg (1999) 12. Boran, F.E., Akay, D.: A biparametric similarity measure on intuitionistic fuzzy sets with applications to pattern recognition. Inf. Sci. 255(10), 45–57 (2014) 13. Garg, H., Kumar, K.: A novel correlation coefficient of intuitionistic fuzzy sets based on the connection number of set pair analysis and its application. Scientia Iranica 25(4), 2373–2388 (2018) 14. Joshi, R., Kumar, S., Gupta, D., Kaur, H.: A Jensen-α-norm dissimilarity measure for intuitionistic fuzzy sets and its applications in multiple attribute decision making. Int. J. Fuzzy Syst. 20(4), 1188–1202 (2017) 15. Li, F., Lu, Z.H., Cai, L.J.: The entropy of vague sets based on fuzzy sets. J. Huazhong Univ. Sci. Technol. (Nat. Sci.) 31, 1–3 (2003) 16. Li, D.F.: Some measures of dissimilarity in intuitionistic fuzzy structures. J. Comput. Syst. Sci. 68(1), 115–122 (2004) 17. Shore, J.E., Gray, R.M.: Minimization cross-entropy pattern classification and cluster analysis. IEEE Trans. Pattern Anal. Mach. Intell. 4(1), 11–17 (1982) 18. Srivastava, A., Maheshwari, M.: Decision making in medical investigations using new divergence measures for intuitionistic fuzzy sets. Iranian J. Fuzzy Syst. 13(1), 25–44 (2016) 19. Szmidt, E., Kacprzyk, J.: Intuitionistic fuzzy sets in intelligent data analysis for medical diagnosis. In: Proceedings of the Computational Science ICCS (2001) 20. Szmidt, E., Kacprzyk, J.: Intuitionistic fuzzy sets in some medical applications. In: Proceedings of the 7th Fuzzy Days, 2206, Computational Intelligence: Theory and Applications, pp. 148– 151. Springer, Berlin (2001) 21. Szmidt, E., Kacprzyk, J.: A similarity measure for intuitionistic fuzzy sets and its application in supporting medical diagnostic reasoning. Artif. Intell. Soft Comput. ICAISC 3070, 388–393 (2004) 22. Umar, A., Saraswat, R.N.: Novel divergence measure under neutrosophic environment and its utility in various problems of decision making. Int. J. Fuzzy Syst. Appl. 9(4), 82–104 (2020) 23. Umar, A., Saraswat, R.N.: New generalized intuitionistic fuzzy divergence measure with applications to multi-attribute decision making and pattern recognition. Recent Adv. Comput. Sci. Commun. (Recent Patents Comput. Sci.) 14(7), 2247–2266 (2021). https://doi.org/10.2174/ 2666255813666200224093221 24. Zhang, F.W., Huang, W.W., Sun, J., Liu, Z.D., Zhu, Y.H., Li, K.T., Xu, S.H., Li, Q.: Generalized fuzzy additive operators on intuitionistic fuzzy sets and interval-valued intuitionistic fuzzy sets and their application. IEEE Access 7, 45734–45743 (2019) 25. Zhang, H., Xie, J., Lu, W., Zhang, Z., Fu, X.: Novel ranking method for intuitionistic fuzzy values based on information fusion. Comput. Ind. Eng. 133, 139–152 (2019) 26. Zhang, L., Zhan, J., Xu, Z., Alcantud, J.C.R.: Covering-based general multigranulation intuitionistic fuzzy rough sets and corresponding applications to multi-attribute group decisionmaking. Inf. Sci. 494, 114–140 (2019) 27. Zhang, L., Zhan, J., Yao, Y.: Intuitionistic fuzzy TOPSIS method based on CVPIFRS models: an application to biomedical problems. Inf. Sci. 517, 315–339 (2020) 28. Zhang, H., Xie, J., Song, Y., Ge, J., Zhang, Z.: A novel ranking method for intuitionistic fuzzy set based on information fusion and application to threat assessment. Iranian J. Fuzzy Syst. 17(1), 91–104 (2020) 29. Saraswat, R.N., Umar, A.: New fuzzy divergence measure and its applications in multi-criteria decision-making using new tool. Math. Anal. II: Optim. Diff. Eq. Graph Theory Springer Proc. Math. Stat. 307, 191–205 (2020) 30. Umar, A., Saraswat, R. N.: Neoteric divergence measure for refined interval-valued neutrosophic sets and its application in decision making. Int. J. Math. Oper. Res. (2020) (In Press). https://doi.org/10.1504/IJMOR.2020.10034000

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31. Pal, N.R., Pal, S.K.: Object background segmentation using new definition entropy. IEE Proc. E 366, 284–295 (1989) 32. Zadeh, L.A.: Probability measures of fuzzy events. J. Math. Anal. Appl. 23, 421–427 (1968)

Development of 2D Axisymmetric Acoustic Transient and CFD Based Erosion Model for Vibro Cleaner Using COMSOL Multiphysics Vipulkumar Rokad

and Divyang H. Pandya

Abstract In cleaning industries, vibro cleaner has been used to remove contamination like dust, dirt, oil, etc., from out of reach or critical surfaces of the objects. This process is completely safe for human and eco-friendly toward nature. The aim of this research is to investigate cavitation erosion phenomenon by presenting pressure acoustic transient model of COMSOL Multiphysics for vibro cleaner. By using COMSOL Multiphysics, acoustic and CFD approaches have been coupled to gather to achieve great results in sound pressure level, acoustic pressure, velocity of fluid flow, particle trajectory and erosion rate. Piezoelectric transducer is attached to generate pressure waves in liquid media by converting electric energy into mechanical vibration through tank wall transience. Due to pressure difference in liquid, cavitation bubbles have been produced which creates turbulence in acoustic streaming fluid flow and can be studded by using bubbly flow module. Also, particle tracing module and finnie erosion module have been used to understand the particle trajectory and erosion phenomenon. Based on cleaning contaminants, different frequencies like 28 and 40 kHz have been applied for evaluating various process parameters. Here, Lead zirconium titanate (PZT-4) piezoelectric transducer is used to improve pressure generation rates and also reduces a cleaning process time. Keywords Pressure acoustic transient · Cavitation erosion effect · Pressure waves · Piezoelectric transducer · Erosion rate · COMSOL Multiphysics

1 Introduction Vibro cleaner (VC) has been designed to work on highly efficient cleaning process considering ultrasonic cleaning technique. In Ultrasonic cleaning, ultrasound energy is used to clean contaminants from surface of the object to be cleaned. It is one of the most powerful method of cleaning over other conventional methods because it can

V. Rokad (B) · D. H. Pandya Kadi Sarva Vishwavidyalaya, Gandhinagar, Gujarat 382024, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_18

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be used effectively to clean non accessible areas or out of reach areas of the components [1, 2]. The combination of pressure waves and cavitation erosion effect is the theoretical fundamentals behind the ultrasonic cleaning process [2, 3]. Piezoelectric transducer converts electrical input signals into mechanical sound waves know as ultrasound energy which strikes on the surface of the tank wall and react as pressure waves itself by means of cavitation effect in the cleaning solvent. Due to cavitation effect, generated bubble has been burst over the surface of the components and clean the surface by the erosion phenomenon [1, 3]. It can be used to clean dust, oil, grease, dirt and other contaminants. Various objects like ornaments, lenses, coins, watch parts, hard disk parts, automobile parts, medical surgery tools and other industrial tools can also be cleaned by this method [2, 4]. For cleaning purpose, generally, 20–400 kHz frequencies [1, 2] have been used with appropriate liquid solvents. This cleaning technique has zero risk for human safety and eco-friendly about the nature [2, 5]. In general, vibro cleaner is a closed vessel which is made for removing contaminants from outer surface of different objects. Each transducer has been designed to operate for a particular frequency. Based on the cleaning contaminants, various frequencies have been used with respect to various frequency transducer to clean different components with different solvents. The results of cleanliness are uncertain [2]. To avoid repeated process, simulation has been introduced for the investigation of sensible cavitation erosion effect considering various parameters to save the process time and avoid damage of object surfaces. COMSOL Multiphysics is one of the most powerful multi coupling module software. It has capability of coupling many modules at same time to investigate the multi physics problems. Here, pressure acoustic transient module is used with frequency domain solver which gives really a stiff solution for the investigation [6, 7]. For the study of erosion phenomenon, fluid flow and particle tracing have been coupled with acoustic transient module [2, 6, 7].

2 Technical Background Ultrasonic cleaning is working based on pressure waves and cavitation erosion effects. Pressure acoustic transient has been introduced by transition of tank wall between transducer and liquid media which creates pressure waves by means of mechanical vibration in liquid.

2.1 Working Principle Ultrasonic cleaning is one of the efficient cleaning techniques to eliminate contaminants from part surface by using ultrasound energy of the transducer. Piezoelectric transducer converts electrical signals into mechanical sound waves which are directly reflect over the tank wall surface to generate pressure waves in liquid. The ultrasound

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Fig. 1 Ultrasonic cleaning working principle

strikes on the tank wall, and these tank wall reacts as a transience between transducer and liquid media and crates pressure waves. Repetitive vibrating surface of transducer generates continuous flow of pressure waves in liquid media which creates millions of small tiny bubbles by means of positive and negative pressure difference. Theses bubbles strike and burst over the surface of cleaning components. The continuous bombarding of such kind of tiny bubbles which creates microscopic implosions in liquid, creates erosion phenomenon. Due to erosion, continuous mass loss of contaminates cleans the surface of the parts. The rapid oscillations of pressure waves cause atomic implosions (cavitation) in liquid media, makes cleaning process so effective [1, 2, 4].

2.2 Cavitation Erosion Effect Bubble is a heart of ultrasonic cleaning. Bubbles have been created by sound waves which have been generated by piezoelectric transducer. Tank wall reacts as a transition between transducer and liquid media. By rapid vibration of transducer surface creates lowest and highest pressure phase change in pressure waves which creates millions of blisters called “Cavitation bubbles.” These blisters are empty spaces and will burst and form very quickly as they are produced. This continuous rise produces the power of a large machine in the form of heat as well as pressure which gives a great deal of cleaning power over the surface of material for removing the impurities which is known as “Cavitation effect.” Erosion is the process of surface wear and material loss due to vacuum pockets generated inside the liquid. Contaminant material has been eroded by the effect of cavitation from the surface of parts so it is also known as “Cavitation Erosion effect” [8, 9].

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2.3 Simulation Equations Pressure Acoustic—Transient. For CAD model development, pressure acoustic transient module with frequency domain is used because transducer is attached at the bottom of tank wall not directly to the liquid media. Due to this tank wall transience, pressure waves have been created in liquid media by a transducer through the tank wall vibration. In boundary condition, sound hard boundaries are applied on the whole tank walls because of external boundaries where sound soft boundaries are applied on whole object which need to be clean because of internal boundaries. Now, spherical wave radiation has been chosen which creates turbulence flow in the liquid media to generate cavitation effect for processing [7].   2 K eq. . pt −1 = Qm ∇. (∇. pt − qd ) − ρC ρC

(1)

Solid Mechanics—Linear Elastic. In model, the term solid mechanics has been applied to the whole tank for processing. Roller terminology transmits sound pressure waves generated from transducer to liquid media through SS tank wall [7]. −ρω2 u = ∇.s + Fv ei∅ , −ik z = λ

(2)

  1 S = Sad + C : el + − trace(C : el ) − ρw l, el =  − inel 3

(3)

Electrostatics—Piezoelectric Transducer. The term electrostatics has been applied to the piezoelectric transducer because it converts electrical signals in to mechanical sound waves and that waves directly transfer to the tank wall in the form of vibration. For electric input, 230 V power is used [4]. −ρω2 u = ∇.(F.S.)T + Fv ei∅ ,

F = I + ∇u, −ik z = λ

−1 −T − E.e, el = S = S0 + Ji Finel (C : el )Ji Finel

  D = Dr + e.εel + ∈0vac ∈r S +J C −1 − 1 .E,

 1 T Fel Fel − 1 , 2

(4)

−1 Fel = F Finel (5)

J = det(F), C = F T .F

(6)

Fluid Flow—Bubbly Flow k−ε model. Cavitation effect creates bubbles due to phase difference of negative and positive pressure. Because of this, bubbly flow has been used to generate turbulence in the moving fluid. Bubbly flow k-ε model gives more stringent than another model [7]. φ1 ρ1 (u 1 .∇)u 1 = ∇.[−ρ2I + K ] + φ1 ρ1 g + F

(7)

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Particle Tracing for Fluid Flow. The particle tracing module is used to trace the trajectories of particles in the presence of an external field. Fluid-based particle tracing helps to investigate the motion of fluid particles in liquid media. The forces like drag and acoustophoretic radiation force have been predefined, and fluid flow has been computed with it to get appropriate results [7]. d(m v V ) = Ft dt FD =  Frad =

−2πr P3

F(αi ) =

  1 m P M u − V τp S

(9)

  fl ∗  1   fl ∗ 1 κs Re f 0 ρ ∇ρ − ρRe f 1 u · ∇u 3 2

m i = 

(8)

ci ρm P Vi2   F(αi ) 4HV 1 + m P r P2 /l P

2 cos αi 2 sin(2αi ) − P

P=

2 P

tan αi > 2 sin αi tan αi ≤

(10)

(11) P 2 P 2

(12)

K 1 + m P r P2 /I P

IP =

2 m P r P2 5

(14)

3 Methodology In this section, research methodology and related parameters have been discussed to develop vibro cleaner for an improved performance.

3.1 Input Parameters for Ultrasonic Cleaning Tank and Other Parts The cylindrical shaped stainless-steel tank has been made for the investigation. The tank size is 125 radius × 200 mm height with 2-mm thick wall. The tank has been made from 2 mm SS material sheet. The capacity is 10 L water volume approx. The properties like good elasticity, durability and anti-corrosiveness, SS has been taken for the tank manufacture. 28 kHz frequency transducer has been attached

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which converts sound waves into pressure waves in liquid medial through tank wall transience by using 230 V power input [3, 10].

3.2 Computational Model and Simulation COMSOL Multiphysics software has been used to develop CAD model. The model is developed by using 2D axisymmetric method. COMSOL is capable to couple many modules to get meaningful results of the study. Here, two modules like acoustic and CFD have been coupled to get efficient results of the process. In acoustic module, piezoelectric transducer (PZT-4) has been used to generate pressure waves into liquid media by converting electric energy into mechanical vibration with tank wall transience. In CFD module, bubbly flow has been chosen to get the turbulence effect in the tank fluid because cavitation effect generates small micro bubbles due to pressure difference. So, bubbly flow is useful to predict the motion of fluid particles. Particle tracing module is also coupled to trace the movement of fluid particles and particle engagement with parts to get the erosion phenomenon for cleaning. Refer Fig. 2a for 2D axisymmetric CAD model [6, 7, 11]. In simulation, accuracy is more depended on the method of node generation in meshing. To get meaning full outputs in wave generation, mesh type extra-fine is applied. It contains maximum and minimum element size are 3 and 0.1, respectively, with 1.7 growth rate and 0.8 curvature factor. Resolution of the narrow regions is applied as 0.3. Refer Fig. 2b for mesh model. For simulation, COMSOL Multiphysics software is used with version 5.3a. The whole study has been carried out mainly using pressure acoustic transient module with considering frequency domain as solver. Because it is designed for studying the sensible effects of sound to pressure waves in a transient condition. For the study,

Fig. 2 a 2D axisymmetric geometric model. b Mesh model

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Table 1 Values of material properties Properties

Stainless steel

Aluminum

Density (kg/m3 )

7850

2700

Young’s modulus (Pa)

20 ×

1010

7.0 × 1010

Poisson’s ratio

0.30

0.33

Bulk modulus (Pa)

11.5 × 1010

5.1 × 1010

Shear modulus (Pa)

7.69 × 1010

2.6 × 1010

SS tank and aluminum have been set as solid domains and water as liquid domain. Piezoelectric transducer is used to transmit sound energy into acoustic properties through tank wall transience into liquid media. After the setting all the values, study applied with frequency domain which gives result of sound pressure level, acoustic pressure and Von Mises stresses, etc. Also, the stationary domain has been used to study the turbulence of fluid particles in bubbly flow which gives velocity in its results and the time dependent domain has been applied to particle tracing module to study the movement of particles and with the help of drag force, acoustophoretic radiation force, erosion phenomenon has been studded which gives the results of particle trajectories and erosion rate. The erosion rate has been calculated based on finnie model. For used material property details, refer Table 1. The data of standard model of transducer BJC-2860 T-59HS has been used which is manufactured by Beijing ultrasonics. It is PZT-4 type transducer [4, 6, 7].

4 Result and Discussion 4.1 Acoustic Approach In this study, computational model has been developed based on 2D axisymmetric method using COMSOL Multiphysics. COMSOL Multiphysics has enough capabilities to give stringent result than other simulation method and can be coupled many modules to study the various approaches. PZT-4 piezoelectric transducer has hard power transmitting material and can work on high preload. Spherical wave radiation has been considered for acoustic streaming due to cavitation effect. Results of sound pressure level, acoustic pressure and stresses have been compared with previous work done by Worapol Tangsopa. Result shows that at a frequency of 28 kHz, the acoustic pressure is 9.64 × 106 Pa and sound pressure level is 260 dB where in previous work done, and it was 101 kPa maximum. Axisymmetric 2D and 3D views have been shown for more clarity. In 3D view, pressure waves generation has been clearly visible around the object which need to be clean. Sound pressure level was not mentioned in previous work study.

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Fig. 3 a Acoustic pressure, Pa. b Sound pressure level, dB (axisymmetric 2D)

Fig. 4 a Acoustic pressure, Pa; b Sound pressure level, dB (axisymmetric 3D)

Also, tank is made of SS material because of high sustainability at high stress, good elasticity and durability. Results show that at frequencies 28 kPa, Von Mises stresses are 3.92 × 108 Pa. Frequency increment is directly proportionate to the stress variation. It shows that pressure acoustic transient module can give more stringent result than HRA technique. The obtained results of this research re-evaluate the previous work done by Worapol Tangsopa.

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Fig. 5 a Von Mises stresses, Pa; b Velocity magnitude, m/s

4.2 CFD Approach Fluid flow and particle tracing approaches have been used for study of particle trajectory movements and correlation with object surfaces. Due to phase difference of negative and positive acoustic pressure, cavitation bubbles have been generated which creates turbulence in fluid media. For the acoustic streaming of the cavitation bubbles, bubbly fluid flow k-ε model and RANS turbulence type have been introduced. All these parameters effect the velocity of fluid. 1.23 × 103 m/s velocity magnitude has been achieved as a result.

Fig. 6 a Particle tracing; b Erosion rate, kg/(m2 s)

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Fluid flow is directly connected with particle trajectories. It can be seen clearly in particle tracing plot. Stokes drag law with Basset rarefaction effect and acoustophoretic radiation force give ample result of particle trajectory which shows the movement of water droplets and striking movement with object. Erosion phenomenon is directly proportional drag force and acoustophoretic radiation force. Finnie erosion model has been applied to get the erosion rate. In results, 1.1 × 10–5 kg/m2 s erosion rate has been found with considering 0.1 kg/s mass flow rate where in previous study, total 95,368 nos. of particles have been removed in 60 s with mass flow rate 7.5 L/min.

5 Conclusion According to obtained results, it is concluded that COMSOL Multiphysics has significant approach for giving more stable and stringent results with compare to other simulation approaches like Ansys-HRA technique. COMSOL Multiphysics is capable to couple many modules to get adequate results. Results may depend on many factors like, geometry of tank, liquid media, power parameters, flow rate, etc. As per results, acoustic pressure depends on the sound pressure level. Increase in frequency from to 28 kHz, density of sound pressure level as well acoustic pressure level have been increases by means of transducer. Here, achieved sound pressure level is 260 dB which is more enough than required sound pressure level of 150 dB and above for such kind of application and acoustic pressure is 9.64 × 106 Pa. Piezo reflectivity and input power are the affecting factors to sound pressure level. By varying different parameters related to piezo and power input, better results can be achieved. High-range frequency or multiple transducers have been applied to get sonic flow of particles. CFD approach has played vital role in getting generous result of particle trajectory. Due to cavitation effect, bubbly flow has been applied which gives improved velocity 1.23 × 103 m/s of fluid flow. Also, particle tracing module supported to predict motion of particles in fluid flow and striking of water droplet to the object has been observed which is used to understand erosion phenomenon and find out erosion rate. As a result, 1.1 × 10–5 kg/m2 s erosion rate has been obtained by using finnie model which is widely used. Erosion rate also depends of the mass flow rate of liquid, increment and decrement in mass flow rate, directly effects the result of the erosion rate because there is propionate relation in between.

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References 1. Tangsopa, W.: Development of an industrial ultrasonic cleaning tank based on harmonic response analysis. Ultrasonics, 91, 68–76 (2019) 2. Rokad, V., Pandya, D.H.: Development of 3D improved acoustic transient model for vibro cleaner using COMSOL multiphysics. Mater. Today Proc. (2020) 3. Tangsopa, W.: A novel ultrasonic cleaning tank development by harmonic response analysis and computational fluid dynamics. Metals 10(335), 1–18 (2020) 4. Tangsopha, W., Thongsri, J.: Simulation of ultrasonic cleaning and ways to improve the efficiency. In: IOP Conference Series—Earth and Environmental Science, ICERE 2018, vol. 159. IOP Publishing (2018) 5. Duran, F., Teke, M.: Design and implementation of an intelligent ultrasonic cleaning device. Intell. Autom. Soft Comput. 25(3), 441–449 (2019) 6. Zhong, L.: COMSOL multiphysics simulation of ultrasonic energy in cleaning tanks. In: COMSOL Conference (2015) 7. Introduction to COMSOL Multiphysics—User’s Guide, Version 5.3a (2017) 8. Bretz, N., Strobel, J.: Numerical simulation of ultrasonic waves in cavitating fluids with special consideration of ultrasonic cleaning. In: IEEE-Ultrasonics Symposium, pp. 703–706 (2005) 9. Lewis, J.P.: A 2D finite element analysis of an ultrasonic cleaning vessel: results and comparisons. Int. J. Modell. Simul. 27(2), 181–185 (2015) 10. DeAngelis, D.A.: Performance of PZT8 versus PZT4 piezoceramic materials in ultrasonic transducers. Sci. Dir. Phys. Proc. 87, 85–92 (2016) 11. Lais, H.: Numerical modelling of acoustic pressure fields to optimize the ultrasonic cleaning technique for cylinders. Ultrason. Sonochem. 45, 7–16 (2018)

Performance Analysis of Nonlinear Isothermal CSTR for Its Different Operating Conditions by Sliding Mode Controller Design N. S. Patil and B. J. Parvat

Abstract In this work, performance analysis of nonlinear isothermal continuous stirred tank reactor (CSTR) control using sliding mode controller is presented. An isothermal CSTR is a benchmark example of nonlinear chemical process which produces inverse response; hence, it is difficult to control using traditional controllers. From steady-state characteristics of CSTR, it is observed that it shows inverse response, negative response, and maximum yield point operating conditions. An observer-based SMC is designed for process models obtained from these three operating conditions and made their comparative study. For estimation of uncertainties and nonlinearities present in CSTR, an extended state observer is used. Designed controllers are tested in simulation study, and their control performance analysis is done on the basis of error indices from obtained results. Keywords Sliding mode control · Advanced control · Isothermal CSTR · Nonlinear process

1 Introduction An isothermal CSTR is highly nonlinear process and has inverse response as a remarkable property. The CSTRs with van de Vusse reaction are commonly used as benchmark examples of non-minimum phase processes for design of control strategies. Commonly, there are two types of the reactions found in the literature, i.e., isothermal [1–4] and non-isothermal [5, 6]. In case of the isothermal reaction, the product concentration is a control variable and inlet reactant feed flow rate is manipulating variable, whereas, in non-isothermal reactor, the product concentration and temperature in the reactor are control variables.

N. S. Patil (B) · B. J. Parvat (B) Maratha Vidya Prasarak Samaj’s Karmaveer Adv. Baburao Ganpatrao Thakare College of Engineering, Nashik 422013, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_19

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In this paper, observer-based sliding mode controller is designed for product concentration control in nonlinear isothermal CSTR for which traditional controllers are inadequate. Therefore, nonlinear controllers have more demands in controlling this kind of processes [7]. For isothermal CSTR control, in [8] and [9], stable estimation of the inversion approach is successfully developed and applied. However, such control schemes are not simple because it requires solving a set of partial differential equations. A simple and an efficient fuzzy predictive control approach is invented by [2] for isothermal CSTR. A robust H∞ control method was successfully applied to the isothermal CSTR by [3]. An adaptive nonlinear control scheme for non-minimum phase processes is based on approximate linearization and backstepping design procedures explained in [8] for isothermal CSTR with van de Vusse reaction. In this work, SMC is designed from estimated states of extended state observer for product concentration control in isothermal CSTR. In this control system, initially extended state observer is independently designed and its stability is proved by Popov criterion [10]. The overall closed-loop stability in connection with the observer is explicitly guaranteed in [11]. In addition, the effects of uncertainties, nonlinearities, and immeasurable disturbances are compensated by enhancing the SMC with extended state observer. The most important advantage of this control scheme is, it yields smooth control input and overcomes the chattering problem occurred in conventional SMC. The rest of paper is organized as, in Sect. 2, nonlinear isothermal CSTR process is described in detail. An observer design with its stability is explained in Sect. 3. In Sect. 4, design of SMC from estimated states of observer is explained. Section 5 presents closed-loop results and its comparative control performance analysis in simulation study following by conclusions in Sect. 6.

2 Process Description: Isothermal CSTR In this process, concentration of component B is a control variable and inlet flow rate is manipulated variable which is shown in Fig. 1. The process is described by material balance equations concerning C A and C B only. F dC A = (C AF − C A ) − k1 C A − k3 C A 2 dt V F dC B = − + k1 C A − k2 C B dt V

(1)

where C A and C B are the concentrations of components A and B, respectively. C AF is the concentration of substance A at the inlet (C AF > C A ), F is the inlet flow, V is constant reactor volume, and k1 , k2 , k3 are the reaction rate constants. From Eq. (1), steady-state concentrations C A and C B can be obtained as:

SMC for Nonlinear Isothermal CSTR

217

Fig. 1 Isothermal CSTR control

(a)

(b)

1.4

8

7

1.2

6

1

Bs

0.8

4

C

CAs

5

3

0.4

2

0.2

1 0 0

0.6

1

2

4

3

6

5

0 0

1

4

3

2

5

6

F/V

F/V

Fig. 2 Steady-state concentrations of A and B as function of F/V



CA =

F − k1 + V

k1 C AS CB = F + k2 V



 +

F k1 + V

2

 + 4k3

F V

 C AF

2k3 (2)

From Eq. 2, steady-state concentration of A and B with respect to the manipulated variable F/V is obtained for k1 = 0.8333, k2 = 1.6667, k3 = 0.1667, and C AF = 10 as shown in Fig. 2. From this, it is observed that the product A has positive gain variation. The product B has both positive and negative gain variations. Input variety occurs at places where F/V = 0.5714 and 2.8744, at which C B results same output 1.117. The steady-state gain is positive at the lower value of F/V and negative gain at larger value. The maximum concentration of B = 1.266 yields at F/V = 1.292. These steady-state operating conditions are tabulated in Table 1.

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Table 1 Steady-state operating conditions of isothermal CSTR Sr.no. Operating condition CA CB 1 2 3

Inverse response Negative response Maximum yield point

3 6.087 4.4949

1.117 1.117 1.266

F/V 0.5714 2.8744 1.2921

Hence, isothermal CSTR is highly nonlinear process with significant different input output characteristics. From its nonlinear material balance given in Eq. (1), a state space process model can be obtained by considering C B S be the desired set point for C B with corresponding system equilibrium at C A = C AS and it can be written as:   F F dx1 = − k1 + 2k3 C As + x1 − k3 x12 + (C AF − C AS − (k1 C AS + k3C 2AS )) dt V V   dx2 F F = k1 x 1 − k2 + x2 − (C Bs + (k1 C AS + k2C B S )) (3) dt V V where x1 = (C A − C AS ) and x2 = (C B − C B S ). Here, closed-loop control system is designed to control C B at desired set point C B S by manipulating inlet feed flow F at constant volume and temperature. Hence, only concentration of component B is taken as feedback.

3 State Observer The advantages of controller design based on state observer are, it compensates process uncertainties and nonlinearities. State observer is quite free of plant model and is easy to implement [12, 13]. The designed SMC gives chatter-free control input. For design of state observer, consider a generalized 2nd order system described as: x˙1 = x2 x˙2 = f (x, d(t)) + bu y = x1

(4)

where x1 , x2 are state variables, f (.) represents the system function, u is the control input, and d(x, t) is unknown bounded lumped uncertainty and matched disturbance. By consideration of d(x, t) is an extended state, extended state system for Eq. (4)can be written as x˙1 = x2 x˙2 = x3 + bu x˙3 = h y = x1

(5)

SMC for Nonlinear Isothermal CSTR

219

where h is the rate of change of uncertainty, and it is assumed to be unknown bounded function and x3 is an extended state. The state observer can be designed for the system Eq. (5) as: z˙ˆ 1 = zˆ 2 + β1 g1 (eo ) zˆ˙ 2 = zˆ 3 + β2 g2 (eo ) + bu z˙ˆ 3 = β3 g3 (eo )

(6)

where eo = y − yˆ = x1 − zˆ 1 and zˆ 3 is an estimate of the uncertainty. βi are the observer gains obtained by pole placement method and gi (.) defined as set of nonlinear gain functions satisfying eo gi (eo ) > 0 ∀eo = 0 and gi (0) = 0. The proper selection of gi (.) and their related parameters estimates zˆ i and converges to states of the systems xi (where i = 1, 2, 3). The generalized expression for this is given by [13],  |eo |αi sgn(eo ) |eo | > δ gi (eo , αi , δ) = (7) eo |eo | ≤ δ δ 1−αi where α ∈ (0, 1), δ > 0. The nonlinear function Eq. (7) yields high gain when eo is small and small gain for eo is large. The value of δ is chosen as very small so that it limits the gain around the origin and defines the range of eo corresponding to high gain. The nonlinear function is incorporated into the observer, in order to improve the transient response of the estimation error and decreases the observer sensitivity to model and external disturbances, such as described in [13] and [11]. An observer given in Eq. (6) can take the form of conventional Luenberger observer if one can choose αi as unity and gives gi (eo ) = eo .

3.1 Stability Analysis of State Observer Stability analysis for above-mentioned extended state observer is well described by [10] using Popov criterion. It is discussed in detail as below. Let e˙i = xi − zˆ i , i = 1, 2, 3. The observer estimation error is defined as e˙1 = e2 − β1 g1 (eo , α1 , δ) .. . e˙2 = e3 − β2 g2 (eo , α2 , δ) e˙3 = −β3 g3 (eo , α3 , δ)

(8)

For simplicity, gi (eo , αi , δ) is noted as gi . Choosing the unity values αi for all components and adding and subtracting the term βi ei to the right side of (8) and it can be expressed as e˙ = Ae + bu (9)

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where u = −ϕ(e) is a function of error and ϕ(e) is defined as ϕ(e) = gi − ei and matrix A and b are given as ⎡

⎤ ⎡ ⎤ −β1 1 0 β1 A = ⎣−β2 0 1⎦ b = ⎣β2 ⎦ β3 −β3 0 0 βi must be computed such that A is Hurwitz and (A, b) is controllable. By applying Popov criterion, it is necessary that ϕ(e) satisfies the sector condition. It is said that the function belongs to sector[0 k] if ϕ(e)[ke − ϕ(e)] ≥ 0, ∀t ∈ R+ and ∀e ∈ Γ ∈ R

(10)

where Γ is the region of the error which guarantees the stability of the observer. In this case, condition is fulfilled if 0 < δ < 1 and k ≥ 1/(δ 1−α − 1). Then, δ is taken from this range. According to the Popov criterion, the system (8) is absolutely stable if there is η ≥ 0 (not corresponding to eigenvalue of A) such that 1 + Re[G( jω)] − ηωIm[G( jω)] > 0, ∀ω ∈ R k

(11)

with suitable selection of parameters (αi , δ) and βi , the state observer will have asymptotic stability SMC is designed from estimated state vector and it explained in the next section.

4 Controller Design The closed-loop control system block diagram is shown in Fig. 3 which clarifies the design procedure and control structure. For design of the sliding mode controller linearized process model between C B (control variable) and F/V (manipulating variable) is obtained. Method for obtaining linearized state space model from material balance equations is explained below.

Fig. 3 Closed-loop control system

SMC for Nonlinear Isothermal CSTR

221

4.1 State Space Model of Isothermal CSTR Process Consider material balance equations of CSTR stated in Eq. (1) at constant temperature and volume as: F dC A = (C AF − C A ) − k1 C A − k3 C A 2 dt V F dC B = − + k1 C A − k2 C B f2 = dt V f1 =

By linearizing the above nonlinear model equations, the state space model can be written in the form: x˙ = Ax + Bu y = Cx T

where the x = [x1 x2 ]T = (C A − C AS ) (C B − C B S ) and u = F/V are state variables and system input, respectively. Matrix A and B are derived as A=

∂ f1 ∂ x1 ∂ f2 ∂ x1

∂ f1 ∂ x2 ∂ f2 ∂ x2

B=

 ∂ f1  ∂u ∂ f2 ∂u

and it gives ⎡

⎤ F − − k − 2k C 0 1 3 AS ⎢ ⎥ A=⎣ V ⎦ F k1 − − k2 V



C AF − C AS B= −C B S



From this three linearized state space models are obtained for three steady-state operating conditions mentioned as in Table 1. For design of SMC, obtained state space models are transformed into controllable canonical form by using transformation matrix [14]. The process models obtained for three cases are given as below. 1. Case-I: F/V = 0.5714, steady-state concentrations are C A = 3, and C B = 1.117 State space model is 

−2.4048 0 A= 0.8333 −2.2381



Controllable canonical form   0 1 A= −5.3821 −4.6432









7 B= −1.117

−1.1174 B= 3.1475

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2. Case-II: F/V = 2.8744, steady-state concentrations are C A = 6.087, and C B = 1.117 State space model is 

−5.7367 0 A= 0.8333 −4.5411



Controllable canonical form   0 1 A= −26.0417 −10.2734



3.913 B= −1.117





−1.1168 B= −3.1458



3. Case-III: F/V = 1.2921, steady-state concentrations are C A = 4.4949, and C B = 1.266 State space model is  A=

−3.6237 0 0.8333 −2.9588



Controllable canonical form   0 1 A= −10.7218 −6.5825

 B=

 B=

7 −1.117



0 −1.2658



4.2 Sliding Mode Controller From Eq.(3), x1 = (C A − C AS ) and x2 = (C B − C B S ) where C AS and C B S are desired concentrations of component A and B. In this application, nonlinear C B is control variable. Hence, sliding mode controller is designed in such way that x2 converges to zero. Let sliding surface for design of SMC is taken as σ = Sx

(12)

where S = [c 1] and |c| > 0. Since the estimated state vector by observer zˆ is converges to plant state vector x. Moreover, the observing sliding surface be σˆ = S zˆ Therefore, we have σ˙ˆ = cˆz 2 + f (ˆz , t) + bu + zˆ 3

(13)

SMC for Nonlinear Isothermal CSTR

223

where zˆ 3 = d(x, t) extended state estimate by observer. Control law is computed from Lyapunov function as 1 2 σˆ 2 V˙ = σˆ σ˙ˆ V =

If V > 0 then V˙ < 0 and vice versa, control u converges state variables at origin and closed-loop control system is stable. σˆ σ˙ˆ = σˆ (cˆz 2 + f (ˆz , t) + bu + zˆ 3 ) = −ρσ 2

(14)

Therefore, u=−

1 (ˆz 3 + cˆz 2 + f (ˆz , t) + ρσ ) b

(15)

where ρ is positive and real.

5 Simulation Study In this section, obtained control low in Eq.(15) is implemented and tested for abovediscussed three cases in simulation study. For simulation, the nominal parameters of the isothermal CSTR are taken as k1 = 0.8333, k2 = 1.6667, k3 = 0.1667, V = 1, and C AF = 10. For controlling of CSTR desired output is taken as:

CBS

⎧ ⎪ 0 ≤ t ≤ 100 ⎨1 = 0.9 100 ≤ t ≤ 200 ⎪ ⎩ 1 200 ≤ t ≤ 300

In this test, three different controllers are designed from state space models obtained for these three mentioned cases. The designed controllers are tested for nonlinear isothermal CSTR and made their comparative control performances analysis. The parameter values of controller are taken as c = 2, ρ = 3, β = [12 21 10], α = [1 1 1], and δ = 10−10 . From obtained results shown in Fig. 4, it is observe that designed controller gives stable and desired output. The comparative error performance indices of controller are tabulated in Table 2 for three cases. From this it is observed that controller designed from Case-II model gives best performance. The robustness of the controller is tested for ±20% variation in C AF and ±10% variations in reaction rate constants. Results for C AF and reaction rate constants variations are shown in Fig. 5 and Fig. 6, respectively. From Fig. 5, it is observed that for increasing of C AF , setpoint is achieved fast with increased overshoot

224

Fig. 4 Closed-loop response for nominal parameters value of CSTR

N. S. Patil and B. J. Parvat

SMC for Nonlinear Isothermal CSTR

Fig. 5 Closed-loop responses for ±20% variation in C AF

225

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Fig. 6 Closed-loop responses for ±10% variation in reaction rate constant

SMC for Nonlinear Isothermal CSTR Table 2 Error performance indices Case-I IAE ISE ITAE ITSE

1.902 0.3273 87.9 2.866

227

Case-II

Case-III

0.5136 0.1285 21.23 0.4726

0.5528 0.1329 23.11 0.5035

and gives stable closed-loop response. Similarly, from Fig. 6, it is observed that for variations in reaction rate constants gives stable desired output. Hence, SMC controller is robust for disturbance changes like inlet concentration C AF and reaction rate constants.

6 Conclusion In this way, observer-based sliding mode controller is designed and tested for product concentration control system in nonlinear isothermal CSTR process using process models obtained at different operating steady-state conditions. The comparative performance control analysis of these three controllers is made from obtained simulation results and summarized it in Table 2. From this, it is found that controllers designed from process models of negative response and maximum yield point conditions are gives better performance as compared to inverse response condition. It is also conclude that, for design of control system for nonlinear isothermal CSTR, any process model obtained for different operating condition can be used. It observed that, observer-based sliding mode controller is robust, chatter-free, and the best choice of controller for controlling of nonlinear isothermal CSTR. From obtained simulation results, it is found that designed controller gives satisfactory results and proved its applicability for nonlinear processes control.

References 1. Chen, C.T., Peng, S.T.: Design of a sliding mode control system for chemical processes. J. Process Control 15(5), 515–530 (2005) 2. Kuure Kinsey, M., Cutright, R., Bequette, B.: Computationally efficient neural predictive control based on a feedforward architecture. In: IEEE American Control Conference (2006) 3. Tsai, S.H.: Robust h∞ control for Van de vusse reactor via T-S fuzzy bilinear scheme. Expert Syst. Appl. 38(5), 4935–4944 (2011) 4. Kuntanapreeda, S., Marusak, P.M.: Nonlinear extended output feedback control for CSTRs with van de vusse reaction. Comput. Chem. Eng. 41, 10–23 (2012) 5. Akesson, B.M., Toivonen, H.T.: A neural network model predictive controller. J. Process Control 16(9), 937–946 (2006)

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6. Graichen, K., Hagenmeyer, V., Zeitz, M.: Design of adaptive feedforward control under input constraints for a benchmark CSTR based on a BVP solver. Comput. Chem. Eng. 33(2), 473–483 (2009) 7. Parvat, B.J., Patre, B.M.: Fast terminal sliding mode controller for square multivariable processes with experimental application. Int. J. Dyn. Control 5, 1139–1146 (2017) 8. Wu, W.: Stable inverse control for nonminimum-phase nonlinear processes. J. Process Control 9(2), 171–183 (1999) 9. Perez, H., Ogunnaike, B., Devasia, S.: Output tracking between operating points for nonlinear processes: Van de vusse example. IEEE Trans. Control Syst. Technol. 10(4), 611–617 (2002) 10. Erazo, C., Angulo, F., Olivar, G.: Stability analysis of the extended state observers by Popov criterion. Theoret. Appl. Mech. Lett. 2(4), 1–4 (2012) 11. Talole, S., Kolhe, J., Phadke, S.: Extended-state-observer-based control of flexible-joint system with experimental validation. IEEE Trans. Ind. Electron. 57(4), 1411–1419 (2010) 12. Qian, J., Xiong, A., Ma, W.: Extended state observer-based sliding mode control with new reaching law for PMSM speed control. Math. Probl. Eng. 1–10 (2016) 13. Wang, W., Gao, Z.: A comparison study of advanced state observer design techniques. IEEE Am. Control Conf. 6, 4754–4759 (2003) 14. Ogata, K.: Modern Control Engineering. Prentice Hall, New Jersey (2002)

Production Inventory Model for Deteriorating Items with Hybrid-Type Demand and Partially Backlogged Shortages Sushil Bhawaria and Himanshu Rathore

Abstract In present article, we have developed a production-inventory model specially focusing on deteriorating with constant rate of deterioration. To put a mark on deteriorating nature, we have also focused on preservation technologies. In the model, demands are accepted according instant level of stock and the selling price. A hybrid function is used to show the variations of demand with respect to selling price. In case of stock out situation, the demands are partially backlogged. The total study is carried under the effect of inflation. The main parameters are production cost, deterioration cost, holding cost, lost sales cost, backlogging cost, and preservation technology. The numerical solution and example are provided to finding this paper, and sensitive analysis of the optimal solutions with respect to main parameters is carried out using the software mathematica-7.0. Keywords Production · Deterioration · Hybrid-demand · Shortages · Inflation

1 Introduction Jaggi and Verma [1] have evolved a two-warehouse inventory model with a linear trend in demand under the inflationary conditions. Lin [2] has developed an EOQ model for items with imperfect quality and quantity discounts under lot-splitting shipment considering this phenomenon appears in real life when the buyer is powerful. Chaudhary et al. [3] investigated an integrated supply chain model with an exponential demand rate, inflation under preservation prime target of this study is to drive the optimal numbers of deliveries with single-manufacturer and single distributor when the integrated joint total cost minimized on the world level. Patel and Parech [4] have developed a two-warehouse an inventory model under time-varying holding cost and linear demand under inflation and permissible delay in payments. Kumar and Kumar [5] have estimated a two-warehouse inventory model

S. Bhawaria · H. Rathore (B) Manipal University Jaipur, Jaipur, Rajasthan 303007, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_20

229

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for non-instantaneous deteriorating items underneath the impact of inflation and demand is stock-dependent. For more research review we can go through following Table 1, in which we have bifurcated the work of different researchers on the basis of various important parameters like demand rate, deterioration, shortages, etc. In present article, we have focused on production inventory model for deteriorating items. For which paper is divided into seven different sections. The assumptions and Table 1 Literature review based on important parameters of inventory modeling References

Demand depend

Deterioration

Shortages

Inflation

Preservation technology

Singh and Rathore [6]

Stock

Non-instantaneous

Partially

∗

∗

Jaggi et al. [7]

Per unit time

–

–

–

–

Mishra et al. [8]

Stock and Selling

Constant

Fully and partially

–

∗

Rasel [9]

Ramp type

Non-instantaneous

Fully

–

–

Sahu et al. [10]

Price

Constant

Fully

–

∗

Shaik et al. [11]

Price and Stock

Non-instantaneous

Fully

–

–

Arif [12]

Selling price

Non instantaneous

Partially

–

–

Mashud [13] Stock and price

Constant

Partially

–

∗

Mishra et al. [14]

Selling price

Controllable

–

–

∗

Rathore [15]

Advertising and selling price

Constant

–

–

∗

Rathore [16]

Advertising and selling price

Constant

–

–

–

Shah et al. [17]

Selling price, Constant time, Stock

–

–

∗

Soni and Suthar [18]

Stochastic

Non-instantaneous

Partially

–

–

Khan et al. [19]

Price and stock

Instantaneous

Fully and partially

–

–

Mashud [20] Price and stock

Instantaneous

Fully backlogged

–

–

In present paper

Non-instantaneous

Partially and fully backlogged

∗

∗

Hybrid type

Production Inventory Model for Deteriorating Items …

231

notations are described in second section. In third and fourth section, mathematical formulation and the cost calculation are well presented. The optimality conditions, numerical verification and sensitivity analysis are described in remaining sections fifth, sixth, and seventh, respectively.

2 Assumptions and Notations Here are main notations and assumption which we used in mathematical formulation of production inventory model.

2.1 Assumption • Production rate is stationary and deterministic. • Production rate is higher than stock dependent demand rate. • Demand rate is taken as directly related to instantaneous stock level and selling price. As given below function: f (p, I(t)) = (D(p) + aI i (t)), where a > 0 and – D( p) = τ (x1 − yp) + (1 − τ )x2 p −γ ; where0 ≤ t ≤ 1, x1 > 0, x2 > 0, y > 0, xy1 ≥ p and y > 1. • • • • • •

Lead time is trivial and zero. Deterioration rate is constant in nature. Preservation technique is used to control the rate of deteriorating. Effect of inflation is considered. Horizon of planning is comprehensive. Partially backlogged shortages are taken in to consideration.

2.2 Notation • • • • • • • •

P: The production rate θ : The constant rate of deterioration r: The inflation rate δ: The backlogging rate D(p): The price dependent hybrid demand function p: The selling price per unit C 1 : The unit production cost of deteriorating items h i : The unit holding cost (per unit for i = 1, 2 according to time t ∈   per time) (0,T p ) for i = 1 and for time t ∈ T p , TH i = 2 • C i : The unit lost sale cost • C B : The unit backlogging cost

232

• • • • • • • • • • • • •

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Cθ : The unit cost of deterioration ξ : The preservation technology cost per time unit A: Setup cost PC: Production cost DC: Deterioration cost HC: Holding cost LSC: Lost sales costs BC: Backlogging cost PTC: Preservation technology T: Total time T p : Time at which production stops T H : The time at which inventory level is zero I i (t): Inventory level; i = 1, 2, 3.

3 Mathematical Model Formulation In the present production inventory system, every cycle starts in the first opening market and stops in the last closing market. The demand rate is increasing with stock and decreasing with the selling price of the items. At the initial stage, the production starts at time t = 0 after some time t = T p , when inventory reaches its maximum level, the production unit stops the process. At times t = TH inventory level becomes zero, due to which shortages start occurring of unfulfilled demands are backlogged some lost. To fulfill the backlogged demands production process began again at time t = T, so we have focused on total cycle length T. During the period (0, T p ) and (T p , TH), an inventory declines because of deterioration as well as demand. The whole functioning of inventory is given in Fig. 1 and also described by the following differential equations: dI1 + (θ + a)I1 = P − D( p) dt

(1)

dI2 + θ I2 = −D( p) + (a I2 ( p)) dt

(2)

dI3 = −(D( p) + a I3 (t))e−δt dt

(3)

Solving (1), (2) and (3), respectively, and applying boundary conditions I 1 (t = 0) = 0; I 1 (t) = I 2 (t) at t = T p ; I 2 (t = T H ) = 0 = I 3 (t = T H ), we get I1 =

P − D( p) (D( p) − P) −(θ+a)Tp + e θ +a θ +a

(4)

Production Inventory Model for Deteriorating Items …

233

Fig. 1 Inventory functioning in production inventory system

D( p)  (θ+a)(TH −t)  −D( p) + e (θ + a) (θ + a)  D( p)  a (e−δt −e−δTH H +H H H h ) eδ −1 I3 = a    P −1   Tp = log (θ + a) P − D( p) 1 − e(θ+a)TH I2 =

(5) (6)

(7)

4 Cost Calculation The calculation of main cost parameters is given as below: I. Setup cost = A II.

(8)

Production cost PC = PC p T p

III.

Holding cost  HC = h 1

 P − D( p) 1  1 − er Tp θ +a r

(9)

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   1 D( p) − 1 1 − e−(θ+a+r )Tp + θ +a θ +a +r    D( p)e−(θ+a)Tp  −r Tp  D( p)  −r TH −r T p −r TH e e + + h2 −e −e r (θ + a) (θ + a)r (10) IV.

Deteriorating cost 

D( p) −r Tp D( p) P Tp − e + r r

−r Tp

P − D( p) e D( p) − P e−(θ+a+r ) −a − − θ +a r θ +a θ +a +r



D( p) − P 1 1 P − D( p) + + r θ +a θ +a (θ + a + r )    D( p)  −r TH D( p)  −r TH e e − e−r Tp − a − e−r Tp + r (θ + a)r

−(θ+a)Tp

   −r TH  e D( p) −r T p e − −e θ +a r

DC = Cθ

V.

Loss sales costs per cycle LSC   −1  −(r +a)T LSC = Ci D( p)eaTH e − e−(a+r )TH (a + r ) −(a+r +δ)T

 e − e−(a+r +δ)TH + (a + r + δ)

VI.

(11)

(12)

Backlogging costs per cycle   −C B D( p) 1  −r TH e − eaTH −(a+r )T a a +r   1  −r T e − e−r TH + r

BC =

VII.

(13)

Preservation cost PTC = ξ TH

4.1 Total Cost Total cost function includes various cost parameters described given below:

(14)

Production Inventory Model for Deteriorating Items …

235

TC = (1/T )[ HC + DC + PC + BC + LSC + PTC]

(15)

5 Optimality of the Model To minimize total cost, we differentiate cost TC (ξ , T, TH ) w.r.t. to T, TH and ξ and for the optimal value necessary conditions are   ∂TC T, TH, ξ = 0, ∂T

  ∂TC T, TH, ξ = 0, ∂ξ

  ∂TC T, TH, ξ =0 ∂ TH

So, by differentiating (15) with respect to suitable decision variable, we formed above mentioned necessary conditions. Solution of these equation gives us the optimal values, provided the determinant of principal minor of hessian matrix are positive definite, i.e., det(H1)> 0, det(H2)> 0, det(H3))> 0 where H1, H2, H3, are the principle minor of the Hessian matrix. Hessian matrix of the total cost function is as follows. ⎡ ⎢ ⎣

∂ 2 TC ∂ 2 TC ∂ 2 TC ∂ξ 2 ∂ξ ∂ TH ∂ξ ∂ T ∂ 2 TC ∂ 2 TC ∂ 2 TC ∂ TH ∂ξ ∂ TH ∂ T ∂ TH2 ∂ 2 TC ∂ 2 TC ∂ 2 TC ∂ T ∂ξ ∂ T ∂ TH ∂T 2

⎤ ⎥ ⎦

6 Numerical Verification Taking suitable numerical values (based on existing literature review) of different parameters in proper units, are given below, for calculating optimal values of TC, T H , T and ξ ; The production rate P = 352 unit per month, the backlogging cost cb = 1, holding costs h1 = 1 and h2 = 1.5 per month, the unit lost sale cost ci = 3, the backlogging rate δ = 0.5, the inflation rate r = 0.01, the price dependent demand D(p) = 62.0051, the unit cost of deterioration cθ = 1.2, per month and parameters a = 0.05, x 1 = 600, x 2 = 300, τ = 0.1, y = 40, γ = 1.5, p = 15.56, respectively. And b = 0.0025, the unit production cost cp = 1.5. The optimal values of ξ ∗ ,T ∗ , and TC∗ have been calculated from equations from below. h 2 = 1.5, cθ = 1.2, a = 0.05, θ1 = 0.05,

P = 352, b = 0.0025,

δ = 0.5, h 1 = 1, r = 0.01, ci = 3, cb = 1, ∗

ξ = 48125.19,

TH∗

∗

D( p) = 62.0051, c p = 1.5,

= 121.77, T = 163.72, TC∗ = 1.00165 × 107

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S. Bhawaria and H. Rathore

Fig. 2 Convexity graph of TC w.r.t. T H and ξ

Note: Superscript * denotes the optimal values of respective parameter.

6.1 Convexity Graph of Total Cost Functions To show the optimality of the model we have taken help of 3D graphs. So the convexity graphs of TC with respect to other decision variables are given as Figs. 2, 3 and 4.

6.2 Sensitivity Analysis To sensitize the present study, we have carried out a sensitivity analysis by varying values of some important parameters. The numerical observation is present in Table 1. The variation in optimal total cost function with respect to small changes in various important parameters are well described in Table 2.

Production Inventory Model for Deteriorating Items …

Fig. 3 Convexity graph of TC w.r.t. T and ξ

Fig. 4 Convexity graph of TC w.r.t. T H and T

237

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Table 2 Sensitivity analysis Parameters

Change

ξ∗

TH*

T*

TC* × 108

a

0.055

51,210.1

110.508

153.568

7.57118

0.050

48,125.2

121.771

163.717

7.27057

0.049

47,489.1

124.446

165.976

7.20638

0.55

50,440.7

114.446

155.987

7.1315

0.50

48,125.2

121.771

163.717

7.27057

0.49

47,647.3

123.375

165.407

7.30026

0.052

46,777.3

120.797

162.024

7.02221

0.050

48,125.2

121.771

163.717

7.27057

0.040

56,565.4

127.429

173.712

8.85921

0.015

48,125.2

121.771

163.717

7.27057

0.010

48,125.2

121.771

163.717

7.27057

0.005

48,125.2

121.771

163.717

7.27057

352

48,125.2

121.771

163.717

7.27057

350

47,728.7

121.486

163.222

7.19749

342

46,153.0

120.336

161.225

6.90787

62.0051

48,125.2

121.771

163.717

6.90787

60.0051

48,826.4

123.407

166.578

7.02590

59.0051

49,189

124.251

168.064

6.90433

650

48,826.4

123.407

166.578

7.02590

600

49,189.0

124.251

168.064

7.27187

590

49,560.0

125.114

169.589

6.78307

350

48,826.4

124.251

168.064

7.27187

300

49,560

125.114

169.589

6.78307

290

49,939.7

125.996

171.154

6.66365

14.56

50,328.6

126.897

172.762

6.54129

13.56

51,134.9

128.764

176.113

6.30294

12.56

51,553.3

129.731

177.86

6.18387

δ

θ1

r

P

D(p)

x1

x2

p

7 Conclusion In this paper, we developed a production inventory model and focused on deteriorating with constant rate of deterioration and also focused on preservation technologies. The optimal values of total cost function and various decision variables like holding cost, deteriorating cost, backlogging cost, loss sale cost are calculated by using software mathematic 7.0. The convexity of the function is presented which shows the optimality of the model. Numerical verification and sensitivity analysis or also done for possibility of the model. Further model can be extended by incorporate by the often useful inventory parameters.

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References 1. Jaggi, C.K., Verma, P.: Two-warehouse inventory model for deteriorating items with linear trend in demand and shortages under inflationary conditions. Int. J. Procure. Manag. 3(1), 54–71 (2010) 2. Lin, T.Y.: An economic order quantity with imperfect quality and quantity discounts. Appl. Math. Model. 34(10), 3158–3165 (2010) 3. Chaudhary, U., Singh, S.R., Sharma, V.: A supply chain model for deteriorating items under inflation with preservation technology (2014) 4. Patel, R., Parekh, R.U.: Inventory model for variable deteriorating items with two warehouses under shortages, time varying holding cost, inflation and permissible delay in payments. Int. Refereed J. Eng. Sci 3(8), 06–17 (2014) 5. Kumar, N., Kumar, S.: Inventory model for non â instantaneous deteriorating items, stock dependent demand, partial backlogging, and inflation over a finite time horizon. Int. J. Supply Oper. Manag. 3(1), 1168–1191 (2016) 6. Singh, S.R., Rathore, H.: A two warehouse inventory model with preservation technology investment and partial backlogging. Sci. Iran. 23(4), 1952–1958 (2016) 7. Jaggi, C.K., Cárdenas-Barrón, L.E., Tiwari, S., Shafi, A.: Two-warehouse inventory model for deteriorating items with imperfect quality under the conditions of permissible delay in payments. Sci. Iran. 24(1), 390–412 (2017) 8. Mishra, U., Cárdenas-Barrón, L.E., Tiwari, S., Shaikh, A.A., Treviño-Garza, G.: An inventory model under price and stock dependent demand for controllable deterioration rate with shortages and preservation technology investment. Ann. Oper. Res. 254(1–2), 165–190 (2017) 9. Rasel, S.: A deterministic inventory system with power distribution deterioration and full backlogged shortage. Asian J. Curr. Res. 2(3), 81–88 (2017) 10. Sahu, S., Gobinda, C.P., Das, A.K.: A fullly backlogged deteriorating inventory model with price dependent demand using preservation technology investment and trade credit policy. Int. J. Eng. Res. Technol. (IJERT) 06(06) (2017) 11. Shaikh, A.A., Mashud, A.H.M., Uddin, M.S., Khan, M.A.A.: Non-instantaneous deterioration inventory model with price and stock dependent demand for fully backlogged shortages under inflation. Int. J. Bus. Forecast. Mark. Intell. 3(2), 152–164 (2017) 12. Arif, M.G.: An inventory model for deteriorating items with non-linear selling price dependent demand and exponentially partial backlogging shortage. Ann. Pure Appl. Math. 16(1), 105–116 (2017) 13. Mashud, A., Khan, M., Uddin, M., Islam, M.: A non-instantaneous inventory model having different deterioration rates with stock and price dependent demand under partially backlogged shortages. Uncertain Supply Chain Manag. 6(1), 49–64 (2018) 14. Mishra, U., Tijerina-Aguilera, J., Tiwari, S., Cárdenas-Barrón, L. E.: Retailer’s joint ordering, pricing, and preservation technology investment policies for a deteriorating item under permissible delay in payments. Math. Probl. Eng. (2018) 15. Rathore: Inventory model for deteriorating item with advertising and quantity discount policies. Int. J. Manag. Appl. Sci. (IJMAS) 4(5), 36–38 (2018) 16. Rathore, H.: A preservation technology model for deteriorating items with advertisementdependent demand and trade credit. In:s Logistics, Supply Chain and Financial Predictive Analytics, pp. 211–220. Springer, Singapore (2019) 17. Shah, N.H., Chaudhari, U., Jani, M.Y.: Optimal control analysis for service, inventory and preservation technology investment. Int. J. Syst. Sci. Oper. Log. 6(2), 130–142 (2019) 18. Soni, H.N., Suthar, D.N.: Pricing and inventory decisions for non-instantaneous deteriorating items with price and promotional effort stochastic demand. J. Control Decis. 6(3), 191–215 (2019)

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19. Khan, M.A.A., Shaikh, A.A., Panda, G.C., Konstantaras, I., Cárdenas-Barrón, L.E.: The effect of advance payment with discount facility on supply decisions of deteriorating products whose demand is both price and stock dependent. Int. Trans. Oper. Res. 27(3), 1343–1367 (2020) 20. Mashud, A.H.M.: An EOQ deteriorating inventory model with different types of demand and fully backlogged shortages. Int. J. Logist. Syst. Manag. 36(1), 16–45 (2020)

Influence of Electric and Magnetic Fields on Rayleigh–Taylor Instability in a Power-Law Fluid Krishna B. Chavaraddi , Praveen I. Chandaragi , Priya M. Gouder , and G. B. Marali

Abstract Electrodynamic RTI (ERTI) with a poor conducting fluid in a thin layer of an incompressible two fluids bounded upper side by a boundary with more dense fluid and lower side with a rigid boundary subjected to linear stability is investigated in the present study. A dispersion relation for growth rate of ERTI using linear stability theory was developed. This relation accounting for the growth disturbance is also obtained. The impact of the consequences of both electric as well as magnetic fields along with surface tension, power-law fluid, layer thickness on the ERTI using the approximations established by Rudraiah et al. (Acta Mech 119:165, 1996 [1]) is investigated. It reveals that electric-field, magnetic-field, power-law fluid and layer thickness stabilizes the system, whereas bond number destabilizes the interface. Keywords Power-law fluid · ERTI · Porous layer · Magnetic field

1 Introduction The instability of boundary occurs between fluids with density variance when the external acceleration and mass gradient are acting in opposite direction [2, 3]. This circumstance is termed as Rayleigh–Taylor instability (RTI). In several areas for instance astrophysics [4], plasmas [5], inertial confinement fusion [6], etc. RTI plays very significant role. K. B. Chavaraddi S.S. Government First Grade College and P.G. Studies Center, Nargund 582207, India P. I. Chandaragi (B) · G. B. Marali KLE Technological University, Hubballi 580031, India e-mail: [email protected] G. B. Marali e-mail: [email protected] P. M. Gouder KLE Dr. M. S. Sheshgiri College of Engineering and Technology, Belagavi 590008, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_21

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This unsteadiness has attracted substantial attention theoretically (see Chandrasekhar [7] and Sharp [8]) as well as practically (see Kull [9]) due to the significance to understand the controlling and exploitation of various processes corresponding to basic chemical, physical and biological. Babchin et al. [10] deliberated the nonlinear RTI in a thin Newtonian fluid film, while wavelength is ample greater compared to the film thickness. Further, this supposition was relaxed by Brown [11] related to the wavelength and considered RTI in a viscous fluid possessing finite thickness by applying stokes and lubrication approximations [10]. In Newtonian fluids, by including the impact of viscosity stratification and oblique magnetic fields, Rudraiah et al. [1] prolonged the work of Brown [11]. The literature revealed above is mostly focused on Newtonian fluids. Rudraiah et al. [12] also deliberated the RTI of a non-Newtonian fluid layer of finite thickness layer. It reveals essence of dispersion relation is driven via both the reciprocal of characteristic length and non-Newtonian parameter, although film width quite affects nature of evolution of RTI. Though, the focus on non-Newtonian fluid by the researchers is not as much in comparison with Newtonian fluid. The electro-rheological RTI in composite couple-stress fluid and porous layer is examined by Rudraiah et al. [13]. Agoor and Eldabe [14] conferred RTI in a porous medium under impact of magnetic field at interface of underlying couplestress Casson-fluid flow. Awasthi and Srinivastava [15] discussed boundary among two dielectric and viscous fluids with effect of a oblique electric field when heat and mass transfer over the interface. Garai et al. [16] observed RTI equilibrium in incompressible non-magnetized dusty plasma which acts as non-Newtonian behavior with a practically tested prototype of shear drift rate depending on viscosity. It was conventional that non-Newtonian stuff has a vital part in RTI aside from velocity shear stabilization in small wavelength region. The importance of non-Newtonian quantities is enormous in greater velocity shear rate system. Rudraiah et al. [17] explored the impact on RTI by electric field in a layer of non-Newtonian fluid. The work of RTI in pseudoplastic fluids by Doludenko et al. [18], established on the basic properties of shear-thinning fluids as well Atwood number, their study visualizes that a direct three-dimensional numerical replication of mixing with numerous rheologies of two media then conquer mixing layer width and kinetic energy spectra. Further, on RTI Chavaraddi et al. [19] show dominance by magnetic field in couple-stress fluid. In Nasehi and Shirani [20]’s work, RTI phenomenon is studied by computing the evolution equation for varying bond numbers. The outcomes demonstrate that the evolution of the free surface thin film for pseudo plastic fluids is varies from that for Newtonian and dilatant fluids. Piriz et al. [21] have shown linear theory of incompressible RTI in elastic–plastic solid wedges is established in a linear elastic (Hookean) early stage, followed by a rigid-plastic phase. Chavaraddi et. al. [22] have discussed the impact of surface roughness on RT unsteadiness of superposed couple-stress fluids on the interface. Through various applications of power-law fluid in modern era, the study of ERTI in power-law fluid in existence of electromagnetic fields subjected to boundary and surface conditions (3.1)–(3.4) confined to approximations given by Rudraiah et al. [1] is the main objective of the present work. In order to attain, this paper is structured

Influence of Electric and Magnetic Fields on Rayleigh–Taylor …

243

as: the governing mathematical equations with boundary and surface coditions are given Sect. 2. Also, electrical conductivity, magnetic field, electric field, electric potential and distribution due to density of charges are accomplished in Sect. 2. In Sect. 3, the velocity distribution is established and by using linear stability analysis, and dispersion relation is analytically derived. The results are discussed in Sect. 4, and necessary conclusions are encountered.

2 Mathematical Model of the Problem A two-dimensional medium having width h with distinct fluids are considered in the physical configuration (see Rudraiah et al. [17]), as explained in Fig. 1. Here, g is acceleration and q = (u, v) velocity. The lower fluid is less dense in comparion with the density of fluid at the upper side (ρ p > ρ f ). This compound scheme contains two incompressible power-law fluids and densely packed porous layer which is detached by plane y = h. As a result of gravity, resultant acceleration acts on the above fluid (positive y-direction), therefore less dense fluid drives more dense fluid. The boundary within fluids remains static. That is completely flat which is perpendicular toward the resultant acceleration, till lighter (power-law) fluid has adequate pressure in controlling the heavy fluid in resistance to the ceiling. On other hand, small perturbations are enclosed to take place at the interface. Whenever there is a occurance of asymmetry, the chunks of boundary lie higher than normal; therefore, extra force out-of less dense fluid is experienced as it necessary to support as a result interface starts to raise in the spots. In portions wherever boundary drops by insignificant extent beneath the normal, it requires strained to maintain stability. Although the unsteadiness is even produced by the smallest amount perturbations. Suppose the pressure-gradient is

Fig. 1 Physical framework of the problem

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inverted, i.e., heavier fluid is supported by less dense one, then portions of the interface grow and dense fluid will reduced back to normal though lower parts tends to grow. This phenomena provide a revealing declaration that scheme is stable if the dense fluid propel lighter one and unstable when light fluid pushes heavy fluid and this unstable approach is called ERTI. To explore such instability in a rectangular co-ordinate system (x, y) accompanied by x-axis parallel to film along with y-axis normal to it and η(x, t) denotes boundary elevation within nano-structured porous media as well as a fluid thin film. For non-Newtonian fluid, the governing basic equations in consideration by poor electrical conductivity for a thin film is specified as (refer Rudraiah and Kaloni [23] and Rudraiah et al. [17]): ∇ · q = 0 ρ

D q = −∇ p + ∇ · τ + ρe E + μ0 ( J × H ) Dt ∂ρe + ( q · ∇)ρe + ∇ · J = 0 ∂t

Maxwell’s Equations ∇ · E =

ρe − → , ∇ × E = 0 or E = −∇ϕ, εe

(2.1) (2.2) (2.3)

   σ = σ0 1 + αh (C − C0 ) J = σ E,

(2.4) where J current density, E electric field, φ electric potential, p pressure, ρ fluid density, ρe mass of charges, εe dielectric constant, C concentration and σ electrical conductivity, σ0 electrical conductivity related to C 0 , μ fluid viscosity, α h volumetric expansion coefficient of σ , stress tensor τi expressed as     n−1 1 1 2    τi = k1  γ˙ : γ˙  γ˙ . 2 2 i i 

(2.5)

Here, k1 is consistency index, and γ˙ is strain rate expressed as  γ˙i, j =

 ∂q j ∂qi γ˙ . + ∂x j ∂ xi

(2.6)

The σ diverges in comparison with concentration C of Deuterium–Tritium (DT) as shown in Eq. (2.3), in addition to advection of concentration, we get d2 C =0 dy 2

(2.7)

Influence of Electric and Magnetic Fields on Rayleigh–Taylor …

245

subject to C = C0 at y = 0

(2.8a)

C = C1 over y = h.

(2.8b)

By employing the given constraints, Eq. (2.7) is solved and using that in Eq. (2.4)(iv), to attain σ = σ0 [1 + αy] ≈ σ0 eαy

(2.9)

here α = αh C/ h and C = C1 − C0 . Supposing frequency of charge distribution remains less comparing to corresponding relaxation rate related to electric field, consequently time-derivative ρe is not more than ∇ · (σ E) in Eq. (2.3), it indicates that ∂ 2φ ∂ 2φ ∂φ = 0. + +α ∂x2 ∂ y2 ∂y

(2.10)

By considering following conditions, the above equation has to be solved ϕ= ϕ = v0

v0 x about y = 0 h

(x − x0 ) through y = h h

(2.11) (2.12)

Due to the embedded electrodes, these conditions arise above boundaries and allows a linear deviation of ϕ and x. Stokes and lubrication approximations as stated in (Rudraiah[1]) are used, presuming the dense fluid in porous lining is nearly static as a result of creeping flow approximation, which leads to governing equations in region of thin film in subsequent way: ∂v ∂u + ∂x ∂y  

 ∂u m−1 ∂u ∂p ∂ 0= + k1   + μ2h σ f H02 u + ρe E x ∂x ∂y ∂y ∂y 0=

0=

∂p + ρe E y ∂x

In the case of fully developed flow, Eq. (2.10) reduces to

(2.13)

(2.14)

(2.15)

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∂ 2φ ∂φ = 0. +α ∂ y2 ∂y

(2.16)

By applying the boundary conditions specified by (2.11) and (2.12), solution of this equation is  v x0 −αy x− (1 − e ) . h 1 − eαh

φ=

(2.17)

Substituting Eq. (2.17) in Eq. (2.4)(a) that brings to ρe = −

v x0 α 2 e−αy h 1 − e−αh

(2.18)

and hence, ρe E x = −ρe

v 2 x0 α 2 e−αy ∂φ = 2 . ∂x h 1 − e−αh

(2.19)

3 Dispersion Relation To calculate a wavelength-frequncy equation, firstly, we find the velocity distribution starting with Eq. (2.14) by employing the appropriate following constraints are u = 0 along y = 0 αp ∂u = −√ u ∂y k v= p = −δη − γ

about y = h

(3.1) (3.2)

∂η towards y = h ∂t

(3.3)

∂ 2η E x2 η ± ε 0 ∂x2 h

(3.4)

along y = h.

Here, η = η(x, y, t) is elevation of interface. Then, Eq. (2.14) relating to above conditions, we obtain u = C1 CosM y + C2 SinM y +

where

∂p v2 α 2 e−αy 1  (3.5)

− M 2 k1 b(m) ∂ x h 2 (α 2 + M 2 )k1 b(m) 1 − e−αh

Influence of Electric and Magnetic Fields on Rayleigh–Taylor … C1 = −

247

∂p α2 v2 1

 + 2 2 M 2 k1 b(m) ∂ x h (α + M 2 )k1 b(m) 1 − eαh

1 (MCosMh + α p σ p SinMh)

⎫ ⎧ 2 ⎪ α2 ∂p ⎪ 1 ⎪ (MSinMh − α σ CosMh) v ⎪ ⎪ ⎪

 − p p ⎪ ⎪ ⎨ h 2 (α 2 + M 2 )k1 b(m) 1 − eαh M 2 k1 b(m) ∂ x ⎬ × ⎪ ⎪ ⎪ ⎪ (α p σ p − α)v 2 α 2 e−αh αpσp ⎪ ⎪ ⎪ ⎪

− ⎩ + 2 2 ⎭ 2 2 αh M k1 b(m) h (α + M )k1 b(m) 1 − e

C2 =

Let us integrate Eq. (2.13) w.r.t. y from y = 0 to y = h and make use of Eq. (3.5), we get h

∂2 p h3 ∂u dy = 3 ∂x M k1 b(m) ∂ x 2 0    MSinMh + (1 − CosMh)α p σ p ∗ (1 − CosMh) × Mh − SinMh − MCosMh + α p σ p SinMh

v(h) = −

(3.6)

Equation (3.3), using (3.6) and (3.4), becomes    M SinMh + (1 − CosMh)α p σ p h3 ∂η Mh − SinMh − = 3 (1 − CosMh) ∂t M CosMh + α p σ p SinMh M k1 b(m) 

 2 2 4 ∂ η εe E x ∂ η +γ 4 (3.7) × δ± h ∂x2 ∂x

In order to obtain growth rate, ω, solution of Eq. (3.7) is assumed in subsequent form η = η(y)eix+ωt

(3.8)

Here, wavenumber is  with amplitude of perturbation of the interface is η(y). Substituting the value from Eq. (3.8) into Eq. (3.7), the corresponding dispersion relation is obtained as    M SinMh + (1 − CosMh)α p σ p h3 Mh − SinMh − (1 − CosMh) M CosMh + α p σ p SinMh M 3 k1 b(m)  

2 εe v (3.9) × − δ ± 2 2 + γ 4 h

ω=

Reducing Eq. (3.9) as non-dimensionalized by using the following  γ ωk1 h v , v∗ = , , ∗ =  ω∗ = √ , h ∗ = √ δ h γδ γ /δ

σp M ∗ = Mh, σ p∗ = h

(3.10)

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to yield  2 1 ω =  (1 ± W e) − B 2

(3.11)

Here, 1 =

h3 M 3 k1 b(m)



 M − SinMh −

 M SinMh + (1 − CosMh)α p σ p (1 − CosMh) M CosMh + α p σ p SinMh μ2 σ f H 2

W e = εe v 2 /δh 3 indicates electric parameter, M = kh1 b(m)0 the Hartmann number, b(m) is a fitting constant, B = δλ2 /γ bond number. The positive or negative sign in front of We in above equation depends on whether potential difference is along or opposing the gravity. Subsequently, the potential difference is opposing gravity, we have preferred negative sign, and Eq. (3.11) reduces to the form  2 1 ω =  (1 − W e) − B 2

(3.12)

Here, b(m) must be picked in a means so that b(1) = 1. So, consider following cases for discussion. Case (a): b(m) = Case (b): b(m) =

m+1 . 2 1 m + 2 4

+

m2 . 4

In current work, we conferred case (a) only. Dispersion relation represented by Eq. (3.12) is numerically evaluated for different values of m, h, We, M, B and σ p with appropriate relation to flows corresponding to shear-thinning and shear-thickening also consequences are revealed in Figs. 2, 3, 4, 5, 6 and 7 for case (a).

4 Concluding Remarks The paper presents an innovative approach for studying the influence of electromagnetic fields on RTI in power-law fluid. The problem is modeled throughout the Cartesian co-ordinate system as usual. A mathematical formulation of this problem is made simple for the problem under Stokes and lubrication approximations by two infinite incompressible non-Newtonian power-law fluids viz., one below with rigid surface and other one above by fluid saturated porous layer due to great importance of the porous media. Using approximations on the power-law stress function and suitable boundary and surface conditions, we get the dimensionless dispersion relation (3.12). The dispersion relation Eq. (3.12) of case (a) for variation of -m, h, We, B and σ p is computed and presented the results graphically in Figs. 2, 3, 4, 5, 6 and 7. It is seen that the nature of growth rate is influenced by both characteristic length γδ and

Influence of Electric and Magnetic Fields on Rayleigh–Taylor …

249

m=0.3

0.02

0.5 0.7

0.01

ω 0.00 0.00

0.05 wavenumber, l

0.10

-0.01

Fig. 2 Different values of power-law index m for h = 5, α p = 0.1, σ p = 4, B = 0.02, M = 5 and We = 0.25

h=20

1.2

0.8

ω 0.4

0.0 0.00

0.05

16 12 8

4 0.10

0.15

wavenumber, l -0.4

Fig. 3 Variation of film thickness h if m = 0.5, α p = 0.1, σ p = 4, B = 0.02, M = 5 and We = 0.25

power-law index, m. That is, an increase in m is to decrease the growth rate as shown in Fig. 2. Power-law fluid has a stabilizing role in reducing the asymmetry of surface deflection. Figure 3 describes film thickness h influencing nature of growth rate of unsteadiness regime, it means there is an increase in growth rate corresponding to an increase in film thickness parameter h and thus, hereby showing that the system has destabilized due to increase in the fluid film thickness.

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We=0.0 0.03

0.02

0.25

0.01

0.5

ω

0.00 0.00

1.0

0.75 0.04

0.08

0.12

wavenumber, l -0.01

-0.02

Fig. 4 Varied the electric parameter We with fixed m = 0.5, h = 5, α p = 0.1, σ p = 4, B = 0.02 and M = 5

M=5 0.016 0.012 0.008

10

0.004

15

ω 0.000 0.00 -0.004

0.04

0.08

20 0.12

wavenumber, l

-0.008

Fig. 5 Different values of Hartmann number M with constant m = 0.5, h = 5, α p = 0.1, σ p = 4, B = 0.02 and We = 0.25

Figure 4 shows the wavelength increases, while the amplitude of the elevation decreases as the parameter, We increases, and hence, frequency shrinks, while related to conventional outcome is very steep for We in scale from 0.5 to 1.0. The impact of magnetic intensity on RT instability for distinct values of Hartmannnumber, M is depicted in Fig. 5. It envisage that ERTI can be reduced by M, and thereby increasing Hartmann ratio results in slight increase in critical wavenumber

Influence of Electric and Magnetic Fields on Rayleigh–Taylor … 0.04

251

B=0.04 0.03 0.02

0.02

ω

0.01

0.00 0.00

0.05

0.10

0.15

wavenumber, l

-0.02

Fig. 6 Varying Bond number B when m = 0.5, h = 5, α p = 0.1, σ p = 4, We = 0.25 and M = 5

σp=4

0.010

8 12

16

20

0.005

ω 0.000 0.00

0.05

0.10

0.15

wavenumber, l -0.005

-0.010 Fig. 7 Effect of porous parameter, α p with fixed m = 0.5, h = 5, α p = 0.1, B = 0.02, We = 0.25 and M = 5

and decrease in maximum evolution. We observe that magnetic field has a stabilization impact on RTI for particular values of input parameters due to increased value of M (Lorentz force to viscous force). Figure 6 demonstrates ω against  for variation of B increases in the range 0.01–0.04 using relation (3.12). The resulting outcomes are envisages the instability among fluids having  smaller compared to critical wavenumber ct are intensified whenever We > 0, which implies instability decreases with a reduction in the values of B. This indicates a growth in surface

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tension due to B is reciprocal of it. In addition, it is established that growth level (n) reduces a rise in surface tension and accordingly system attains stability. Further, it is shown that by increasing the bond number, the nonlinearity effect would be more pronounced. In this situation, the imbalance between the gravitational and surface tension forces makes the film unstable and drops are created. Therefore, the bond number has a destabilizing role on RTI due to bond number is reciprocal of surface tension. The implications of Darcy’s coefficents due to porous layer on the stability picture are depited in Fig. 7, it shows the variation of porous parameter σ p in the range from 4 to 20, and it is noted that the porous layer has stabilized the system. In conclusion, we have investigated ERTI in power-law fluid and in contrast, the stability of composite system is enhanced by ratios of W e , M, m and σ p , whereas h and B have a destabilizing influence on the same. Acknowledgements Authors (PMG, GBM, PIC) are greatful to KLE Society Management, University, and Principal for the generous help in carring out research work. Author (KBC) is thankful to DCE and UGC for the encouragement and everlasting help.

References 1. Rudraiah, N., Krishnamurthy, B.S., Mathad, R.D.: The effect of oblique magnetic field on the surface instability of a finite conducting fluid layer. Acta Mech. 119, 165 (1996) 2. Rayleigh, L.: Investigation of the character of the equilibrium of an incompressible dense fluid of variable density. Proc. Lond. Math. Soc. 14, 170–177 (1883) 3. Taylor, G.I.: The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. Proc. R. Soc. A: Math. Phys. Eng. Sci. 201(1065), 192–196 (1950) 4. Modestov, M., Bychkov, V. Marklund, M.: The Rayleigh-Taylor instability in quantum magnetized plasma with para- and ferromagnetic properties. Phys. Plasmas 16, 032106 (2009) 5. Faganello, M., Califano, F., Pegoraro, F.: Competing mechanisms of plasma transport in inhomogeneous configurations with velocity shear; the solar wind interaction with earth’s magnetosphere. Phys. Rev. Lett. 100, 015001 (2008) 6. Betti, R., Sanz, J.: Bubble acceleration in the ablative RTI. Phys. Rev. Lett. 97, 205002 (2007) 7. Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Oxford University Press, Oxford (1961) 8. Sharp, D.H.: An over view of Rayleigh-Taylor instability. Phys. D 12, 3 (1984) 9. Kull, H.J.: Theory of Rayleigh-Taylor instability. Phys. Rep. 206, 198 (1991) 10. Babchin, A.J., Frenkel, A.L., Levich, B.G., Shivashinsky, G.I.: Nonlinear saturation of Rayleigh-Taylor instability in thin films. Phys. Fluids 26, 3159 (1983) 11. Brown, H.C.: Rayleigh-Taylor instability in a finite thickness layer of a viscous fluid. Phys. Fluid A 1(5), 895 (1989) 12. Rudraiah, N., Sridharan, P., Bhargava, S.: Rayleigh-Taylor ins tability in a finite thickness layer of a non-Newtonian fluid. Appl. Mech. Eng. 5, 315 (2000) 13. Rudraiah, N., Kalal, M., Chandrashekara, G.: RTI at the interface between a porous layer and thin shell with poorly conducting couple-stress fluid. Int. J. Nonlin. Mech. 46, 57–64 (2011) 14. Agoor, B.M., Eldabe, N.T.M.: Rayleigh-Taylor instability at the interface of superposed couplestress casson fluids flow in porous medium under the effect of a magnetic field. J. Appl. Fluid Mech. 7(4), 573–580 (2014) 15. Kumar, A., Srinivastava: Study on electrodynamic RTI with heat and mass transfer. Sci. World J., 1–8 (2014)

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16. Garai, S., Banerjee, D., Janaki, M.S., Chakrabarti, N.: Stabilization of Rayleigh-Taylor instability in a non-Newtonian incompressible complex plasma. Phys. Plasmas 22, 033702 (2015) 17. Rudraiah, N., Chavaraddi, K.B., Shivakumara, I.S., Shankar, B.M.: Electrohydrodynamic Rayleigh-Taylor instability in a non-Newtonian fluid layer bounded above by a porous layer. Int. J. Appl. Mat. Eng. Sci. 1(2), 45–58 (2016) 18. Doludenko, A.N., Fortova, S.V., Son, E.E.: The Rayleigh–Taylor instability of Newtonian and non-Newtonian fluids, Phys. Scr. 91, 104006 (2016) 19. Chavaraddi, K.B., Awati, V.B., Gouder, P.M., Nandeppanavar, M.M.: The effect of the magnetic field on the Rayleigh-Taylor instability in a couple-stress fluid. Int. J. Appl. Mech. Eng. 23(3), 611–622 (2018). https://doi.org/10.2478/ijame-2018-0033 20. Nasehia, R., Shirani, E.: Evolution of thin liquid film for Newtonian and power-law nonNewtonian fluids. Sci. Iran. B 25(1), 266–279 (2018) 21. Piriz, A.R., Piriz, S.A., Tahir, N.A.: Stability boundaries for the Rayleigh-Taylor instability in accelerated elastic-plastic solid slabs. Phys. Rev. E 100, 063104 (2019) 22. Chavaraddi, K.B., Gouder, P.M., Kudenatti, R.B.: The influence of boundary roughness on Rayleigh–Taylor instability at the interface of superposed couple-stress fluids. J. Adv. Res. Fluid Mech. Therm. Sci. 75(2), 1–10 (2020) 23. Rudraiah, N., Kaloni, P.N.: Flow of Non-Newtonian Fluids, Encyclopaedia of Fluid Mechanics. Gulf Publishing Company, USA (1990) (Chapter 1, 9, 1)

An Eco-Epidemic Dynamics with Incubation Delay of CDV on Amur Tiger Jyoti Gupta, Joydip Dhar, and Poonam Sinha

Abstract Retrieving the tiger population in the world ecosystem is an essential requirement for maintaining ecological balance. Canine distemper virus (CDV) is a deadly infection found in the Amur tiger in the Russian Far East, one of the crucial causes of tiger extinction. We study the impact of the canine distemper virus on the Amur tiger and other tiger populations in the World. A four-compartment Amur tigerdomestic dog and wild carnivores delayed eco-epidemiological model is developed taking incubation period as delay parameter. The existence and boundedness of the solutions are derived. The basic reproduction number R0 is determined. The local and global stability of coexistence equilibrium are established. The system exhibits Hopf bifurcation at coexistence equilibrium for the critical value of incubation delay. Keywords Eco-epidemiological model · Stability · Incubation delay

1 Introduction The tiger population is rapidly decreasing in the worldwide ecosystems, with the estimated number of breeding females are approximately 1000 [1]. Canine distemper virus (CDV) is one reason behind these tiger’s mortality and the mortality of various carnivores. In recent years, this virus became a danger to Amur tigers in the Russian Far East and many tigers in India as many CDV cases have been reported [1]. The canine distemper virus is a contagious infection transmitted through aerosol droplets and exposure to contaminated body fluids. This virus is the most important cause of infection, and high mortality in Amur tiger and dog [2]. The radiotelemetry studies on four tigers show that the mean rate of predation of dogs and wild carnivores per tiger per year is 1.66 and 3.87 respectively [1]. Hence, this virus spread in tigers via predation of unvaccinated domestic dogs [2]. In addition to this, infected females J. Gupta (B) · P. Sinha S.M.S. Government Model Science College, Jiwaji University, Gwalior 474002, India J. Dhar ABV-Indian Institute of Information Technology and Management, Gwalior 474015, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_22

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transfer the infection to their dependent litters [1]. The majority of infected female Amur tiger does not contribute to population growth as they reach sexual maturity at 42–54 months with an inter-birth interval of 22 months [1] and CDV remain in species less than two months with either recovery or death [3]. Further, the incubation period of CDV is from 1–2 weeks to 4–5 weeks or even more [4]. In recent years, many mathematicians studied the dynamic behavior of systems where prey-predator both have infection[5]. The article [6] consider that healthy prey is more active than infected ones so that they are not easily catchable for a predator. Some mathematicians analyzed a model considering that infected predators are not capable of hunting healthy prey [7]. In addition to this, some researchers also considered that healthy predators got the infection through predation of infected prey and became infected [8]. Delay differential equation exhibits much more complicated dynamics than ordinary differential equation as time delay could become a reason for stability or instability of system [9]. Many mathematicians analyzed that how infection delay affects the prey-predator dynamics[6]. In the following Sect. 2, the mathematical model is developed and formulated. In Sect. 3 we analyzed the eco-epidemiological model and study that all solutions of the system are positive and lies in a bounded domain, in Sect. 4 basic reproduction number has been obtained, in Sect. 5 we classified the equilibrium states and obtained the conditions of their existence. In the next Sect. 6, we have done the mathematical examination of the delay model. Here, we will discuss the stability and bifurcation behavior of the model at coexistence equilibrium. In Sect. 7 we present a sensitivity analysis of reproduction number. Section 8 contains a numerical simulation of the result in support of the analytical findings. We will discuss and conclude the quantitative results in Sects. 9 and 10, respectively. In Sect. 11, we have discussed the future scope of our model.

2 Development of the Model This section contains the development of eco-epidemiological model is with following hypothesis: 1. The entire prey (Dog and wild carnivores) and predator (Amur tiger) population are divided into four parts, the healthy prey X 1 , the infective prey I1 , the susceptible predator X 2 and the infected predator I2 . 2. At time t, the total prey population N (t) = X 1 (t) + I1 (t) satisfies the logistic = r X 1 (1 − KN ) = r X 1 (1 − X 1K+I1 ), where K and r are differential equation dN dt described in Table 1. Due to the high mortality rate of CDV, the infected domestic dogs and wild carnivores die after few days of getting the infection. So we have not considered the birth term in infected prey class. 3. When predator population is absent, the disease is transmitted horizontally through infected prey to healthy prey with mass action incidence rate αS, where α is described in Table 1.

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4. CDV infected prey population do not rapidly show the symptom of the disease; there is a time lag (incubation period) between infection and the occurrence of the symptoms, which is considered a delay. 5. There is no recovery of infected prey from the CDV and dies with disease death rate μ1 . 6. The susceptible predator predates on healthy prey at predation rate of β with Holling-II functional response [11]. The susceptible predator and infected predator predate on infected prey at the rate of β1 and β2 with linear function response [11]. Moreover, an infected predator predates only on an infected prey because it weakens due to disease and incapable of hunting a healthier one. 7. We have assumed that the infected female Amur tiger does not contribute to population growth as CDV has an infectious period of fewer than two months with either recovery or death [10]. Again, since the susceptible Amur tiger feeds on CDV infected dogs, a portion (m 2 β1 X 2 I1 ) of newborn cubs are infected at birth and helps in the growth of the infected predator class and, the rest of the part (m 1 β1 X 2 I1 ) contributes to the growth of the healthy predator class. Here, m 1 and m 2 are conversion coefficients of infected prey to healthy predator and infected predator, respectively. In addition to this, healthy prey also helps grow a healthy predator class with a conversion coefficient m 3 . 8. A very few numbers of infected predators are recovered [3] naturally and transfer to the healthy predator class. Further, both healthy and infected predators die due to natural and disease death rate μ2 . All the above assumption leads to the following set of the non-linear autonomous differential equation:   ⎫ X 1 + I1 β X1 X2 ⎪ dX 1 = r X1 1 − − α X 1 (t − τ )I1 (t − τ ) − ,⎪ ⎪ dt K H + X1 ⎪ ⎪ ⎪ ⎪ dI1 ⎪ ⎬ = α X 1 (t − τ )I1 (t − τ ) − β1 I1 X 2 − β2 I1 I2 − μ1 I1 , dt dX 2 X1 X2 ⎪ ⎪ = m 1 β1 I1 X 2 + m 3 β − μ2 X 2 + ηI2 , ⎪ ⎪ ⎪ dt H + X1 ⎪ ⎪ dI1 ⎪ ⎭ = m 2 β1 I1 X 2 − ηI2 − μ2 I2 , dt

(1)

with initial condition: X 1 (0) ≥ 0, I1 (0) ≥ 0, X 2 (0) ≥ 0, and I2 (0) ≥ 0 for all t ≥ 0. The interpretation of all parameters of (1) is given in Table 1.

3 Positivity and Boundedness Theorem 1 The model (1) with initial condition always have positive solution, for all t ≥ 0. Proof We have used theory of differential equation and shown that the solution of model in t ∈ [0, τ ] can be written in the following form;

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Table 1 Interpretation of parameters of model 1 Parameter Interpretation r K α β

Per capita intrinsic growth rate Environmental carrying capacity Disease transmission coefficient Predation rate on susceptible prey by susceptible Amur tigers Predation rates on infected prey by susceptible and infected Amur tigers Conversion rate from infected prey to susceptible predator and infected predator Conversion rate from susceptible prey to susceptible predator Half saturation constant Total mortality rate of infected prey Total mortality rate of susceptible and infected Amur tigers Incubation period of CDV Recovery rate of infected predators

β1 , β2 m1, m2 m3 H μ1 μ2 τ η

X 1 (t) ≥

X 1 (0)

t

r 0 K

X 1 (0)  t r X1 0 ( K + α X1 +

dv + exp −

β X2 )du H +X 1

,

t I1 (t) ≥ I1 (0)ex p{−

(β1 X 2 + β2 I2 + μ1 )du}, 0

X 2 (t) ≥ X 2 (0) exp{−μ2 t}, I2 (t) ≥ I2 (0) exp{−(η + μ2 )t.} Similarly, this solution of (1) is positive for all the intervals [τ, 2τ ] . . . [nτ, (n + 1)τ ], ∀n ∈ N . Hence, by mathematical induction X 1 (t), I1 (t), X 2 (t) and I1 (t) are positive for all t ≥ 0. Lemma 1 All the non-negative solutions of the model are start and always remains in the bounded domain Δ = {(X 1 , I1 , X 2 , I2 ) : 0 ≤ X 1 (t) + I1 (t) + X 2 (t) + I2 (t) ≤ rK } for any initial value, where δ = min{r, μ1 , μ2 }. δ Proof We consider, V (t) = X 1 (t) + I1 (t) + X 2 (t) + I2 (t) and differentiate V (t) with respect to t,

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d V (t) X 1 + I1 β X1 X2 − β1 I1 X 2 − β2 I2 I1 + (m 1 + m 2 )β1 I1 X 2 = r X 1 (1 − )− dt K H + X1 m3β X1 X2 + − μ1 I1 − μ2 (X 2 + I2 ). H + X1

For, m 1 + m 2 , m 3 < 1 and δ = min{r, μ1 , μ2 .}, the above equation transformed to, d V (t) dt

+ δV (t) ≤ 2r X 1 −

r X 12 K

⇒

d V (t) dt

+ δV (t) ≤ r K .

We further solve the above expression and get, 0 < V (t) < V (0)e−δt + rδk . If we take t → ∞ and we have, 0 ≤ V (t) < rδk . Hence, all solutions of system (1) always remain in the bounded domain Δ.

4 Basic Reproduction Number The basic reproduction number R0 for the system (1), derived by next-generation matrix method [12], is given by the following expression: R0 =

H K αμ2 (m 3 β − μ2 ) >0 H m 3rβ1 (−μ2 (H + K ) + K m 3 β) + K μ1 (−m 3 β + μ2 )2

provided the following condition holds. m 3 β > max{ μ2 (HK+K ) , μ2 } =

μ2 (H +K ) . K

5 Equilibrium Classification and Existence The equilibriums of the proposed model are: (i) The trivial equilibrium: E 0 (0, 0, 0) always exists. (ii) The axial equilibrium: E 1 (K , 0, 0, 0) always exists. (iii) The predator-free equilibrium E 2 ( Xˆ1 , Iˆ1 , 0, 0) where, Xˆ1 = μα1 , Iˆ1 = exist provided, K α > μ1 . μ2 , (iv) The disease-free equilibrium E 3 ( X˙1 , 0, X˙2 , 0) where, X˙1 = m 3Hβ−μ 2 m 3 Hr (m 3 Kβ−(H +K )μ2 ˙ X2 = exist only when, 2

r (K α−μ1 ) α(r +K α)

K (−m 3 β+μ2 )

m3β >

μ2 (H + K ) K

(2)

(v) Through the method of isoclines [13], we see that the nontrivial equilibrium point E 4 (X 1∗ , I1∗ , X 2∗ , I2∗ ) exists if and only if following conditions hold:

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α X 1∗ > μ1 −

(3)

m 2 β2 β(α X 1∗ − μ1 )(η + μ2 ) r −α+ μ2

(4) (5)

6 Mathematical Analysis of the Model The jacobian of the system (1) at coexistence equilibrium E 4 (X 1∗ , I1∗ , X 2∗ , I2∗ ) is, ⎛

c11 − e−λτ d c12 − e−λτ c ⎜ e−λτ d c22 + e−λτ c V (E 4 ) = ⎜ ⎝ c31 c32 0 c42

c13 c23 c33 c43

⎞ c14 c24 ⎟ ⎟. c34 ⎠ c44

With the help of matrix row operation, With the help of matrix row operations, R2 → R2 + R1 , R2 → c34 R2 − c24 R3 , R3 → c44 R3 − c34 R4 , R1 → d23 R1 − c13 R2 , R2 → d33 R2 − d23 R3 , and R1 → (d33 d22 − d23 d32 )R1 − (d23 (c12 − ce−λτ ) − c13 d22 )R2 , respectively, we reduce the last matrix, ⎛

Π 0 ⎜ d33 d21 − d23 d31 d33 d22 − d23 d32 ⎜ V (E 4 ) = ⎝ d32 d31 0 c42

0 0 d33 c43

⎞ 0 0 ⎟ ⎟, 0 ⎠ c44

(6)

where, Π = (d23 c11 − e−λτ d − c13 d21 )(d33 d22 − d23 d32 ) − (d23 (c12 − e−λτ c) − c13 d22 )(d33 d21 − d23 d31 ), d21 = c11 c34 − c31 c24 , d22 = c34 (c22 +c12 ) − c24 c32 , d23 = = c44 c32 ∗− c34 c42 , d33 = c44 c33 − c34 c∗43 , c34 (c23 + c13 ) −∗ c24 c33 , d31 ∗= c∗ 44 c31 , d32 rX X +I β X 1∗ X 2∗ β X2 r X 1∗ βX , c13 = − H +X1 ∗ , and c11 = − K 1 + r (1 − 1K 1 ) + (H +X ∗ 2 − ∗ , c12 = − H +X 1 K 1) 1 c14 = 0, c21 = e−λτ d, c22 = −β1 X 2∗ − β2 I2∗ − μ1 , c23 = −β1 I1∗ , c24 = −β2 I1∗ , m β X∗ X∗ m β X∗ m β X∗ c31 = − (H3 +X2∗ )21 + H3+X ∗2 , c32 = m 1 β1 X 2∗ , c33 = H3+X ∗1 + m 1 β1 I1∗ − μ2 , c34 = η, 1 1 1 c42 = m 2 β1 X 2∗ , c43 = m 2 β1 I1∗ , c44 = −η − μ2 , c = α X 1∗ , d = α I1∗ . The characteristic equation at coexistence equilibrium E 4 = (X 1∗ , I1∗ , X 2∗ , I2∗ ) of the system (1) with positive delay reduce to the following transcendental equation, λ4 + F1 λ3 + F2 λ2 + F3 λ + F4 + (F5 λ3 + F6 λ2 + F7 λ + F8 )e−λτ = 0. Here,

(7)

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F1 = −(c11 + c22 + c33 + c44 ), F2 = c11 (c22 + c44 + c33 ) − c13 c31 − c23 c32 + c22 c33 − c24 c42 − c34 c43 +c22 c44 + c33 c44 , F3 = c13 c31 (c22 + c44 ) − c12 c23 c31 + c23 c32 (c11 + c44 ) − c11 c22 (c33 + c44 ) +c24 c42 (c11 + c33 ) − c23 c34 c42 − c24 c32 c43 + (c11 + c22 )(c34 c43 − c33 c44 ), F4 = c13 c24 c31 c42 − c11 c24 c33 c42 + c11 c23 c34 c42 − c12 c24 c31 c43 + c11 c24 c32 c43 −c11 c22 c34 c43 − c13 c22 c31 c44 + c12 c23 c31 c44 − c11 c23 c32 c44 + c11 c22 c33 c44 , F5 = −c + d, F6 = c(c11 + c21 + c33 + c44 ) − d(c12 + c22 + c33 + c44 ), F7 = (c33 + c44 )((c22 + c12 )d − c(c21 + c11 )) + (c13 + c23 )(cc31 − c32 d) +(c14 + c24 )(cc41 − c42 d) + (d − c)(c33 c44 − c34 c43 ), F8 = (c33 c44 − c34 c43 )(c(c11 + c21 ) − (c12 + c22 )d) + (c13 + c23 )(c44 (c32 d − cc31 ) +c34 (cc41 − c42 d)) + (c14 + c24 )(c43 (cc31 − c32 d) + c33 (c42 d − cc41 )). We applied the lemma described in [6] for the transcendental polynomial equation to discuss the local stability of E 4 and obtained the following theorems. Theorem 2 The following conditions assure the local asymptotic stability of coexistence equilibrium E 4 (X 1∗ , I1∗ , X 2∗ , I2∗ ) of the system (1) for all τ ≤ τ + : (c22 + c12 ) < 0 , c34 > max{

c24 c31 c44 c33 , }, c11 c43

c11 + d < 0 , c < min{|

c13 d22 c13 d32 − c11 |, |c12 − |}. d23 d33

(8) (9) (10) (11)

Theorem 3 The coexistence equilibrium E 4 (X 1∗ , I1∗ , X 2∗ , I2∗ ) of the system (1) exists and asymptotically unstable for all τ ≥ τ + if following conditions hold: c11 + d > 0, c > max{

c13 d32 c13 d22 − c11 , − c12 }. d33 d23

(12) (13)

Moreover, the system (1) undergoes a Hopf bifurcation at E 4 when τ = τ + provided L(ω0 )U (ω0 ) − M(ω0 )V (ω0 ) > 0. Proof (Theorem 2 and 3) The characteristic equation for the matrix (6) is as follows, F(λ)F1 (λ) = 0 ,

(14)

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where, F(λ) = (λ − c44 ){λ − (c44 c33 − c34 c43 )}{λ − (d23 d22 − d23 d32 )} = 0,

(15)

2 F1 (λ) = λ + c12 d21 d23 d33 − c13 d21 d22 d33 − c12 d23 d31 + c31 c22 d23 d31 − 2 c11 d22 d23 d33 + c13 d21 d22 d33 + c11 d23 d32 − c13 d21 d23 d32 + 2 2 [c{d23 d31 − d21 d23 d33 } + d{d22 d33 d23 − d23 d32 }]e−λτ = 0.

(16)

The characteristic roots of (15) are λ1 = c44 , λ2 = c44 c33 − c34 c43 , λ3 = d23 d22 − d23 d32 . Here, d23 d22 − d23 d32 = (c22 + c12 )c33 c34 c44 + c24 c32 c34 c43 − c13 c32 c34 c44 − c23 c32 2 2 c34 c44 + (c13 c42 − c12 c43 )c34 + c34 (c23 c42 − c22 c43 ). It can be clearly seen from (8) and (9) that eigenvalues λ1 , λ2 and λ3 of (15) are negative. Further, the equation (16) can be written in the following form; F(λ) = (λ + v + qe−λτ ) = 0 ,

(17)

2 d31 + c31 c22 d23 d31 − c11 d22 d23 d33 where, v = c12 d21 d23 d33 − c13 d21 d22 d33 − c12 d23 2 2 + c13 d21 d22 d33 + c11 d23 d32 − c13 d21 d23 d32 , q = c(d23 d31 − d21 d23 d33 ) + d(d22 d33 2 d23 − d23 d32 ). We observe from (8)–(11) that q + v and v 2 − q 2 both are positive. Then, according to lemma described in [6], the equation (16) has negative real root or root with negative real parts so as the transcendental polynomial (14) has negative real roots or roots with negative real part. This shows the local stability of the endemic equilibrium E 4 of the model (1). Further, from (9) and (12)– (13) it is clear that q + v is positive and v 2 − q 2 is negative. Then, lemma described in [6] referred that endemic equilibrium E 4 become unstable and the characteristic equation (16) have a couple of purely imaginary root so as the transcendental equation (14) also have a couple purely imaginary roots. To find stability switches of equilibrium E 4 , substitute λ = iω in (7) and comparing real and imaginary parts of the right side of (7) to the left hand side, and solving equation we have,

Cos[τ ω] = −

−(F3 ω − F1 ω3 )(F7 ω − F5 ω3 ) + (−F8 + F6 ω2 )(F4 − F2 ω2 + ω4 ) , (F8 − F6 ω2 )(−F8 + F6 ω2 ) − (F7 ω − F5 ω3 )2 (18)

ω8 + (−2F2 + F12 − F52 )ω6 + (F22 − 2F4 − 2F1 F3 − 2F5 F7 − F62 )ω4 + (F32 − F72 − 2F2 f 4 + 2F5 F8 )ω2 + F42 − F82 = 0. (19) It is clear that (19) has at least one positive root if and only F42 − F82 < 0. Let, ω0 is root of the equation (19); therefore, ±iω0 is root of the characteristic equation (7).

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From equation (18) we have, τk+ =

(F6 −F1 F5 )ω06 +(F3 F5 +F1 F7 −F6 F2 −F8 )ω04 +(F2 F8 +F4 F6 −F7 F3 )ω02 −F4 F8 1 k ] + 2π ω0 [arccos[ ω0 . (−F6 ω2 +F8 )2 +(−F5 ω3 +F7 ω0 )2 0

0

Clearly, τ ∗ is the function of ω0 . On differentiating (7) and substituting λ = iω0 , (

dλ −1 LU − V M ) = 2 , dτ (L + M 2 )

where, L = F5 ω03 − F7 ω0 , M = (F8 − F6 ω02 ), U = (B4 ω03 − 2B2 ω0 sin[ω0 τ0 ] + (−3B1 ω02 + B3 )Cos[ω0 τ0 ] + (−3F5 ω02 + F7 ), V = −(4ω03 − 2F2 ω0 ) cos[ω0 τ0 ] + (−3F1 ω02 + F3 ) sin[ω0 τ0 ] + 2F6 ω0 . dλ −1 )τ =τ ∗ = 0. We know that the system undergoes Hopf bifurcation at τ = τ ∗ if Re( dτ Which is possible only when LU = V M. This completes the proof of the theorems (2) and (3).

6.1 Global Stability Analysis Theorem 4 The following conditions assure the global asymptotic stability of coexistence equilibrium E 4 (X 1∗ , I1∗ , X 2∗ , I2∗ ); 1. β2 I1∗ > η. 2. m 1 β1 X 2∗ (H + X 1∗ )2 < m 3 β X 2∗ H . 2r X ∗ r X∗ 3. K 1 + β2 I2∗ + β1 X 2∗ + μ1 = max{2 K 1 − 2r (1 − αe−λτ I1∗ ,

2r X 1∗ K

X 1∗ +I1∗ ) K

+

+ β2 I2∗ + β1 X 2∗ + μ1 , 2β X 1∗ + β1 I1∗ + μ2 −

2β X 2∗ H + (H +X 1∗ )2 m 3 β X 1∗ − m 1 β I1∗ }. H +X 1∗

Proof We have applied method of geometric approach [14] to prove the global stability of E 4 (X 1∗ , I1∗ , X 2∗ , I2∗ ) under the condition (1)–(3). This completes the proof.

7 Sensitivity Analysis We have performed sensitivity analysis using the parameters values r = 4, K = 50, m 1 = 0.9, m 2 = 0.5, m 3 = 0.9, α = 0.01, μ1 = 0.9, μ2 = 1.7, η = 0.009, β = 2, β2 = 2, β1 = 2, H = 2. It can be easily seen from the Table 2 that parameters α, H , μ2 have positive impact on R0 , while r , K , m 3 , β, β1 and μ1 have negative impact on R0 . Further, R0 does not affected by rest of the parameters. Moreover, parameters m 3 , β, μ2 are highly sensitive parameters to R0 than other parameters.

264 Table 2 The sensitivity indices γ yRj 0 =

J. Gupta et al. ∂ R0 ∂yj

×

yj R0

with respect to the parameters y j

Parameter (y j )

γ yRj 0

r K α m1 m2 m3 β β1 β2 H μ1 η μ2

−0.980 −2.084 1 0 0 −38.842 −37.862 −0.9813 0 2.103 −0.019 0 38.842

8 Numerical Simulation Here, the dynamic behavior of the system (1) is studied through extensive numerical simulation. The parameter value taken in the numerical simulation are hypothetical values based on the ecological fact such that as far as possible these values do not violate the real scenario. We have shown the stability figure of the system (1) for parameter values r = 4, K = 50, m 1 = 0.2, m 2 = 0.8, m 3 = 0.9, α = 2.3, μ1 = 0.9, μ2 = 1.1, η = 3, β = 0.4, β2 = 0.5, β1 = 2, H = 25, τ = 1.5 which is shown in Fig. 1. Keeping parameter values same as in Fig. 1, if we increase the value of delay parameter τ , the system switches its stability for critical value of τ = τ + = 2.32 and undergoes Hopf bifurcation around coexistence state. We have shown Hopf bifurcation behavior of system (1) in Fig. 2. Hence, we can conclude that above this threshold value τ = 2.32 < τ = 2.7, the model system (1) become locally unstable and Hopf bifurcation occurs and below this value τ = 1.5 < τ = 2.32 system is stable.

9 Discussion This article proposed and investigated a four compartments eco-epidemiological prey-predator model for susceptible and infected domestic dogs or wild carnivores as prey and susceptible Amur tigers and infected Amur tigers predator. We consider CDV infection in both prey and predator and incubation delay for CDV infected dogs or wild carnivores, which helps us understand the incubation period’s effect on

An Eco-Epidemic Dynamics with Incubation …

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50 45 40

Populations

35

S(t) X(t) P(t) Y(t)

30 25 20 15 10 5 0

0

100

200

300

400

500

600

700

800

900

1000

Time t (days)

Fig. 1 Stability figure of coexistence equilibrium, when r = 4, K = 50, m 1 = 0.2, m 2 = 0.8, m 3 = 0.9, α = 2.3, μ1 = 0.9, μ2 = 1.1, η = 3, β = 0.4, β2 = 0.5, β1 = 2, H = 25, τ = 1.5, with the initial condition [X 1 (0),I1 (0),X 2 (0),I2 (0)]=[25 5 15 15]

disease dynamics. We have shown that the solutions of system are exist and bounded in a specified region. The local and global stability of coexistence equilibrium with specified conditions are shown under bounded region Δ. Further, the system exhibits Hopf bifurcation at coexistence state at the critical value of incubation period τ = τ+ = 2.32, above this threshold value, coexistence state is locally unstable, and periodic solution occurs in the system. Below this threshold value, the system becomes stable at coexistence equilibrium. Further, sensitivity analysis shows that β and μ2 are highly sensitive parameters to R0 . We know that system becomes infection-free at R0 < 1. Thus, if we increase the predation rate of healthy prey β, the value of R0 started decreasing, and R0 < 1 can be achieved. Further, we can also get R0 < 1 via decreasing predator’s death rate as μ2 has a positive impact on R0 .

10 Conclusion We can conclude from the last section that the incubation period destabilizes the Amur tiger-prey coexistence system and, for extensive delay system is unstable around the coexistence state. Moreover, the large incubation period increases the chance of infection in Amur tigers, thereby causes their extinction. The last section also results that an infection-free environment can be achieved via increasing the predation rate of healthy prey β. We can increase β by making healthy prey available for the Amur tiger

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Suceptible Prey

45 40 35

S(t)

30 25

0

100

200

300

400

500

600

700

800

900

1000

700

800

900

1000

Time t (days)

(a) 5 X(t)

4.5

Infected Prey

4 3.5 3 2.5 2 1.5 1 0.5 0

0

100

200

300

400

500

600

Time t (days)

(b) 30 P(t)

Suceptible Predator

25 20 15 10 5 0

0

100

200

300

400

500

600

700

800

900

1000

Time t (days)

(c) Fig. 2 Hopf bifurcation figure for (a) susceptible prey, (b) infected prey, (c) susceptible predator, (d) infected predator, when r = 4, K = 50, m 1 = 0.2, m 2 = 0.8, m 3 = 0.9, α = 2.3, μ1 = 0.9, μ2 = 1.1, η = 3, β = 0.4, β2 = 0.5, β1 = 2, H = 25, τ = 2.7, with the initial condition [X 1 (0),I1 (0),X 2 (0),I2 (0)]=[25 5 15 15]

An Eco-Epidemic Dynamics with Incubation …

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16

Infected Predator

14 Y(t)

12 10 8 6 4 2 0

0

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200

300

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500

600

700

800

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Time t (days)

(d) Fig. 2 (continued)

and reducing its mortality rate. Further, we can also get a disease-free environment via decrement in the predator death rate. The predator is always inclined to predate on infected prey, and at the moderate predator death rate, the predator predates almost all the infected prey, and hence, we get a disease-free environment. The predator death rate can be decreased via proper regulation of diseased predator death rate and other measures like hunting, poaching, etc.

11 Future Scope We have ignored horizontal transmission of CDV among tigers that may negatively impact the tiger population. We can also take different death rates for healthy and infected Amur tiger. It will signify which reason more influences the extinction of the Amur tiger, the death rate due to CDV, or other reasons like poaching, human conflict, and lack of habitat.

References 1. Gilbert, M., Miquelle, D., Goodrich, J., Reev, R., Cleaveland, S.: Estimating the potential impact of canine distemper virus on the Amur Tiger Population (Pathera tigirs altaica) in Russia. PloS ONE 9(10), e110811 (2014) 2. Beineke, A., Baumgartner, W., Wohlsein, P.: Cross-species transmission of canin distemper virus—an update. One Health 1, 49–59 (2015) 3. Van de Bildt, M.W., Kuiken, T., Visee, A.M., Lema, S., Fitzjohn, T.R., Osterhaus, A.D.: Distemper outbreak and its effect on African Wild Dog conservation. Emerg Infect. Dis. 8(2), 211–213 (2002)

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4. Mullin, C. (Ed.): Canine Distemper (2009) 5. Hadeler, K., Freedman, H.: Predator-prey populations with parasitic infection. J. Math. Biol. 27(6), 609–631 (1989) 6. Singh, H., Dhar, J., Bhatti, H.: Dynamics of a prey generalized predator system with disease in prey and gestation delay for predator. Model. Earth Syst. Environ. 2, 52 (2016) 7. Beluisi, S., Venturino, E.: An eco-epidemic model with diseased predators and prey group defence. Simul. Model. Practice Theory 34, 144–155 (2013) 8. Khalifa, K., Majeed, A.A., Naji, R.: The local bifurcation and the Hopf bifurcation for ecoepidemiological system with one infectious disease. Gen. Math. Notes 31(1), 18–41 (2015) 9. Meng, X., Qin, N., Huo, H.: Dynamic analysis of a predator-prey system with harvesting prey and disease in prey species. J. Biol. Dynamics 12, 342–374 (2018) 10. Boerlijst, M., de Roos, A.: Competition and Facilitation between a disease and a predator in a stunted prey population. PLoS ONE 10(7), e0132251 (2015) 11. Murray, J.: Mathematical Biology, vol. 19, 2 edn. Springer, Berlin (1993) 12. Heffernan, J., Smith, R., Wahi, L.: Perspective on basic reproduction ratio. J. R. Soc. Interface 2(4), 281–293 (2005) 13. Roy, P., Upadhyay, R.K.: Assessment of rabbit hemorrhagic disease in controlling the population of red fox: a measure to preserve endangered species in Australia. Ecol. Complex. 26, 6–20 (2016) 14. Sharma, S., Samanta, G.: A Leslie-Gower predator-prey model with disease in prey incorporating a prey refuge. Chaos Solitons Fractals 70, 69–84 (2015)

Optimization and Statistical Analysis

Comparison of Optimization Results of RSM Approaches for Transesterification of Waste Cooking Oil Using Microwave-Assisted Method Catalyzed by CaO Nirav Prajapati, Pravin Kodgire, and Surendra Singh Kachhwaha Abstract The present study describes the comparison of optimization process parameters determined by employing response surface methodology (RSM)-based full factor design (FFD) and Box-Behnken design (BBD) methods for synthesis of biodiesel from waste cooking oil. Using experimental yield values, quadratic polynomial equations were obtained. Parameters considered to maximize the yield are: MeOH: oil molar ratio (A), CaO quantity (B), reaction interval (C), for the optimization process. The consequence of these variables on biodiesel yield was analyzed by various plots. From perturbation plots, the analysis showed that the effect of catalyst quantity is highly influential on biodiesel yield than the remaining two parameters. By employing FFD and BBD methods, the determined optimum results for A are 9.6:1 and 9.33:1, for B are 1.33 (w/w %) and 1.37(w/w %), for C are 9.7 min and 9.28 min, and the corresponding yield is 90.41% and 90.49%, respectively. Based on analysis of variance (ANOVA) and various statistical plots, BBD method will prefer over FFD method. Keywords Biodiesel · Microwave-assisted method · RSM · Optimization · Comparison

N. Prajapati · P. Kodgire (B) Department of Chemical Engineering, Pandit Deendayal Energy University, Gandhinagar, Gujarat 382426, India N. Prajapati e-mail: [email protected] S. S. Kachhwaha Department of Mechanical Engineering, Pandit Deendayal Energy University, Gandhinagar, Gujarat 382426, India e-mail: [email protected] N. Prajapati · P. Kodgire · S. S. Kachhwaha Centre for Biofuels and Bioenergy Studies, Pandit Deendayal Energy University, Gandhinagar, Gujarat 382426, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_23

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1 Introduction Currently, more than 80% of global energy needs are met with petroleum-based fuel, in which 60% of the share is consumed by the transport sector [1]. Continued exploitation of fuel depots to meet current energy needs has led to the rapid depletion of these energy sources. Continued growth and fluctuations in crude oil prices as well as a major impact of greenhouse gas emissions (GHGs) through their use, are creating harmful impacts on humans and also increasing pollution in the environment. Therefore, it is important to find other new energy sources that must be renewable, sustainable, environmentally friendly, efficient, and economically viable. Among the various energy sources, biodiesel has received much attention worldwide because biodiesel is considered to be the most stable and environmentally friendly energy source [2]. Biodiesel is a renewable fuel, biodegradable, and does not exhaust harmful emissions. It can also help to reduce the impact of greenhouse gases and have no impact on global warming being less polluting. Biodiesel does not contain any carcinogens and sulfur content. The national biofuels policy (2018) sets a goal of 5% biodiesel by the year 2030 but till 2019, India has accomplished only 0.14% mixing of biodiesel in mineral diesel [3]. Currently, the biodiesel is produced from non-edible feedstock like castor oil, neem, karanja, jatropha, microalgae, and Ccerbera Cerbera odollam [4]. Waste/used cooking oil (WCO/UCO) as a raw material can greatly moderate the price of biodiesel synthesis due to its lower price [5, 6]. As per the Indian financial express report of the year 2017, in India, every year, around 1.2 MT cotton-seed oil is manufactured and of that, 60–65% production is done in Gujarat state alone, which will produce a huge volume of waste cotton-seed cooking oil (WCCO) per year [3]. There are many processes to produce enhanced quality biodiesel, viz. transesterification process, micro-mixtures of oils, pyrolysis of plant oils, and mixing of oils [7]. Transesterification is modest and most commonly applied for the production of biodiesel. The catalysts such as an acid, base, enzyme, homogeneous, heterogeneous, and ionic liquids are being used for biodiesel synthesis [8–10]. The homogeneous catalyst is sensitive toward FFA that leads to soap formation, slow reaction rate, and difficult catalyst separation. The effect of enzymatic catalyst on biodiesel production is that it is sensitive to methanol. Also, it has a slow reaction rate, and its cost is high [11]. The heterogeneous catalyst has advantages over homogeneous catalyst as it can be easily separated and having a regenerative capacity which reduces downstream processing [12]. In this study, heterogeneous catalyst, CaO, is used. Among recently developed techniques, microwave assisted methods and ultrasound- assisted methods enhance biodiesel yield. The use of microwave method reduces the reaction time and improves separation steps. Microwave irradiation accelerates the warming level of oil molecules, methanol, and catalyst mixture [3]. Microwave irradiation method delivers energy directly to the reactant, unlike conventional heating. Biodiesel production usually starts by mixing oil, methanol, and CaO

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in the appropriate ratio. Then after, this mixture is processed under microwave irradiation. The last step, separation and purification, is done for producing biodiesel. A disadvantage of the microwave method is that this method is costly because of its high capital cost. Aim of the present study is to optimize biodiesel yield corresponding to three parameters: MeOH: oil ratio (A), CaO quantity (B), reaction interval (C). Among DOE methods, Box-Behnken designs (BBD) usually have lesser design points than full factorial design (FFD) and central composite design (CCD) methods [13]. In this work, an assessment study of FFD and BBD methods is presented. Also, the effectiveness of the models is related by coefficient of determination (R2 ), R2 adjusted, and R2 -predicted values and built on the analysis of variance (ANOVA).

2 Materials and Methodology 2.1 Materials Raw material of biodiesel, i.e., WCCO was procured from a local eatery in Ahmedabad, India. Methyl alcohol and CaO were purchased from Merck, India. Here, all the experimental work was performed using research grade chemicals.

2.2 Design of Experiment (DOE) The biodiesel production process was optimized using 33 (three levels and three factors) FFD and BBD. The required number of experimental runs is 27 for FFD method and 15 for BBD method, correspondingly. Table 1 shows the values of the factors for three levels and three factors used for both FFD and BBD methods. DOE runs were generated applying design expert software (V-11; Stat-Ease, Inc., USA). Tables 2 and 3 show all number of experimental runs for FFD and BBD methods, respectively, and the maximum and minimum predicted biodiesel yield for BBD are 89.94% and 69.04%, respectively. Table 1 Coded values and real values for the selected variables for DOE

Factors

Levels −1

0

1

MeOH: oil molar ratio, (A)

8

10

12

CaO quantity, (B); (w/w %)

0.5

1.25

Reaction interval, (C); (min)

6

9

2 12

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Table 2 Experiment and predicted yield of biodiesel for FFD method (bold numbers are the highest and the lowest biodiesel yields) Run order

A

B

C

Actual yield

Modeled yield

1

−1

1

2

0

1

0

84.35

83.03

1

77.89

3

−1

80.52

0

−1

83.01

82.88

4 5

−1

−1

1

72.07

71.33

−1

0

0

89.06

89.54

6

1

0

1

84.13

86.21

7

1

1

1

78.59

77.06

8

1

−1

1

76.45

75.84

9

0

−1

0

78.32

79.47

10

1

1

−1

62.62

64.97

11

0

1

−1

75.22

72.82

12

−1

1

1

78.91

78.20

13

1

−1

0

76.52

76.64

14

−1

−1

−1

71.34

70.36

15

0

−1

−1

69.34

70.30

16

1

0

0

86.92

86.56

17

1

0

−1

75.17

74.84

18

−1

0

1

83.12

84.26

19

0

0

1

90.50

84.86

20

0

1

0

83.01

82.88

21

0

0

−1

81.19

81.06

22

−1

1

−1

73.44

75.43

23

1

1

0

78.91

78.20

24

0

−1

1

76.36

76.20

25

−1

−1

0

76.32

77.06

26

0

0

0

89.94

90.67

27

1

−1

−1

65.49

65.05

2.3 Microwave-Assisted Method A net batch volume of 50 mL of reaction ingredients in a 3 neck flask (250 mL size) was used for performing a transesterification reaction. The mixture of MeOH and CaO was prepared first. A measured amount of oil which is preheated, and a concoction of methyl alcohol and catalytic agent CaO was added. The pulse power (30 s sequence, i.e., ~26% on and 74% closed) of 180 W at low stirring and temperature of reaction at 50 °C was used for each experimental run. The measured pH was found to be in range of 6–9. After end of the reaction, by using centrifugation, the catalyst

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Table 3 Experiment and predicted yield of biodiesel for BBD method (bold terms are the highest and the lowest biodiesel yields) Run order

A

B

C

Actual yield

Modeled yield

1

0

1

1

77.89

80.52

2

0

0

0

89.94

89.94

3

1

0

1

84.13

84.41

4

0

−1

1

76.36

75.91

5

−1

0

−1

83.01

82.73

6

−1

1

0

84.35

84.17

7

0

0

0

89.94

89.94

8

0

1

−1

75.22

75.67

9

0

0

0

89.94

89.94

10

1

−1

0

76.52

76.70

11

1

1

0

78.91

78.34

12

1

0

−1

75.17

75.29

13

0

−1

−1

69.34

69.04

14

−1

−1

0

76.32

76.90

15

−1

0

1

83.12

83.00

was separated. Using funnel, remaining reaction mass was separated by forming two layers and removed as per the standard procedure described in the literature. The experiment conducted under optimized conditions was triplicated.

2.4 Response Surface Methodology (RSM) RSM is an optimizing tool which increases the process efficiency. It is an assortment of mathematical and statistical techniques [14]. The goal of the RSM study is to comprehend the nature of the response and to identify the area where the appropriate response occurs. In RSM, the errors are assumed to be random [15]. There are four steps for RSM: (1) process to move to the optimum region, (2) in this optimum region, behavior of the response, (3) finding the optimized result, and (4) verify result [14]. Based on linearity or polynomial behavior, RSM has two types of model: First-Order Model Generally, the first-order model is represented as Y = Ao + A1 X 1 + Ao X 2 + · · · . If there is interaction between the variables, the model can be written as,

(1)

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Y = Ao + A1 X 1 + Ao X 2 + A12 X 1 X 2 . . . .

(2)

Second-Order Model The function with two variables is known as second-order model. Y = Ao + A1 X 1 + A2 X 2 + A11 X 12 + A22 X 22 + A12 X 1 X 2 + · · · .

(3)

Here, As are a set of unidentified parameters. Second-order model also contains the squared terms which indicated the interaction between that parameter only. In general, polynomial models are the linear functions of As unknown, known as linear regression analysis [16]. The data collected from the above equations for both BBD and FFD methods were analyzed using RSM method. ANOVA analysis offers an assessment of the deviations caused in run data and measurement errors. This is completed by analyzing statistical parameters such as the mean square, the sum of the square root, F-test, and the lack of fit test. R2 value, i.e., coefficient of determination is used to evaluate the model and also for evaluating the accuracy of the obtained quadratic polynomial equation. The p-value and F-test of the model signify the importance of model.

3 Results and Discussion 3.1 RSM Empirical Models and Analysis The best fitted model in the form of coded factors or regression equation for FFD and BBD method based on the biodiesel yield is given below: For FFD: YFFD = 90.67 − 1.49A + 1.71B + 3.40C − 2.62 A2 − 9.50B 2 − 6.21C 2 + 2.47A · C − 1.28A · B + 0.45B · C

(4)

For BBD: YBBD = 89.94 − 1.51A + 2.23B + 2.35C − 2.13A2 − 8.79B 2 − 6.45C 2 + 2.21A · C − 1.41A · B − 1.09B · C (5) where A, B, and C are the coded notations of MeOH: oil molar ratio, CaO quantity, and reaction interval, respectively. Here, A · C, A · B, and B · C terms indicate interactions, and A2 , B2 , and C 2 indicate the second-order terms of the independent parameters. For FFD method (refer to Table 4), the F-value of model is 48.88 that shows the model is significant. If P-values are less than 0.05, then it specifies model terms are

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Table 4 ANOVA analysis for the FFD method Source

Sum of squares

F-value

p-value

Model

1209.27

Df 9

Mean square 134.36

48.88

5 ≤10 there is medium risk and if >10 there is a high risk.

3 Results and Analysis Results of groundwater analysis of Amreli district [8] are shown in Table 1. pH of groundwater lies between 7 and 8.2. The fluoride concentration in Punjapa, Victor, Kadiyal and Luvariya villages is found to be 1.7, 1.9, 2.05 and 2.2 ppm, respectively, and Goradka village has nitrate concentration of 330 ppm. Groundwater of Bhuva village has very high-chloride concentration of 3408 ppm and sulfate concentration of 408 ppm. There is a high correlation between Na-Cl (0.94), Na–SO4 (0.8), K–Cl (0.799) and moderate correlation for Na–HCO3 (0.69) and Ca–SO4 (0.63), while a negative correlation of (−0.18) between F–Ca (Table 2). In order to identify the contributing sources on the basis of the chemical signatures, PCA analysis of the

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Fig. 2 Dendrogram of hierarchical cluster analysis for groundwater of Amreli district using Ward Method Table 1 Descriptive statistics N pH Ca Mg Na K HCO3 Cl SO4 NO3 F EC

34 34 34 34 34 34 34 34 34 34 34

Minimum

Maximum

Mean

Std. deviation

7.00 12.00 2.00 45.00 0.00 220.00 36.00 2.00 3.00 0.08 580.00

8.20 320.00 161.00 2354.00 146.00 1391.00 3408.00 408.00 330.00 2.20 12150.00

7.94 75.52 52.20 345.68 11.05 536.68 399.09 78.32 80.08 0.67 2228.1

0.27 61.77 37.99 439.21 29.58 293.57 622.61 84.22 90.20 0.62530 2147.01

groundwater quality data was attempted by using the varimax rotation method. A three-factor model gives a total variance of 81.78% (Table 3). The principal components PC1, PC2 and PC3 have eigen values of 5.80, 2.30 and 0.88, respectively. Results of rotated component matrix using show that PC1 has very high-factor loading for sodium (0.97), potassium (0.78), bicarbonates (0.66), chlorides (0.93) and sulfates (0.83) and EC (0.96), for PC2 calcium (0.57), magnesium (0.68), nitrate (0.69) and fluoride (-0.82) has very high-factor loading and pH (-0.92) for PC3 (Table 4). Hierarchical agglomerative cluster analysis using Ward’s method is applied using the squared Euclidean distances (as a measure of similarity) for thirtyfour samples. Dendrogram shows Cluster I (Ca, Mg, K, SO4 , NO3 , F and pH) and Cluster II (Na, Cl, HCO3 ) and EC related to cluster-I and cluster-II (Fig. 2). The risk level for cal-

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Table 2 Correlation matrix

Table 3 Total variance explained Principal component

Initial eigenvalues

Extraction sums of squared loadings

Rotation sums of squared loadings

Total

% of variance

Cumulative %

Total

% of variance

Cumulative %

Total

% of variance

Cumulative %

PC1

5.80

52.74

52.74

5.80

52.74

52.74

5.10

46.40

46.40

PC2

2.30

20.97

73.72

2.30

20.97

73.72

2.37

21.57

67.98

PC3

0.88

8.05

81.78

0.88

8.05

81.78

1.51

13.80

81.78

Table 4 Rotated component matrix using Varimax Method Principal components PC1 PC2 Ca Mg Na K HCO3 Cl NO3 SO4 F EC pH

0.52 0.40 0.97 0.78 0.66 0.93 0.23 0.83 0.23 0.96 −0.16

0.57 0.67 −0.09 0.06 −0.54 0.18 0.69 0.27 −0.82 0.10 −0.05

PC3 0.51 0.18 0.11 0.19 −0.02 0.21 0.39 0.14 0.20 0.20 −0.92

cium, magnesium, potassium, chloride, nitrate and fluorides are shown in Table 5. Calcium and magnesium in groundwater have to H Q ≤ 1 hence no risk. Potassium and fluorides have concentration in groundwaters with HQ between 1 and 5 there is low risk. The hazard quotient for nitrates is more than 5 for some villages and the risk of chlorides on human health is high with HQ greater than 10.

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Table 5 Chronic Daily Intake, Hazard Quotient indices and Risk level for some cations and anions in groundwater Cation/Anion Statistics Groundwater Risk level CDI HQ Ca+2

Mg+2

K+

Cl−

NO− 3

F−

Min Max Mean SD Min Max Mean SD Min Max Mean SD Min Max Mean SD Min Max Mean SD Min Max Mean SD

0.33 8.89 2.10 1.72 0.06 4.47 1.45 1.06 0.00 4.06 0.31 0.82 1.00 94.67 11.09 17.29 0.08 9.17 2.22 2.51 0.00 0.06 0.02 0.02

0.01 0.21 0.05 0.04 0.01 0.41 0.13 0.10 0.00 4.06 0.31 0.82 14.93 1412.94 165.46 258.13 0.05 5.73 1.39 1.57 0.04 1.02 0.31 0.29

No risk No risk

No risk No risk

No risk Low risk

High risk High risk

No risk Medium risk

No risk Low risk

4 Conclusion Human health risk assessment is carried out by analyzing groundwater quality data of Amreli district. Results of principal component analysis and hierarchical cluster analysis provide deep insights into correlations between the chemical constituents. Dendrogram shows clusters of cations and anions which suggests the possible relationships among water quality parameters. The chronic daily intake and hazard quotient for some chemical species in groundwater are calculated to understand the possible human health risk in case of groundwater use for drinking purpose. The mean chronic daily intake values for anions Cl >NO3 > F. While the mean chronic daily intake values for cations is Ca > Mg > K. The mean hazard quotient values for anions is Cl > NO3 > F and for cations K > Mg > Ca. The results of investigation

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indicate high risk of chlorides in ground water of Amreli district. The mean chlorides in groundwater is 399.09 ppm. Water quality managers can plan to implement appropriate remedial strategies for supplying potable groundwater to the users.

References 1. Dhiman, S.D., Keshari, A.K.: GIS based correlation between groundwater quality parameters and geological units, In: Proceeding of 5th Annual International Conference Map India, , New Delhi, India, 6–8 Feb 2002, pp. 187–190 2. Dhiman, S.D.: GIS based geochemical modeling of fluoride contaminated aquifers and exposure risk assessment. Ph.D. thesis, Indian Institute of Technology, Delhi (2003) 3. Dhiman, S.D., Keshari, A.K.: Quantifying uncertainties using fuzzy logic for groundwater driven contaminant exposure assessment. In: Mishra, S. (ed.) Probabilistic Approaches to Groundwater Modeling Symposium at World Environmental and Water Resources Congress 2003 published in ASCE Book on “Groundwater Quality Modeling and Management Under Uncertainty” (2003). ISBN 0-7844-0696-0. https://doi.org/10.1061/40696(2003)22 4. Dhiman, S.D.: Factoring groundwater quality. Geospat. Today 4(1), 50–54 (2005) 5. Dhiman, S.D., Keshari, A.K.: Hydrogeochemical evaluation of high fluoride groundwaters: a case study from Mehsana district. Gujarat, India. Hydrol. Sci. J. 51(6), 1149–1162 (2006). https://doi.org/10.1623/hysj.51.6.1149 6. Dhiman, S.D., Keshari, A.K.: GIS assisted inverse geochemical modeling for plausible phase transfers in aquifers. Environ. Geol. 50(8), 1211–1219 (2006) 7. Dhiman, S.D.: Groundwater quality assessment of Amreli District, Gujarat, India. In: Proceeding of Indo-German Workshop on Water and Wastewater Management for Sustainable Development, pp. 42–43. Department of Civil Engineering, IIT Delhi (2014) 8. Groundwater Year Book. (2014-15). Gujarat State and UT of Daman and Diu, Regional Office Data Centre, Central Groundwater Board, West Central Region Ahmedabad, http://cgwb.gov. in/Regions/GW-year-Books/GWYB-2014-15/GWYB%20WCR%202014-15.pdf. Accessed on 2 Nov 2020 9. http://gis2.nic.in/cgwb/Gemsdata.aspx. Groundwater Information System, year 2009, CGWB, GOI 10. Kaledhonkar, M.J., Keshari, A.K.: Assessing the effects of sodic water irrigation on soil and groundwater quality. In: Ramanathan, A.L., Ramesh, R. (eds). Recent Trends in Hydrogeochemistry, pp. 129–137. Capital Publishing Company, New Delhi (2003) 11. Keshari, A.K.: Recent trends in groundwater sector in India. In: Yoshida, M. (ed.) Japan International Cooperation Agency Sector Study Working Papers, p. 186 (2003) 12. Krishna, A.K., Mohan, K.R., Dasaram, B.: Assessment of groundwater quality, toxicity and health risk in an industrial area using multivariate statistical methods. Environ. Syst. Res. 8, 26 (2019). https://doi.org/10.1186/s40068-019-0154-0 13. USEPA: Guidelines for exposure assessment, EPA/600/Z-92/001. Risk Assessment Forum, Washington, DC (1992) 14. USEPA (1999) Guidance for performing aggregate exposure and risk assessment office of pesticide programs, Washington, DC 15. Zhang, Y., Xu, B., Guo, Z., Han, J., Li, H., Jin, L., Chen, F., Xiong, Y.: Human health risk assessment of groundwater arsenic contamination in Jinghui irrigation district, China. J. Environ. Manage. 237, 163–169 (2019) 16. Bodrud-Doza, Md., Didar-Ul Islam, S.M., Rume, T., Quraishi, S.B., Safiur Rahman, M., Bhuiyan, M.A.H.: Groundwater quality and human health risk assessment for safe and sustainable water supply of Dhaka City dwellers in Bangladesh, Groundwater for Sustainable Development, ISSN: 2352-801X, vol. 10, p. 100374 (2020)

Tuning P max in RED Gateways for QoS Enhancement in Wireless Packet Switching Networks N. G. Goudru

Abstract In wireless packet switching networks, random early detection (RED) gateways are used to control the congestion formation. By computing mean queue length of the bottleneck link, the random early detection gateway detects the incipient congestion. The RED gateways notify the congestion formation to the source TCP connections either by setting a bit in the header of the acknowledgment packet or by dropping the incoming packets with a certain probability. Through this means RED gateways try to keep the queue length below the maximum threshold value. Tuning RED parameters is a challenging task for network engineers. In this paper, we tried to estimate a stable boundary for Pmax . We use stochastic models representing sender window dynamics, average queue length and packet dropping probability. The non-linear equations have been linearized using transformation technique. By constructing Hermite matrix for time-delay control system, analysis has been made to determine a stable boundary for Pmax . The performance of sender window, queue length, packet dropping probability and throughput has been analysed using statistical data and graphs obtained by MATLAB programming. The results revel that the queue length, sender window size and random packet dropping probability converge to smaller range over their values. A comparative study has been made by conducting experiments for assumed Pmax value (0.05) and for stable Pmax value. Application of stable Pmax value gives an excellent throughput and better network performance. Keywords Bottleneck link · Cwnd packet dropping · Queue length · RED gateway · Stability · TCP · RTT

N. G. Goudru (B) Department of Information Science and Engineering, Nitte Meenakshi Institute of Technology (Affiliated to Visvesvaraya Technological University), Bangalore, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_28

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1 Introduction RED gateways are designed to support TCP for congestion control in wired and wireless networks. RED gateway notifies the congestion to a sender by randomly dropping a packet. The amount of reduction in the sending rate made by a source is roughly proportional to the bandwidth shared by that connection through the gateway. When queue overflows at the gateway, packets are dropped randomly for many connections, and these TCP connections reduce their window size resulting in loss of throughput. The performance of RED gateway depends on the tuning parameters. Important parameters are queue level minimum threshold (qmim ), maximum threshold (qmax ), and maximum packet dropping probability (Pmax ). The selection of the parameter values qmin and qmax depends on the average queue length. When the network traffic is burst, then the selection of qmin is allow the high-link utilization. In this research work, we discuss on tuning of the maximum packet dropping probability, Pmax. The literature review reveals that many researchers prefer to assign some value for Pmax to carry out their experiments. During burst traffic, the improper selection of Pmax leads to an oscillatory situation in the performance of gateway. Practically, most of the times, network traffic is burst, and the oscillatory nature causes degradation in the performance of RED gateway. To overcome this situation, we propose to tune the Pmax to a stable value, so that the gateway functions are smooth and stable.

2 Literature Review Literature review says that many researchers have studied on random early detection as a technique to avoid congestion at the gateway. C.V. Hollot and others linearize the non-linear dynamic models of TCP behavior and queue length along the bottleneck link. By using RED scheme, designed an AQM control system and studied the results in terms of network parameters by stabilizing the queue length [2]. Kuusell et al. [3] present a method for deriving necessary condition on local stability of the TCP-RED congestion control. The authors treat m homogenous TCP sources with constant rtt value. The analysis is based on differential equation model of the system. The system stability is characterized by means of certain stability parameters. Sanjeewa Athuralia and others propose a new AQM scheme called REM (random exponential marking). The authors analyze the similarities of performance in drop tail, RED and REM in wired link. REM is able to provide high-bandwidth utilization, negligible delay and low-packet losses [4]. Sirisena et al. present an auto-tuning RED algorithm developed using control theoretic principles for tuning the important RED parameters automatically so that the average queue length is preserved. They conduct performance control theoretic analysis for setting RED parameters. The results reveal that the adaptive algorithm is faster and accurate than previous methods [5]. Sridharan et al. [6] use classical control tools to analyze the system stability. They tried to optimize the performance of satellite networks by using results achieved. In [7], the

Tuning Pmax in RED Gateways for QoS Enhancement …

323

authors, Erich and others demonstrate that the non-linear drop function perform well than the linear RED function. Kai Jiang et al. made a theoretical analysis and show that the performance of RED can be greatly enhanced by adjusting RED parameter via delayed feedback controller [8]. Tadeusz and others discuss on a network composed of wired and wireless for TCP-DCR variant (TCP-delay control rate) and UDP flows. The existing non-linear flow model has been linearized to study the performance of network in the presence of UDP in closed-loop control [9]. Byun et al. [10] developed AVQRED algorithm by considering queue length as input to the RED algorithm, with an purpose of improving the overall performance of the network by keeping delay low, packet drop rate low and link utilization high. The active queue management parameters are tuned based on emulation. Hui Zhang et al. describe a discrete-time non-linear dynamical model for understanding the working of TCP and RED gateways and using the model, capture the dynamical behavior of TCP/RED [11]. The literature review reveals that less work has been done by the researchers to improve the performance of RED gateway by tuning the parameter, Pmax over a stable boundary. This research work deals with establishing a stability condition for Pmax to eliminate the oscillatory behavior of RED gateway.

3 System Modeling A stochastic model [12] representing TCP window dynamics in wireless networks is, ∂w 1 w(t)w(t − R(t)) w(t − R(t)) = − p(t − R(t)) + β ∂t R(t) 2R(t − R(t)) R(t − R(t))

(1)

In right side of Eq. (1), the first term defines additive increase, second term defines multiplicative decrease, and third term controls the transmission loss in wireless network. A stochastic model calculating the average queue length, q(t) of RED gateway is given by N w(t) ∂q(t) = − Cd ∂t R(t)

(2)

In right side of Eq. (2), first term indicates the rate of incoming packets, second term C d indicates the service rate, and N denotes number of active TCP flows. The round trip delay calculation model is given by R(t) =

q(t) + Tp C

(3)

T p is the delay due to signal propagation, whose value is fixed at 5 ms for 1000 km distance.

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3.1 Wireless Loss-Predictor: Normalized Delay Gradient (NDG) f NDG

   RTTi − RTTi_1 Wi + Wi_1   = RTTi + RTTi_1 Wi − Wi_1

Using f NDG , is packet loss was because of transmission or congestion can be predicted. NDG has lower value when traffic load is less. NDG has higher value when traffic load is high. f NDG uses this information to increasing or decreasing the sender window size [14]. (i) (ii)

If f NDG > 0, the next packet loss is because of congestion. if f NDG ≤ 0, the next packet loss is because of wireless transmission error.

Thus, If (f NDG > 0) % congestion loss. β = 0. end If (f NDG ≤ 0) % Wireless transmission error. β = 0.1 end.

3.2 RED Algorithm RED gateways apply implicit congestion notification (ICN) by randomly dropping a packet. The RED algorithm use two queue level threshold values called minimum threshold denoted as qmin and maximum threshold denoted as qmax . The average queue length, q(t) is compared with qmin and qmax . When q(t) ≤ qmin , no packet is dropped. When q(t) ≥ qmax , every incoming packet is dropped. When q(t) is between qmim and qmax , the arriving packets are dropped with a probability of Pq (t). The probability term Pq (t) is a function of q(t). The lines of pseudo code representing the algorithm for RED gateway [1] are, for the arrival each packet, estimate the average queue length, q(t) If 0 ≤ q(t) ≤ qmin . Packet dropping, Pq (t) = 0. else if qmin ≤ q(t) ≤ qmax . calculate probability of packet dropping min P max Pq (t) = qq(t)−q max −q min else Pq (t) = 1. A step function representing the probability of packet dropping by the RED gateway is given by

Tuning Pmax in RED Gateways for QoS Enhancement …

⎧ ⎪ q(t) ∈ [0, qmin ] ⎪0 ⎨ q(t) − q min Pq (t) = Pmax q(t) ∈ [qmin , qmax ] ⎪ ⎪ qmax −qmin ⎩ 1 q(t) ≥ qmax

325

(4)

3.3 Selection of qmin and qmax Values The selection qmax value depends on the maximum queuing delay that can be allowed by the gateway. The RED gateway performs efficiently when (qmax − qmin ) is larger than the increase in the average queue length. If (qmax − qmin ) is smaller, then the average queue becomes oscillatory. A thumb rule applied by many researchers for the selection of the values is qmax to be at least twice the qmin . The parameter Pmax gives the maximum packet dropping probability when q(t) reaches qmax .

3.4 Stability Analysis of Pmax Define a non-linear function x(t), such that x(t) = f (u(t), v(t), t) where u(t) denotes the source window size, v(t) denotes the dynamics of queue along the bottleneck link. Let us assume that f (u(t), v(t), t) possess continuous derivatives around the equilibrium point Q0 = (W 0 , R0 , q0 , P0 ). Expand f applying Taylor’s series discarding derivatives of second and higher order and substituting the partial derivatives at Q0 [13] we get from (4), Pq (t) =

Pmax (q(t) − qmin ) qmax − qmin

Let B = qmax − qmin; K = Pmax Pq (t) =

K (q(t) − qmin ) B

Define, δ P(t) = P(t) − P0 K K δq(t − R0 ) + (q0 − qmin ) − P0 B B  k R0 Cd2 β 2N δ w(t) ˙ =− δq(t − R0 ) δw(t) − + 2 R0 2N 2 B R0 C d δ P(t − R0 ) =

+

k R0 Cd2 R0 Cd2 P0 − q + ) (q min 0 2N 2 B 2N 2

(5)

(6)

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δ q(t) ˙ =  Denote x(t) =

δw(t) δq(t)

N 1 δw(t) − δq(t) R0 R0



 => x(t) ˙ =

δ w(t) ˙ δ q(t) ˙

(7)



x(t) ˙ = Ax(t) + E x(t − R0 ) + F where =

A

K R0 Cd2 2N 2 B (qmin

 − Rβ0 + R C2 P

N R0

2N R02 Cd

0 − R10

(8)

,

E

=

K R C2

0 − 2N02 Bd 0 0

,

− q0 ) + 02Nd2 0 . 0 Equation (8) is a linear differential equation. Applying Laplace transform technique with L{x(t)} = x(s), the solution of (8) is,   F s I − A − Ee−R0 s x(s) = s

(9)

The characteristic equation of (9) is,   s I − A − Ee−R0 s  = 0

(10)

After simplification,  s + 2

 LCd2 −R0 s β β 2N 1 2N s+ =0 e + 2 + + 3 + R0 R0 2N B R0 C d R02 R0 C d

(11)

Denote,    β 2N 1 P s, e−R0 s = s 2 + s + 2 + R0 R0 R0 C d .  LCd2 −R0 s β 2N e + + 3 + 2N B R02 R0 C d Let β 2N 1 e−R0 s = z, and a0 = 1, a1 = + 2 + , R0 R0 R0 C d a2 =

LCd2 −R0s β 2N e + 3 + 2 2N B R0 R0 C d P(s, z) = a0 s 2 + a1 s + a2

For time-delay control system, the Hermit matrix of (12) is

(12)

Tuning Pmax in RED Gateways for QoS Enhancement …



(0, 1) (0, 2) H= (0, 2) (1, 2) (0, 1) = 2a0 a1 =

327



2(β + 1) 4N + R0 R0 2Cd

(0, 2) = −2a2 Im(z) =

−Lcd2 Im(z) NB

(1, 2) = 2a1 Re(a2 )   LCd2 β β 2N 1 2N + 2 + + + =2 R0 R0 2N B R0 C d R02 R03 Cd Put, x1 =

LCd2 β 2N 1 β 2N + 2 + , x2 = , x3 = 2 + 3 . R0 R0 NB R0 C d R0 R0 C d

Let, z = eiω , z = cos ω + i sin ω, Re(z) = cos ω, Im(z) = sin ω, then   H eiω =



ω  2x1  −x2 sin x2 −x2 sin ω 2x1 x3 + 2 cos ω



The time-delayed control system presented in (1) to (4) perform stable behavior if and only if the following two conditions are satisfied. For all ω[0, 2π ], det H eiω > 0,   x2 cos ω − x22 sin 2ω > 0 (2x1) 2x1 x3 + 2   H eiω = x22 cos2 ω + 2x12 x2 cos ω + 4x1 2x3 − x22 > 0

(13)

Equation (13) is a quadratic equation in cos ω. The necessary condition for (13) for a real solution is the discriminate,  > 0 cos ω =

−x12 ±

   x12 x12 − 4x3 + x22 x2

According to the cosine function properties, for ω[0, 2π ], cos ω[−1, 1]. The conditions can be written as,

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−x12 −

   x12 x12 − 4x3 + x22 x2

−x12 +

   x12 x12 − 4x3 + x22 x2

>1

(14)

< −1

(15)

We cannot determine a solution for (15). We discard it. From (14), we obtain  0 < pmax
N, then go to 10, else go to 6 10 .Compute total downtime for given month, H = sum(daily downtime, T) 11. Compute frequency of failures in given month, F = count(no. of days with daily downtime > 0) 12. Compute MTBF and MTTR as per Eqs. (1) and (2) 13. Compute Availability, A; Unavailability, U; Reliability, R; Unreliability, Z; Annual Failure Rate, Y as per Eqs. (3) to (7) 14. Compute average values and % values of MTBF, MTTR, A, U, R, Z and Y 15. Output average and % values of MTBF, MTTR, Availability, Unavailability, Reliability, Unreliability and Annual Failure Rate 16. End

MTTR = H/F

(2)

MTBF P

(3)

Unavailability U = 1−A

(4)

Availability A =

Relaibility,

P R = e[−( MTBF )]

(5)

Unreliability Z = 1−R  Annual Failure Rate Y = F ∗

(6) 52 (V ∗4)

G

 (7)

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4 An Indian Case Study Spreadsheet based monthly consolidated datasheets shared by the Indian utility contained daily communication data in form of sample data packets received from each smart meter. The entire field dataset has been characterized in seven different categories, and one smart meter in each category has been selected for study. From the same, the failures in data packets have been calculated on daily and monthly basis and the performance metrics—MTBF, MTTR, availability, reliability, etc., have been calculated and presented in Tables 2, 3, 4 and 5 and depicted graphically in Figs.1, 2 and 3. In this work, RF and GPRS installed in NAN network with AMI critical application has been studied and analysed.

4.1 Results Referring Tables 2, 3, 4 and 5 and Figs. 1, 2, and 3, following points have been observed: (i) (ii) (iii) (iv) (v) (vi)

Reliability and availability of communication have been observed better in urban areas as compared to that of rural areas. Residential and commercial consumers have relatively larger variations in performance that of others and industrial type have been stable and better. Communications with large consumer base categories—LT and HT have been found relatively inferior as compared to DTR. DCU-Smart meter connectivity is found stable and sufficiently available. In terms of consistency of performance, others, commercial, residential and DCU-SM connectivity have outperformed the remaining customer types. Large drops have been observed during peak monsoon and winter seasons.

5 Conclusions, Recommendations and Future Directions 5.1 Conclusions Relative under-performance in rural, residential and commercial consumers specifically with LT and HT supplies can lead to dissatisfaction and unpopularity of smart grid as such customers are significant in terms of market share. Further, sudden fall in performance during climatic variations can also be a concern.

Based on economic classification

Based on consumer type

3

Commercial

Others

6

Industrial

Residential

Urban

Rural

5

4

2

1

528.8373

602.8755

567.1873

585.5461

447.7833

613.1569

MTBF (h)

7.5959

4.2127

5.4981

4.7334

14.7451

3.9910

1.5896

1.8710

1.8527

1.8783

1.1511

1.8527

25.0341

29.5201

27.0359

28.4665

21.4768

30.1351

72.3660

82.4738

77.4536

80.3370

61.4096

51.9296

27.6340

17.5262

22.5464

19.6630

38.5904

48.0704

MTTR (h) Failure rate (Y, %) Reliability (R, %) Availability (A, %) Unavailability (U, %)

Sr. No. Characterization Classification Performance metrics of installed communication technology

Table 2 Validation of communication data—based on economic classification of installation location

Development of a Mathematical Framework to Evaluate … 341

Based on type of service connection

1

3

2

Characterization

Sr. No.

LT

HT

DTR

Classification

20,799

148

448

Total no. of smart meters in service

1655

38

127

Average smart meters failing communication

Performance metrics of installed communication technology

Table 3 Validation of communication data—based on type of service connection

7.9571

25.6757

71.6518

Availability (A)

92.0429

74.3243

28.3482

Unavailability (U)

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Development of a Mathematical Framework to Evaluate …

343

Table 4 Validation of communication data—based on DCU to smart meter connectivity Sr. No.

Characterization

Month

Availability (A)

Unavailability (U)

1

Based on DCU to smart meter connectivity

January

21.6216

78.3784

2

February

19.7876

80.2124

3

March

28.3348

71.6652

4

April

28.7387

71.2613

5

May

11.5083

88.4917

6

June

10.9910

89.0090

Average

20.1637

79.8363

5.2 Recommendations To cater the present and future needs, the currently installed RF and GPRS should be replaced by LTE or Wi-Fi or Satellite or WiMAX in the order of preference, for satisfying high-data rate requirements with better latency performance after application of assessment [3].

5.3 Future Directions The work presented is based on MTBF-MTTR based availability and reliability algorithm and limited to validation of communication performance of RF and GPRS technologies only. This work could be extended by considering different types validation algorithms, data of other communication technologies obtained from different locations subjected to different economic and climatic conditions.

Climate

Summer

Monsoon

Winter

Sr. No.

1

2

3

Jan

July

May

Month

89.5161

83.8710

74.5296 49.4624

99.7984

64.8006 89.6147

70.8042

77.8472

Residential

Rural

Urban

Availability (A) based on customer type

Availability (A) based on economic classification of location

Table 5 Validation of communication data—based on impact of climate changes

99.6461

0.0694

98.0242

Industrial

93.8306

96.6420

93.4028

Commercial

98.1362

96.5659

95.3338

Others

21.6216

18.9189

11.5083

DCU to smart meter

Availability (A)

344 J. Bhatt et al.

Development of a Mathematical Framework to Evaluate …

345

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 1 a–f Graphical representation of data of Table 2

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(a)

(b)

Fig. 2 a and b Graphical representation of data of Tables 3 and 4, respectively

Fig. 3 Graphical representation of data of Table 5

Acknowledgements The authors express their sincere thanks to Dr. Chetan Bhatt, Prof.-IC Engineering and Principal, Government MCA College, Ahmedabad for his valuable advice and cooperation. The authors thankfully acknowledge the timely support and help provided by Mrs. Kumud Wadhwa, Sr. General Manager, NSGM, Ministry of Power, Government of India for issuing permission to access data and publish our interpretations. The cooperation received from the Indian utility company by sharing communication data has been acknowledged with thanks.

Development of a Mathematical Framework to Evaluate …

Appendix 1: Nomenclatures and Abbreviations

Nomenclature

Description

N

No. of days in given month

P

Operational availability

S

No. of samples/hour

K

Total no. of daily data packets receivable/day

X

No. of data packets received on dayi

D

No. of data packets dropped on dayd

T

Communication downtime on dayd

H

Total downtime for given month

F

Frequency of failures in given month

i, d

Indexes

Abbreviation

Description

ADR

Automated demand response

AMI

Advanced metering infrastructure

DCU

Data concentrator unit

DR

Demand response

DSM

Demand side management

DTR

Distribution transformer

FAN

Field area network

GPRS

General packet radio service

HAN

Home area network

HT/LT

High tension/low tension

IAN

Industrial area network

ITS

Instrumentation telemetry system

LTE

Long term evolution

MTBF

Mean time between failures

MTTR

Mean time to repair

NAN

Neighborhood area network

WAN

Wide area network

WiMAX

Worldwide interoperability for microwave access

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References 1. Bhatt, J., Jani, O.: Smart grid: energy backbone of smart city and e-democracy. In: E-Democracy for Smart Cities, pp. 319–366. Springer Singapore (2016). https://doi.org/10.1007/978-981-104035-1_11 2. Bhatt, J., Shah, V., Jani, O.: An instrumentation engineer’s review on smart grid: critical applications and parameters. Renew. Sustain. Energy Rev. 40, 1217–1239 (2014). https://doi.org/ 10.1016/j.rser.2014.07.187 3. Bhatt, J., Jani, O., Harish, V.S.K. V: Development of an assessment tool to review Communication Technologies for Smart Grid in India. In: 1st International conference on innovations in clean energy technologies (ICET-2020), pp. 1–11. Springer, Bhopal (2020) 4. India Smart Grid Forum (ISGF): Smart Grid Bulletin, 1(11), 9 (2014) 5. SCE-Cisco-IBM SGRA Team: Smart Grid Reference Architecture (2011) 6. Gupta, B.B., Akhtar, T.: A survey on smart power grid: frameworks, tools, security issues, and solutions. Ann. des Telecommun. Telecommun. 72, 517–549 (2017). https://doi.org/10.1007/ s12243-017-0605-4 7. Wang, Y., Yemula, P., Bose, A.: Decentralized communication and control systems for power system operation. IEEE Trans. Smart Grid 6, 885–893 (2015). https://doi.org/10.1109/TSG. 2014.2363192 8. Mazumder, S.K., Acharya, K., Tahir, M.: Towards realization of a control-communication framework for interactive power networks. In: 2008 IEEE Energy 2030 Conference, pp. 1–8. IEEE, Atlanta (2008). https://doi.org/10.1109/ENERGY.2008.4781037 9. Kumar, Y.V.P., Bhimasingu, R.: Key aspects of smart grid design for distribution system automation: architecture and responsibilities. Procedia Technol. 21, 352–359 (2015). https://doi.org/ 10.1016/j.protcy.2015.10.047 10. Uslar, M., Rohjans, S., Neureiter, C., Andrén, F.P., Velasquez, J., Steinbrink, C., Efthymiou, V., Migliavacca, G., Horsmanheimo, S., Brunner, H., Strasser, T.I.: Applying the smart grid architecture model for designing and validating system-of-systems in the power and energy domain: a European perspective. Energies 12 (2019). https://doi.org/10.3390/en12020258 11. Neumann, P.: Communication in industrial automation-What is going on? Control Eng. Pract. 15, 1332–1347 (2007). https://doi.org/10.1016/j.conengprac.2006.10.004 12. Dileep, G.: A survey on smart grid technologies and applications. Renew. Energy. 146, 2589– 2625 (2020). https://doi.org/10.1016/j.renene.2019.08.092 13. Thakur, J., Chakraborty, B.: Intelli-grid: Moving towards automation of electric grid in India. Renew. Sustain. Energy Rev. 42, 16–25 (2015). https://doi.org/10.1016/j.rser.2014.09.043 14. Beg, F.: An auxiliary study of the smart grid deployment in India. Philosophy and key drivers. Int. J. Smart Grid Green Commun. 1, 38 (2016). https://doi.org/10.1504/ijsggc.2016.077288 15. Shukla, P.: Smart cities in India (2015) 16. Sinha, A., Neogi, S., Lahiri, R.N., Chowdhury, S., Chowdhury, S.P., Chakraborty, N.: Smart grid initiative for power distribution utility in India. In: IEEE Power and Energy Society General Meeting. pp. 1–8. IEEE (2011). https://doi.org/10.1109/PES.2011.6038943 17. Harish, V.S.K.V., Kumar, A.: Planning and implementation strategy of demand side management in India. In: 1st International Conference on Automation, Control, Energy and Systems— 2014. ACES 2014, pp. 1–6. IEEE, Kolkata (2014). https://doi.org/10.1109/ACES.2014.680 8001 18. Harish, V.S.K.V., Kumar, A.: Demand side management in India: action plan, policies and regulations. Renew. Sustain. Energy Rev. 33, 613–624 (2014). https://doi.org/10.1016/j.rser. 2014.02.021 19. Malik, S., Harish, V.S.K.V.: Integration of automated demand response and energy efficiency to enable a smart grid infrastructure. In: 2019 2nd international conference on power energy environment and intelligent control, PEEIC 2019, pp. 371–377. IEEE, Greater Noida (2019). https://doi.org/10.1109/PEEIC47157.2019.8976747 20. Moslehi, K., Kumar, R.: A reliability perspective of the smart grid. IEEE Trans. Smart Grid. 1, 57–64 (2010). https://doi.org/10.1109/TSG.2010.2046346

Development of a Mathematical Framework to Evaluate …

349

21. Kjolle, G.H., Vadlamudi, V.V., Kvistad, S., Tutvedt, K.A.: Potential for improved reliability and reduced interruption costs utilizing smart grid technologies. In: 22nd International Conference and Exhibition on Electricity Distribution (CIRED 2013), pp. 1–4. IET, Stockholm (2013). https://doi.org/10.1049/cp.2013.0870 22. Isa, N.B.M., Wei, T.C., Yatim, A.H.M.: Smart grid technology: communications, power electronics and control system. In: 2015 International Conference on Sustainable Energy Engineering and Application (ICSEEA), pp. 10–14. IEEE (2016). https://doi.org/10.1109/ICSEEA. 2015.7380737 23. Niyato, D., Dong, Q., Wang, P., Hossain, E.: Optimizations of power consumption and supply in the smart grid: analysis of the impact of data communication reliability. IEEE Trans. Smart Grid. 4, 21–35 (2013). https://doi.org/10.1109/TSG.2012.2224677 24. Aamir, M.: Estimating reliability of telecommunication and electronic devices. Presented at the (2015) 25. Rohani, H., Kamali Roosta, A.: Calculating total system availability (2014) 26. Pokorni, S., Jankovi´c, R.: Reliability estimation of a complex communication network by simulation. In: 19th Telecommunications Forum, TELFOR, pp. 226–229. IEEE, Belgrade, Serbia (2011). https://doi.org/10.1109/TELFOR.2011.6143532 27. Rai, S., Soh, S.: Telecommunication network reliability. In: Bellavista, P. (ed.) Telecommunication Systems and Technologies, pp. 1–12. Encyclopedia of Life Support Systems (EOLSS), Paris, France (2009) 28. Yang, J., Teng, H., Yao, C., Liu, N., Sun, B., Yuan, H., Liu, M., Bai, J.: The reliability evaluation method study of power system communication networks in case of ice storm. Engineering 5, 30–36 (2013) 29. Rebaiaia, M.L., Ait-Kadi, D., Merlano, A.: A methodology for modeling and evaluating the reliability of a radio-communication network. IFAC Proc. 43, 92–97 (2010). https://doi.org/10. 3182/20100701-2-PT-4012.00017 30. Niyato, D., Wang, P., Hossain, E.: Reliability analysis and redundancy design of smart grid wireless communications system for demand side management. IEEE Wirel. Commun. 19, 38–46 (2012). https://doi.org/10.1109/MWC.2012.6231158 31. Dominiak, S., Bayer, N., Habermann, J., Rakocevic, V., Xu, B.: Reliability analysis of IEEE 802.16 mesh networks. In: 2nd IEEE/IFIP International Workshop on Broadband Convergence Networks, pp. 1–12. IEEE, Munich (2007). https://doi.org/10.1109/BCN.2007.372739 32. SUNSEED: Traffic modelling, communication requirements and candidate network solutions for real-time smart grid control (2015) 33. Speaks, S.: Reliability and MTBF overview (2002) 34. ITEM Software Inc.: Reliability prediction (2007) 35. Torell, W., Avelar, V.: Performing effective MTBF comparisons for data center infrastructure (2015) 36. Torell, W., Avelar, V.: Mean time between failure: explanation and standards (2004) 37. Krasich, M.: How to estimate and use MTTF/MTBF would the real MTBF please stand up? In: 2009 Annual Reliability and Maintainability Symposium, pp. 353–359. IEEE, Fort Worth, Texas, USA (2009). https://doi.org/10.1109/RAMS.2009.4914702 38. OPAH: Introduction to operational availability (A0). In: Operational Availability Handbook (OPAH), pp. 1–10. Reliability Analysis Center (RAC), New York (2000). 13440-6916 39. Stanley, S.: MTBF, MTTR, MTTF & FIT: explanation of terms (2015). https://doi.org/10.2105/ ccdm.2745.156. 40. Wilson, D.C.: MTBF: understanding its role in reliability engineering (2017) 41. Snow, D.A., Weckman, G.: Protecting critical telecommunications and networking infrastructure. In: The Twelfth International Conference on Networking ICN 2013, pp. 1–200. Ohio University, Ohio (2013) 42. Kounev, V., Lévesque, M., Tipper, D., Gomes, T.: Reliable communication networks for smart grid transmission systems. J. Netw. Syst. Manag. 24, 629–652 (2016). https://doi.org/10.1007/ s10922-016-9375-y

Early Prediction of Cardiovascular Disease Using One-vs-All Model Sarita Mishra , Manjusha Pandey , Siddharth Swarup Rautaray , and Mahendra Kumar Gourisaria

Abstract The rapidly evolving human lifestyle, including eating and drinking habits, workout and resting patterns, consumption of alcohol, smoking habits, work stress, etc., often call for several diseases as such irregularities in lifestyle lead to the improper functioning of many organs. One of the common disease caused by such abnormalities is the cardiovascular disease (CVD). Today, the mortality rate all over the world has drastically increased due to CVD. This drastic increase can be controlled by the early prediction of chances of heart disease in a person using various machine learning technologies. Many data scientists have developed models for the early detection of CVD, however, of them focus on binary classification of the patients, i.e., either having CVD or not having CVD. Such a broad classification is no doubt helpful in segregating the healthy and unhealthy people but not sufficient to provide the most suitable form of treatment to the patients. In this research work, we have suggested the multiclass classification of CVD affected people into five different classes (0–4) with 0 indicating the absence of CVD and 4 indicating the presence of the most critical form of the disease. We have recommended the use of the one-vs-all approach for the multiclass classification of people based on the extent to which a person is affected. In this paper, we have compared the performances of the SVM and KNN algorithms implemented with and without the one-vs-all approach showing better performance with OVA approach and further enhancing the performance applying the principal component analysis technique. The results of this study have shown that the best performance that is the highest accuracy of 99.56% was provided the by KNN algorithm.

S. Mishra (B) · M. Pandey · S. S. Rautaray · M. K. Gourisaria Kalinga Institute of Industrial Technology (Deemed To Be University), Bhubaneswar, Odisha, India e-mail: [email protected] M. Pandey e-mail: [email protected] S. S. Rautaray e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_30

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Keywords CVD · One-vs-all algorithm (OVA) · Support vector machine (SVM) · K-nearest neighbors (KNN)

1 Introduction The human heart, with the size as that of a fist, is a muscular organ that plays a vital role of the circulatory system. It takes up the role of pumping fresh, oxygenated blood throughout the body after purification of the deoxygenated blood in the lungs [1]. The purification and transmission activities are carried out with the help of a number of capillaries, veins, and arteries. Figure 1 explains the normal functioning of the human heart. Cardiovascular disease is one of the fundamental causes behind the rapidly increasing mortality rate all over the globe. The World Health Organization (WHO) has indicated a 24% mortality rate in India [2].

Fig. 1 Working of the heart

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The normal functioning of the human heart can be obstructed by numerous reasons including lack of rest, work stress, unhealthy diet, and many more. These activities cut off the smooth flow of blood to and from the heart via the blood vessels. Sometimes, the blood vessels may narrow down or create a swelling which prevent the clear passing of blood from one organ to another. Sometimes obstruction in the flow of blood may be caused due to the formation of a blood clot in the blood vessel. Such abnormal functioning of the heart causes heart attack that contributes significantly in increasing the mortality rate. This increase in the death rate can be brought down by the early detection and prediction of chances of occurrence of heart disease in a person [3]. Many data scientists have developed several models for detecting the presence or absence of heart disease in a person. But, only its detection will not be sufficient to provide the best treatment for the patient. The impact level of the disease in the patient also plays a significant role in giving appropriate services to the patient. In this paper, we have compared the performances of the SVM and KNN algorithms that are implemented without and with the one-vs-all approach, followed by the application of the principal component analysis technique that enhances the performance of the algorithms. The algorithms are compared in terms of the accuracy they provide in classifying the CVD patients into five different categories. Accuracy in prediction is defined as the ratio between the number of data points that are classified correctly to the total number of data points that are classified. This paper is further organized as follows. Section 2 summarizes a few previous works on the prediction CVD in people. Section 3 explains working of the one-vs-all model, the dataset and the algorithms used in this research. Section 4 presents the results obtained with the conclusion and future works in Sects. 5 and 6, respectively.

2 Literature Review Many researchers have come up with a large number of heart disease prediction models to study the presence or absence of CVD in people. Most of these models are binary classifiers with a few multiclass classifiers. Few of their research works are discussed here. Varpa et al. [4] have used the KNN and SVM algorithms to make predictions on an otoneurological dataset. These algorithms are implemented with the one-vs-all and one-vs-one techniques. They have shown that using these techniques provide higher accuracy than without using them. Lui et al. [5] have built a multiclass classification model to predict myocardial infarction disease using convolutional neural network and recurrent neural network. The ECG model can be incorporated to wearable gadgets and provided an accuracy of 97.2%. Verma et al. [6] have developed a model for classification of skin diseases in people into six different classes. They have used six different algorithms to create their ensembles using ensembling techniques like bagging, Adaboost, and gradient

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boosting. Their experiments have shown that the ensembled models provide much better accuracy when compared to individual algorithms.

3 Dataset and Algorithms Used This paper is based on a dataset of CVD patients classifying them into five distinct categories. Figure 2 depicts the proposed model’s workflow. Figure 2 shows the work flow of this research in which the first task is data collection. The Cleveland Heart disease dataset that is taken from the UCI repository is used for analysis during this research work. The dataset includes 13 predictor variables. Reference [7] that are explained as follows. Age, gender, indicate the age and gender of the person, respectively. The CP variable indicates the type of chest pain the person is having. Trestbps attribute indicates the blood pressure of a person when he is at rest. The range 60–100 beats per minute trestbps denotes a healthy heart functioning. The next feature, cholesterol, when increases exceeds a certain limit, may lead to problems in the proper functioning of the heart. Fasting blood sugar (fbs) is a test conducted to detect the amount of sugar in the blood of a patient after an overnight fast. Fbs value should preferably be less than 120 mg/dl for the normal functioning of the heart. Restecg and thalach are measures of the electrical functionality and maximum heart rate achieved. Some other features that are considered in the prediction of heart disease include oldpeak, slope, exang, thalassemia, and ca [1].

Fig. 2 Workflow diagram of the proposed model

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Table 1 Relevant features for CVD prediction Age

Age of the patient

Sex

Gender of the patient

cp

Type of chest pain experienced by the patient based on factors such as chest tightness, shortness of breath, sweating, etc.

trestbps

Blood pressure of a patient, while at rest. Proper range is from 60 to 100 beats per minute

chol

Bad cholesterol level should remain below a certain threshold value for proper functioning of the heart (measurement in mg/dl)

fbs

A test conducted to detect the amount of sugar in the blood of a patient after an overnight fast. Normal when less than 120 mg/dl

restecg

Pattern generated as a result of recording the electrical functionality of the heart

Thalach The maximum rate of heart beat achieved by the person’s heart Exang

It refers to exercise-induced angina which is caused due to a reduction in blood supply to the heart muscles as a result of heavy exercising

Oldpeak ST depression induced by exercise relative to rest (‘ST’ relates to positions on the ECG plot) Slope

The slope of the peak exercise ST segment

Ca

Number of major vessels (range 0–3)

Thal

Thalassemia is a disease caused due to lack of hemoglobin in the body

Target

Heart disease detection result

Table 1 summarizes the aforementioned features that act as predictors to identify the heart disease status of a person. This dataset is then preprocessed to handle missing values and outliers, followed by partitioning of the dataset into two sections: training dataset and testing dataset in the ratio of 7:3. Finally, the predictive models are built on the training dataset using the support vector machine algorithm and the K-nearest neighbors algorithm with and without the one-vs-all approach. At last, the principal component analysis technique is applied combined with the one-vs-all approach to further enhance the prediction results. The used algorithms in this research work that is the SVM, KNN, principal component analysis, and one-vs-all are explained in brief as follows. Support Vector Machine, a supervised machine learning based algorithm, is typically used to solve classification problems. It considers each data point to be a point in an n-dimensional coordinate system, and its primary objective is to find a separating plane, known as a hyperplane that partitions the different data points belonging to different classes into distinct zones [8]. The hyperplane can be defined by the following equations: px1 + q x2 ⇐ r

(1)

px1 + q x2 > r

(2)

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SVM aims at finding suitable values for variables p, q, and r such that both the equations represent the two distinct classes. K-Nearest Neighbors classification algorithm has widespread use due to its simplicity and accuracy. It accepts all the tuples of the dataset to train the model. Any new data point can be classified into one of the classes based on the measure of the distance between the new point and the classified data points [9]. In this method, the value of K plays a vital role as it denotes the number of nearest neighbors required to be classified into any particular class. The various methods of distance measurement are Euclidean distance given by d=

1/2  , (xi − yi )2

(3)

Manhattan distance, given by d=



(xi − yi ),

(4)

Minkowski distance, given by d=

1/r  (xi − yi )r

(5)

Principal Component Analysis technique helps to eliminate the attributes with smaller variance from the dataset thus reducing its dimension and enabling the model make predictions using a reduced number of features [9]. PCA attempts to estimate the strength of relationship between the existing features and identify the pairs of features that have a stronger association. PCA finds the attributes producing a maximum variance in the high-dimensional dataset and project it onto a smaller dimensional subspace, while retaining most of the relevant information. One-vs-All technique based multiclass classification is implemented by developing ‘n’ number of binary classification models using the SVM and KNN algorithms, where ‘n’ denotes the number of classes in the target variable of the dataset [4]. For this research, ‘n’ is taken to be 5 as there are five classes in the target variable (0–4). Each binary classifier model developed is dedicated to a single category, which the classifier considers to be 1 and all other categories to be 0. The binary classifier thus makes predictions about a data point belonging to a particular category.

4 Results The results of this research work clearly indicate that the use of the one-vs-all technique improves the accuracy of multiclass classification by a significant amount. Also, after the application of the principal component analysis technique, the KNN algorithm outperforms SVM by providing an accuracy of 99.56%.

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Fig. 3 Coefficient matrices without one-vs-all

The accuracies provided by both the algorithms are calculated from the confusion matrices created. A confusion matrix is an n × n matrix where ‘n’ is the number of classes or categories into which the patients are to be categorized. In the confusion matrix, each row implies the actual number of patients belonging to that row and each column implies the number of patients predicted to belong to that column. To calculate the accuracy from the confusion matrix, the diagonal elements are added up and divided by the sum of all entries in the matrix. In the one-vs-all approach, the accuracies of the 2 × 2 confusion matrices are calculated using the formula in Eq. 6, and these accuracies are averaged to calculate the overall accuracy of the algorithm. Figures 3 and 4 show the confusion matrices generated by both the algorithms without and with the One-vs-all approach, respectively. The coefficient matrices created after applying PCA are similar to the ones with only OVA, having slightly better values than the later which improve the accuracy of the algorithms.   Accuracy = (Sum of diagonal elements)/(Sum of all elements) ∗ 100

(6)

Table 2 represents the accuracies obtained by both the algorithms without OVA, with OVA, and with OVA and PCA. Figure 5 shows the bar plot comparison of the accuracies obtained by the implementing the algorithms in their ordinary form, with OVA, and OVA combined with PCA for the SVM and KNN algorithms.

5 Conclusion The number of people dying every year due to heart disease is increasing at a rapid rate. The only way to control this swiftly increasing number is the early prediction of the chances of occurrence of CVD in a person. This prediction can be made with the help of machine learning models that are built using different algorithms. So, in order to enable the prediction process, this paper has compared two different machine

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Fig. 4 Coefficient matrices with one-vs-all Table 2 Accuracies of the algorithms used

Fig. 5 Bar graph for the accuracies obtained

Algorithm

Accuracy (%)

Accuracy OVA (%)

Accuracy PCA (%)

Support vector machine

69.565

89.565

89.565

K-nearest neighbors

72.826

89.782

99.560

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learning algorithms, support vector machine and K-nearest neighbors that are implemented with and without the one-vs-all technique the former showing exceedingly better predictive results than the latter. Further enhancement in the predictive results are obtained by applying the principal component analysis technique on the dataset. The results obtained have shown that KNN outperforms SVM algorithm by providing an accuracy of 99.56% when the principal component analysis is combined with the one-vs-all technique.

6 Future Work In future, data scientists can apply the one-vs-all technique on many other machine learning algorithms or their ensembles and aim to achieve a more optimized predictive model with even better accuracy. Further, several feature extraction techniques can be applied on the dataset to improve the model’s performance.

References 1. Mishra, S., Pandey, M., Rautaray, S.S., Gourisaria, M.K.: A survey on big data analytical tools & techniques in healthcare sector. Int. J. Emerg. Technol. 11(3), 554–560 2. Beulah Christalin Latha, C., Carolin Jeeva, S.: Improving the accuracy of prediction of heart disease risk based on ensemble classification techniques. Inf. Med. Unlocked 16, 100203. https:// doi.org/10.1016/j.imu.2019.100203 3. Baitharu, T.R., Pani, S.K.: Analysis of data mining techniques for healthcare decision support system using liver disorder dataset. In: International Conference On Computational Modelling And Security (CMS 2016). Procedia Comput. Sci. 85, 862–870 (2016) 4. Varpa, K., Joutsijoki, H., Iltanen, K., Juhola, M.: Applying one-vs-one and one-vs-all classifiers in K-nearest neighbors method and support vector machines to an otoneurological multi-class problem. Stud. Health Technol. Inf. (2011). https://doi.org/10.3233/978-1-60750-806-9-579 5. Lui, H.W., Chow, K.L.: Multiclass classification of myocardial infarction with convolutional and recurrent neural networks for portable ECG devices. Inf. Med. Unlocked 13, 26–33 (2018) 6. Verma, A.L., Pal, S., Kumar, S.: Comparison of skin disease prediction by feature selection using ensemble data mining techniques. Inf. Med. Unlocked 16, 100202 (2019). https://doi.org/ 10.1016/j.imu.2019.100202 7. Purushottam, Saxena, K., Sharma, R.: Efficient heart disease prediction system. Procedia Comput. Sci. 85, 962–969 (2016). https://doi.org/10.1016/j.procs.2016.05.288 8. Bagley, S.C., White, H., Golomb, B.A.: Logistic regression in the medical literature: standards for use and reporting with particular attention to medical domain. J Clin Epidemiol 54(10), 979–985 (2001). https://doi.org/10.1016/s0895-4356(01)00372-9 9. Salmi, N., Rustam, Z: Naïve Bayes classifier models for predicting the colon cancer. IOP Conf. Ser. Mater. Sci. Eng. 546, 052068. https://doi.org/10.1088/1757-899X/546/5/052068

Signature Analysis of Series–Parallel System Akshay Kumar , Mangey Ram , Soni Bisht , Nupur Goyal , and Vijay Kumar

Abstract This paper explores the idea of complex system having components in the form of series–parallel combination. Authors extend the idea of general law of addition of probabilities. They use this idea in the system of components having conditional mutual independent events. In this current study, the concept of general law of addition of probabilities is to find the system reliability; signature of components has been discussed. A numerical example has been taken to demonstrate the proposed techniques. Keywords Reliability · Independent events · Conditional probability · Signature · Series–parallel system

1 Introduction The tools of modern engineering are different methods for network modeling, system analysis, device repair, life cycle optimization, and decision-making. The likelihood of components is a critical factor in the study of device reliability. For these purposes, under conditions of uncertainty, all of the above tools operate, and as a result, risk is expected. In this light, an important aspect of modern practice is the role of probability and statistics in technological engineering [1–3]. In terms of translating probability and statistics into practical engineering, planning and design principles, and implementations, probability concepts in engineering are special. To find the performance A. Kumar (B) Department of Mathematics, Graphic Era Hill University, Dehradun, Uttarakhand, India M. Ram · N. Goyal Graphic Era Deemed to be University, Dehradun, Uttarakhand, India e-mail: [email protected] S. Bisht Eternal University, Baru Sahib, Himachal Pradesh, India V. Kumar AIAS, Amity University, Noida, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_31

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of the components having conditional probability and independent mutually exclusive events, reliability plays a major role in engineering fields which is strongly affected by the human source. The fields of law and probability have much assessment for the competence and credibility of human witness in system reliability [4]. Schrödinger [5] assessed the relationship between probabilities with different kind of systems in real life, and author discussed probability of two physical systems. Gupta and Kapoor [6] discussed fundamental operations and mathematical statistical methods for calculating probability of various systems using different methods such as extension formula of addition theorem. Authors give the detailed description about the random variables, mathematical expectations, theory of estimation and give the theory of distributions which is also useful in reliability theory. Anders [7] described the role of probability in various electric power system and many engineering fields on the base of unit. Apostolakis [8] determined the safety of some technological systems based on Bayesian theory. Probability and statistical methods also discussed some expert advice with experimental results. Suo et al. [9] discussed the reliability of the series–parallel system from failure probability of the unit without using classical probability method and obtain probability distribution in lower and upper forms. Frankel [10] evaluated series–parallel system’s reliability using some basic method of probability with failure rate of the units. Reliability of the non-binary system of seven units is assessed by Malik et al. [11] by connecting the maximum number of units to a single unit. Using the logic diagram approach, researchers also translated non-binary systems into simple binary systems. In the field of signature reliability, Navarro and Spizzichino [12] include a comparative analysis of a binary system with stochastic units and using a coherent system with common copula. Signature of coherent module such as series, parallel, and complex systems in terms of cumulative signature is obtained [13]. Eryilmaz [14] studied the coherent system with multiple failure elements, and the system is not working having various failure states. Author discussed signature of multistate system those who have exchangeable elements. Da et al. [15] calculated coherent module’s signature having sub-modules with huge number of units. Authors computed some basic formulae for signature reliability analysis. Coolen and CoolenMaturi [16] introduced the theory of signature analysis based on some methods like order statistic and structure systems. The survival signature of the failure system also determines combination of multi-state systems from its unit’s elements. Eryilmaz [17] studied the useful formula for signature of binary system as well as network system reliability and discussed some combinatorial problems based on the system signature analysis. The different reliability characteristics were evaluated by Bisht and Singh [18]: reliability, expected lifespan, signature reliability, and the networks were also compared on the source of different flows. Using the ugf, the authors measured reliability using various algorithms. Chahkandi et al. [19] study the signature of the repairable coherent system having statistically order method which is homologous from Samaniego’s notation. Authors consider Poisson processes and order statistics methods which consist identically elements. Franko and Yalcin [20] discussed signature of coherent system having series–parallel combination and

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solving problems based on signature and minimal signature system having nondisjoint modules. Kumar and Singh [21, 22] computed signature reliability and its various measures from reliability function of linear multi-state sliding window system and complex multi-state coherent systems having independent identically distributed with the help of statistical method of probability analysis and universal generating function. Suárez-Llorens [23] linked to the fact that the nature of a coherent system is obtained to test its reliability quantification. The reliability function of a repairable coherent structure of n-elements is also discussed here. Bisht and Singh [24] explore the idea of signature reliability in the field of multistage interconnection networks. Here, researchers discuss the signature reliability in the context of Benes network, i.e., rearrangeable non-blocking network. Authors also evaluated the reliability characteristics of considered networks in the three contexts, viz. terminal, broadcast, and network, with the use of universal generating function. From above discussion, it is clear that previously various researchers had computed the reliability function with the help of universal generating function. The novelty of this paper is authors extend the idea of general law of addition of probabilities. Authors also find the reliability function with the use of general law of addition of probabilities and analyze the signature of components in the system.

2 Importance of Signature Analysis There are various approaches for determining the importance of components in the system. With the help of signature analysis, we come across different failure probabilities of each component, which allows reliability engineers to optimize the system’s efficiency based on component value. The impact of the failure probabilities of components in the system is often used to analyze it.

3 Signature Analysis of Series–Parallel System Signature is an important tool for coherent system and plays a key role between comparisons of various stochastic systems. Signature reliability is useful both in binary and in complex system having large number of units. In earlier, some researchers obtained signature from order statistic and structure function methods as Dl P(1), m = 0, ..., n and signature S = sm−1 − sm , m = tail signature sl = (n−m)! n! 1, ..., n (Boland [25]; Navarro et al. [26, 27] and Marichal and Mathonet [28]) of system and also coherent system with independent identically distributed elements. Series–parallel system is the combination of units of order (r, s) which contains r in parallel and s in series form. Probability and structure function of binary system can be evaluated as. • Probability of series system having n-units is expressed as

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φ(r ) =

n 

ri .

i=n

• Probability of parallel system having n-units is defined as φ(s) = 1 −

n 

(1 − si ).

i=n

4 Numerical Example Consider an electronic circuit system having six components arranged in series and parallel manner with performance 0 and 1. The arrangement of complex system is having independent mutually events with conditional probability exposed by Fig. 1. Let C i be the events of the system, i = 1, 2, 3, 4, and let Di be the events from 1 to 6 units expressed as C 1 = D1 ∩ D2 C 2 = D2 ∩ D3 C 3 = D4 C 4 = D5 ∩ D6 Using the extension of general law of addition for n-events is P=



p(Ci ) −

i



p(Ci ∩ C j ) +

i< j



p(Ci ∩ C j ∩ Ck )

i< j 0. Also, 1/ p  n−1  |μ(Pi )| p i=0

λi

= lim

k→∞

 n−1 1/ p  |μk (Pi )| p i=0

λi

≤ lim V p (μk ) ≤ M < ∞ k→∞

Therefore, μ ∈ F p BV (X, F ). Since {μn } is Cauchy, for given ε > 0 there exists a natural number n 0 such that 1/ p  n−1  |(μk − μl )(Pi )| p 0, the total population is N = C s + C A + C I + C H + C R with initial conditions Cs (0) > 0, C A (0) > 0, C I (0) ≥ 0, C H (0) ≥ 0, C R (0) ≥ 0. Now, we have from model (1), N = μρ ≥ 0. We can summarize the above results as   5 Theorem 1 The closed region = (Cs , C A , C I , C H , C R ) ∈ R+ : 0 < N ≤ μρ is positively invariant set of system (1).

2.3 Existence and Uniqueness Let u 1 , u 2 , u 3 , u 4 , u 5 be the elements of the vector field f and w(t) = 5 . (Cs , C A , C I , C H , C R ) ∈ R+  Then, the system (1) can be written as w = f (w) which has the algebraic polynomials of state variables. Thus, each u i is continuous autonomous function on ∂u i ∂u i ∂u i ∂u i ∂u i , ∂C , ∂C , ∂C , ∂C exist and are continuous. Hence, R, and partial derivatives ∂C s A I H R  the system w = f (w) has a unique solution for any initial condition w(0) ∈ R 5 by existence and uniqueness theorem [31].

2.4 Fuzzy SAIHR Model There are different degree of susceptibility, infectivity, isolation and recovery from the disease of a person in the society. So, the concept of susceptibility, asymptomatic infectivity, symptomatic infectivity, isolation and recovery all are uncertain. In the model, we incorporate the heterogeneity of the population and consider some parameters as fuzzy number. To change the deterministic SAIHR model to fuzzy SAIHR model, we presume that the population heterogeneity is given by the virus loads of infected individuals. Thus, higher the virus loads, higher the chance of the disease transmission. So, it is assumed that β = β(x) represents the possibility of virus transmission from infected to the susceptible individuals with the virus load x. Here, the degree of transmission of the disease is uncertain, so,β(x) is the membership function of a fuzzy number [23, 26]. Suppose that, when the virus load is comparatively low that is, the minimum virus loads vmin , the possibility of transmission of virus is negligible. When the virus load increases from vmin , the possibility of transmission of the disease increases. For the fixed virus loads v M , the possibility of the transmission of the disease is equal to one. However, we presume that the amount of virus loads in each infected individual is limited that is denoted by vmax . Hence, β(x) (Fig. 2) is defined as follows

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Fig. 2 Fuzzy coefficient of virus transmission rate β(x)

β(x) =

⎧ ⎪ ⎨ ⎪ ⎩

0 x−vmin v M −vmin

1

if x < vmin if vmin ≤ x < v M if v M ≤ x ≤ vmax

The infection rate k = k(x) is also function of virus load x. Moreover, it is an increasing function of x, because higher the virus load in the asymptomatic infectious individual, higher will be the chance of onset of the symptom of the disease. So, we define k(x) by (Fig. 3). k(x) =

(1 − k0 ) x + k0 , if 0 < x < vmax where 0 < k0 < 1. vmax

The recovery rate from infection from asymptomatic infected humans is α = α(x) and from the symptomatic infected humans is γ = γ (x), and both are function of virus loads x. Higher virus load reduces the recovery. So, α(x) and γ (x) both are decreasing function of x as defined below (Fig. 4). α(x) =

(α0 − 1) x + 1 if 0 < x < vmax vmax

Here, α0 is the lowest recovery rate in asymptomatic infection class. γ (x) =

(γ0 − 1) x + 1 if 0 < x < vmax . vmax

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Fig. 3 Fuzzy coefficient of infection rate k(x) from asymptomatic to symptomatic

Fig. 4 Fuzzy coefficient of recovery rates α(x) and γ(x) from infection of asymptomatic and symptomatic, respectively

Here, γ0 is the lowest recovery rate in symptomatic infection class.

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2.5 Basic Reproduction Number and Equilibrium Points The fuzzy SAIHR model can be written as dCs dt dC A dt dC I dt dC H dt CR

β(x) Cs (C A + C I ) − μCs N β(x) = Cs (C A + C I ) − (k(x) + α(x) + μ)C A N

=ρ−

= k(x)C A − (η + γ + μ)C I = ηC I − (ξ + μ)C H = N − (Cs + C A + C I + C H )

(2)

The compartmental models are able to predict the different ways of the spread of virus, can estimate the duration of epidemic and can be used to understand the results of the spread. The basic reproduction number plays crucial role to obtain these results. The number is an expected number of secondary infections produced by a single infective during his/her whole infectious lifetime [27, 28] and denoted by R0 . The disease-free equilibrium P0 is asymptotically stable, and the epidemic tends to vanish that is, the disease dies out when R0 < 1. If R0< 1, then P0 is unstable, and the disease persists in the population. The point P0 = μρ , 0, 0, 0 can be obtained from the system (2) using next-generation matrix method [27, 28]. Let T1 denote the matrix of transmission terms, and T2 denotes the matrix of transition terms of the system (2) at P0 . ⎡

⎡ ⎤ ⎤ Cs β(x)/N Cs β(x)/N 0 p(x) 0 0 T1 = ⎣ 0 0 0 ⎦, T2 = ⎣ −k(x) q(x) 0 ⎦ 0 0 0 0 0 m Here, p = k(x) + α(x) + μ, q = η + γ (x) + μ and m = ξ. Therefore, the basic reproduction number R0 is  Cs β(x)(k(x) + q(x))  R0 = ρ T1 T2−1 = N p(x)q(x) The endemic equilibrium point of the system (2) P1 = (Cs∗ , C ∗A , C I∗ , C H∗ ) where

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ρ N q(x)μ(R0 − 1) , , C ∗A = μR0 β(x)(q(x) + k(x)) N k(x)(R0 − 1) N q(x)k(x)μ(R0 − 1) , C H∗ = C I∗ = β(x)(q(x) + k(x)) mβ(x)(q(x) + k(x))

Cs∗ =

Therefore, the endemic equilibrium point exists when R0 > 1, and to control the transmission of the virus, we can impose R0 < 1. Now, we assume that each infected individual has different amount of virus loads such as low, medium and high virus load. Each of them represents a triangular shape fuzzy number. Its membership function (Fig. 5) is defined as follows:  (x) =

1− 0

|x−x| ε

if x ∈ [x − ε, x + ε] if x ∈ / [x − ε, x + ε]

where x the central is value and ε denotes the dispersion of each one of the fuzzy sets assumed by (x). The fuzzy basic reproduction number is defined by [29]. f

R0 =

1 FEV{γ0 R0 (x)} γ0

Since R0 (x) can be greater than 1, it is not a fuzzy set. But, γ0 ≤ 1, so 0 ≤ f f γ0 R0 (x) ≤ 1. Thus, γ0 R0 (x) is a fuzzy set, and R0 is well defined. Here, R0 is defined as the expected number of secondary cases of an infectious individual

Fig. 5 Membership function (x)

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Fig. 6 Low, medium and high virus loads with membership function (x).

produced by a completely susceptible class. We have to define a fuzzy measure μ to obtain FEV{γ0 R0 (x)} by using possibility measure [23]. FEV{γ0 R0 (x)} = sup{inf(θ, h(θ ))}, 0 ≤ θ ≤ 1

(3)

And h(θ ) = μ{x : γ0 R0 (x) ≥ θ } = μ(X ) which is a fuzzy measure. μ(X ) = sup (x), ∀x ∈ X, X ⊂ R. We know that R0 (x) is not decreasing with x, from FEV{γ0 R0 (x)}, we have X = [x, vmax ]. Where x is the solution of the equation γ0 R0 (x) = θ . Now, we assume that the virus loads x in an individual is a linguistic variable, which will be classified as low virus loads, medium virus loads and high virus loads. The three fuzzy sets are obtained for the three different cases by the help of the membership function (x) (Fig. 6): Low virus load, if x + ε < vmin Medium virus load, if x − ε > vmin and x + ε < v M High virus load, if x − ε > v M and x + ε < vmax . f

3 Relation Between R0 (x) and R0

We have the classical reproduction number R0 (x) and fuzzy reproduction number f R0 [29].

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R0 (x) =

433

1 Cs β(x)(k(x) + q(x)) f , R0 = FEV{γ0 R0 (x)} N p(x)q(x) γ0

Case (a): Low virus load x + ε < vmin f

FEV{γ0 R0 (x)} = 0 < γ0 ⇐⇒ R0 < 1 This implies that the disease will vanish. Case (b): Medium virus load x − ε > vmin and x + ε < v M f

R0 (x) ≤ R0 ≤ R0 (x + ε) Since R0 (x) is continuous increasing function of x, using intermediate value theorem, there exists a virus load x < x + ε such that 

  f R0 = R0 x > R0 (x) 

  f f This implies that there exists x such that R0 = R0 x and R0 > R0 (x) 

Case (c): High virus load x − ε > v M and x + ε < vmax f

f

R0 (x) ≤ R0 ≤ R0 (x + ε) ⇒ R0 > 1 This implies that the disease will be endemic.

4 Stability Analysis Theorem 2. If R0 > 1, then the disease-free equilibrium is locally asymptotically stable. Proof We prove local stability of the model (1) by the help of its Jacobian matrix J . The matrix J at the disease-free equilibrium P0 is ⎡

− β(x)ρ −μ − β(x)ρ μN μN ⎢ β(x)ρ β(x)ρ ⎢ 0 − p + μN μN J =⎢ ⎣ 0 k(x) −q 0 0 q

⎤ 0 ⎥ 0 ⎥ ⎥ 0 ⎦ −m

The characteristic equation of the matrix J is   β(x)ρ β(x)ρk(x) =0 − (−μ − λ)(−m − λ) − p − λ + μN μN

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Two eigenvalues of J are λ1 = −μ and λ2 = −m. Remaining two eigenvalues are roots of the quadratic equation λ2 + aλ + b = 0 where a = p + q − β(x)ρ and b = pq(1 − R0 ). μN For any virus load x, a < 0 if p + q
0 if (1 − R0 ) > 0 ⇒ b > 0 if R0 < 1 Thus, if 0 < R0 < 1, the real parts of two roots of the equation are negative. Therefore, the real parts of all the eigenvalues of J are negative if 0 < R0 < 1. Hence, P0 is stable if 0 < R0 < 1.

5 Results We have discussed the numerical results of the fuzzy SAIHR model with different virus loads 641 copies/mi, 5 × 107 copies/ml and 1.34 × 1011 copies/ml [5]. Also, the influence of coronavirus loads on the transmission dynamics of COVID-19 disease with fuzzy behavior has been analyzed. To illustrate the dynamics of the model, we need the parameters values (Table 1). After 2–14 days of coronavirus infection, the symptoms of the disease start to appear. About 80% infected population are asymptomatic [5]. The researchers of SARS-CoV-2 suggest that the infected individual has highest virus load before the symptoms onset and decreases thereafter [30]. Normally, the virus loads peaked within first week or at 10 days after onset of the symptom. After 15 days, the virus loads get lower (Table 2). Figures 7, 8 and 9 express the dynamics of susceptible, asymptomatic and symptomatic infected population with different amount of virus loads. When the virus load is less than minimum amount (641 copies/ml), there is nominal infection of the Table 2 Values of the parameters Parameters

μ

ρ

ξ

η

N

Values

1/(70 × 365)

μ×N

0.08

0.09

3 × 105

Units

Day−1

Day−1

Day−1

Day−1

Dimensionless

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Fig. 7 Susceptible population with different virus loads

Fig. 8 Asymptomatic infected population with different virus loads

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Fig. 9 Symptomatic infected population with different virus loads

disease, and the transmission is high, when virus load is maximum amount (1.34 × 1011 copies/ml). The transmission rate of the disease increases with the increase in the virus loads x in infected individual. So, the susceptible population size decreases to its least value (Fig. 7), and infected population increases (Figs. 8, 9). Initially, due to high transmission rate, asymptomatic and symptomatic infected population increase. Later on, they start to decrease as some people start to recover from the disease, and some people may die due to the natural causes (Figs. 8 and 9).

6 Conclusion In this work, we have analyzed the dynamical behavior of the SAIHR deterministic and fuzzy epidemic models. We considered some parameters, namely transmission rate β, progression rate k, recovery rates α and γ on the model as the functions of virus loads x and defined the membership functions of these parameters. Also, we looked at the stability of the disease-free equilibrium point and obtained the classical basic reproduction number R0 (x). We accessed the fuzzy basic reproduction number f f R0 and made up the relation between R0 (x) and R0 with different virus loads. Because of natural immunity, COVID-19 could not be spread in the population if the infected individual has very low amount of virus load. If amount of virus load is very high, it will be endemic. These phenomena could be considered only in the fuzzy system, not in the crisp system. So, the fuzzy system is more realistic and flexible than the crisp system.

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Both symptomatic and asymptomatic infected humans may have different amount of virus loads. Symptomatic infected individual gets isolated or died or recover from the disease. So, there are less chance of the transmission of disease by symptomatic infected than asymptomatic infected. Since, asymptomatic infected individual does not onset the symptom but they will have high virus loads and 80% of infected individuals are asymptomatic. So, there is more risk on that kind of infected individuals. Thus, to avoid the infection of COVID-19, we must be aware of the disease such as using face mask, regular hand wash, social distancing, etc.

References 1. World Health Organization: Situation update. View corona virus disease 2019 (COVID 19) (2020). https://www.worldmeters.info/coronavirus 2. Bastola, A., Sah, R., Morales, A.J.R., Chu, D.: The first 2019 novel corona virus in Nepal. Natl. Libr. Med. 20, 279–280 (2020) 3. MoHP: Nepal’s Latest Update on COVID-19. Ministry of Health and Population, Kathmandu, Nepal (2020) 4. World Health Organization: Modes of transmission of virus causing COVID-19: Implication for IPC precaution recommendation, 29 March 2020. https://www.who.int/newsroom/com mentaries/detail/mods-of-transmission-of-virus-causing-covid-10-implication-foripc-precau tion-recommendations 5. Pan, Y., Zhang, D., Yang, P., Poon, L.L.M., Wang, Q.: Viral load of SARS-CoV-2 in clinical samples. Lancet Infect Dis. 20(4), 411–412 (2020) 6. Polonsky, J.A., et al.: Outbreak analytics: a developing data science for informing the response to emerging pathogens. Philos. Trans. R. Soc. B 374, 1–11 (2019) 7. Bhuju, G., Phaijoo, G.R., Gurung, D.B.: Sensitivity analysis of COVID-19 transmission dynamics. Int. J. Adv. Eng. Res. Appl. (IJA-ERA) 6(4), 72–82 (2020) 8. Qasim, M., Ahmad, W., Yoshida, M., Gould, M., Yasir, M: Analysis of the worldwide corona virus (COVID-19) pandemic trend: a modeling study of predict its spread, medRxiy (2020) 9. Quasim, M., Ahmad, W., Zhang, S., Yasir, M., Azhar, M.: Data model to predict prevalence of COVID-19 in Pakistan, medRxiy (2020) 10. Singh, J., Ahluwalia, P.K., Kumar, A.: Mathematical model based COVID-19 prediction in India and its different stated. medRxiy (2020) 11. Tang, Y., Wang, S.: Mathematical modeling of COVID-19 in the United States. Emerg. Microbes Infect. 9(1), 827–829 (2020) 12. Bhuju, G., Phaijoo, G.R., Gurung, D.B.: Mathematical study on impact of temperature in malaria disease transmission dynamics. Adv. Comput. Sci. 1(2), 1–8 (2018) 13. Bhuju, G., Phaijoo, G.R., Gurung, D.B.: Fuzzy approach analyzing SEIR-SEI sengue dynamics. Biomed. Res. Int. 2020, 1–11 (2020) 14. Bhuju, G., Phaijoo, G.R., Gurung, D.B.: Modeling transmission dynamics of COVID-19 in Nepal. J. Appl. Math. Phys. 8, 2167–2173 (2020) 15. Phaijoo, G.R., Gurung, D.B.: Sensitivity analysis of SEIR-SEI model of dengue disease. GAMS J. Math. Math. Biosci. 6(a), 41–50 (2018) 16. Phaijoo, G.R., Gurung, D.B.: Mathematical model of dengue disease transmission dynamics with control measures. J. Adv. Math. Comput. Sci. 23, 1–12 (2017) 17. Kermack, W.O., MacKendrick, A.G.: Contribution to the mathematical theory of epidemic. Bull. Math. Biol. 53(1–2), 33–55 (1927) 18. Li, Y., Wang, B., Peng, R., Zhan, Y., Liu, Z., Jiang, X., Zhao, B.: Mathematical modeling and epidemic prediction of COVID-19 and its significance to epidemic prevention and control measures. Ann. Infect. Dis. Epidemiol. 5(1), 1–9 (2020)

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19. Phaijoo, G.R., Gurung, D.B.: Mathematical study of dengue disease transmission in multi-patch environment. Appl. Math. 7, 1521–1533 (2016) 20. Souleiman, Y., Mohamed, A., Ismail, L: Analysis the dynamics of SIHR model: COVID-19 case in Djibouti. Math. Comput. Sci. Appl. Math. 239(1), (2020) 21. Zadeh, L.A.: Fuzzy set. Inf. Control 8, 338–353 (1965) 22. Mondal, P.K., Jana, S., Haldar, P., Kar, T.K.: Dynamical behavior of an epidemic model in a fuzzy transmission. Int. J. Univ. Fuzziness Knowl. Base Syst. 23, 651–665 (2015) 23. De Barros, L.C., Ferreira Laite, M.B., Bassanez, R.C.: The SI epidemiological models with a fuzzy transmission parameter. Int. J. Comput. Math. Appl. 45, 1619–1628 (2003) 24. Massad, E., Ortega, N.R.S., De Barros, L.C., Struchiner, C.J.: Fuzzy logic in action: application in epidemiology and beyond. Stud. Fuzzyness Soft Cimput. 232, 97–110 (2008) 25. Ahmad, S., Ullha, A., Shah, K., Salahshour, S., Ahmadian, A., Ciano, T.: Fuzzy fractional-order model of the novel coronavirus. Adv. Differ. Eqn. 472, 1–17 (2020) 26. Barros, L.C., Oliveira, R.Z.G., Leite, M.B.F., Bassanezi, R.C.: Epidemiological model of directly transmitted disease: An approach via fuzzy sets theory. Int. J. Univ. Fuzzyness Knowl. Based Syst. 22(5), 769–781 (2014) 27. Diekmann, O., Heesterbeek, J.A.P., Metz, J.A.J.: On the definition in Heterogeneous populations. J. Math. Biol. 28(4), 365–382 (1990) 28. Driessche, P., Watmough, J.: Reproduction number and sub-threshold endemic equilibria for compartment models for disease transmission. Math. Biosci. 180, 29–48 (2002) 29. Verma, R., Tiwari, S.P., Ranjit, U.: Dynamical behavior of fuzzy SIR epidemic model. Confer. Pap. Adv. Intel. Syst. Comput. 10, 482–492 (2018) 30. Supriya, L.: Virus Load Peak Before Symptom Onset in COVID-19. News Medical Life Science. https://www.news-medical.net/news/20201005/Viral-loads-peakbeforesymptomonset-in-COVID-19.aspxl 31. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, Texts in Applied Mathematics, 2nd edn. (2003)

Cryptanalysis of Fuzzy-Based Mobile Lightweight Protocol Scheme Nishant Doshi

Abstract With the advancement of technology evolvement in this twenty-first century, the use of mobile devices also increased with a rapid growth. Due to pluses of mobile devices like lightweight, mobility, features on day-to-day routines makes it a handy choice for today’s society. With the increasing use of the mobile technology, need of securing the data which were communicated among the devices becomes prevalent factor for research. Among various technologies for achieving the data security, the authentication protocol is more suitable because it provides authentication in addition to data security. The authentication protocol is broadly classified into three categories, i.e., one factor (identity), two factor (identity and text password), and three factor (identity, text password, and fuzzy biometric password). Among these, the three factor is widely used due to more secure as to others. Recently, Qiu et al. in 2020 proposed the authentication scheme using three factor for the mobile lightweight devices and proved safe against attacks. However, we prove that the Qiu et al.’s scheme is yet susceptible to the attacks like replay, session-specific temporary information (SSTI), denial of service, and stolen verifier. In addition, the Qiu et al.’s scheme yields overhead as to other conventional three-factor schemes. Keywords Three factor · Fuzzy · Authentication · Mobile lightweight device · Attacks

1 Introduction In this era of technology, the virtual distance between two persons is zero even though the physical distance is large. This happens due to the rapid improvement in a device which is tiny and efficient, i.e., mobile. As per the report (Fig. 1) from the Radicati Group [1], the number of mobile users will be growing year by year. This leads to the other challenges like more power, more efficient, data security, authentication of N. Doshi (B) Pandit Deendayal Energy University, Gandhinagar, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_38

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Fig. 1 Forecast of mobile user trend in world from year 2020–2024 [1]

device, etc. Among other challenge, the security of communication data as well as authentication is of more important as data are valuable of these devices. In [2], Lamport firstly proposed the remote user authentication protocol in which a session key is established between two users even though the physical distance is so far. In [3–60], the authors have proposed the improvements over the earlier scheme and showcase the security against the various attacks. Recently in [61], Qiu et al. mentioned the scheme for mobile device and show safe against many attacks. However, we proved that the scheme [61] is yet vulnerable to the various attacks.

1.1 Our Contributions We have cryptanalysis the scheme of Qiu et al. and prove that the scheme is susceptible to the following vulnerabilities. • Replay: Forwarding the earlier message cannot be detected or takes more computation for entity. • SSTI attack: Giving short-term keys (i.e., nonce) can help to get the session key. • Overhead computation: With the increasing session as well as user, system requires unnecessary computations. • Denial of service: System becomes unresponsive for multiple requests.

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• Stolen verifier: Server stores the credentials related to users is going to be compromised.

1.2 Paper Organization In Sect. 2, we have given the basic terminology to be useful throughout the paper. Section 3 deals with the various phases of the Qiu et al. scheme. In Sect. 4, the cryptanalysis of Qiu et al. scheme is given. Conclusion is given in Sect. 5.

2 Literature Survey This section deals with notations to be useful for Qiu et al.’s scheme (Table 1). In 2013, Tsai et al. [29] proposed the two-factor authentications scheme and proved its security against various attacks. In 2015, Zhu et al. [22] proposed the improved scheme that shows safe by many attacks. In 2017, Wazid et al. [3] also proposed the more secure scheme. In similar line, the authors from [10–50] proposed the secure scheme as to its predecessor. Table 1 Notations

Symbol

Meaning

S

Server

Ui

ith user in the system

I Di , P Wi , B I Oi

Secret credentials for Ui

p

Large prime

Tn (x)

Chebyshev chaotic maps for n ∈ Z + and x ∈ [−1, 1]

s, Ts (x)

Private and public key for S

A

Adversary of system

a||b

Concatenation operation

a⊕b

XOR operation between a and b

H (), H0 ()

One-way hash function

GEN()

Fuzzy extraction algorithm for B I Oi

REP()

Fuzzy reproduction algorithm for B I Oi

SCi

Smart card reader

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Fig. 2 Registration of user phase [61]

3 Scheme of Qiu et al. The main phases of the scheme are as follows.

3.1 System Setup Phase The following procedure will be done by user Ui and server S in the system over secure channel (Fig. 2).

3.2 Login and Authentication The following procedure will be done between Ui and server S over public channel (Fig. 3).

3.3 Password and Biometric Update Phase This phase is between user Ui and smart card reader SCi (Fig. 4).

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Fig. 3 Login and authentication phase [61]

4 Cryptanalysis of Qiu et al.’s [61] This section deals with the attacks with detailed steps for the same in Qiu et al.’s scheme.

4.1 SSTI Attack The system is said to be vulnerable to this attack if compromise of session-specific temporary variables leads to compromise of session key. An attacker A is having u, einew , v. • Computes C2 = Tu (Ts (x)) as Ts (x) is public key of S • Computes I Di and B0 by C3 ⊕ H0 (C2 ) as C3 is available from public channel messages. • Gets C I Di from public channel message

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Fig. 4 Password and biometric update [61]

• Computes C6 = Tv (C1 ) • Computes B0new by C7 ⊕ H (C6 ||B0 )   • Finally computes session key S K = H I Di C I Di B0 B0new C2 C6 . Thus, the Qiu et al. are prone by SSTI.

4.2 Replay Attack In the Qiu et al.’ scheme, S fails to identify the repetition of {C I Di , C1 , C3 , C4 }. Thus, the Qiu et al. are prone by the replay.

4.3 Overhead Computation In the conventional authentication protocol schemes, once the smart is given to users by server, it remains as it is until the change of any credential. However, in this scheme, authors have updated the credentials in smart of Ui for every session with server.

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Table 2 Analysis References [3] [4] [5] [21] [22] [23] [61]

EC1 √

EC2 √

EC3 √

√

√

√

√

√

√

√

√

√

√

√

√

√ √

× ×

√ √

EC4 × × × × × √ ×

EC5 √ √ √ √ × √ √

EC6 √ × × √ × √ ×

EC7 √

EC8 √

EC9 √

√

√

√

√ √ √ √ √

× × × √ ×

√ √ √ √ √

Consider that if there are 1 billion devices and if each devices do the ten communication with server than for the 10 billion session a day following computation overhead on the server and each individual user is imposed. • Server S (30 billion hash functions to be run) • User Ui (ten hash function, ten XOR operation, ten modular arithmetic, and ten random number generation).

4.4 Denial of Service The scheme is secure against this attack if system will detect the fake requests at early stage without much computation overhead. In the Qiu et al. scheme, server S requires more amount of time to verify the user as well as extra steps for generating updated parameters like B0new , C I D new .

4.5 Stolen Verifier Attack System is said to be secure if server stores only necessary data. In the Qiu et al. scheme, server S requires to store the I Di , ri , Honey_list. With compromise of these data, the identity of all users will be revealed too which compromise the anonymity of the system. During the login and authentication phase, server S requires to identify the I Di by exhaustive search in the list and extract related ri . With increasing number of users, this lead to the searching overhead (Table 2).

5 Conclusion and Future Work Mobile device is the key to the future for its pluses with its requirement with people’s need. In addition, the data generated through these mobility devices also need to be

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secure as well as authenticated too. In 2020, Qiu et al. proposed the scheme with three-factor remote user authentications and prove to be efficient as well as safe by many attacks. However, here, we proved that the scheme is unsafe against various attacks. In addition, it also embarks the additional overhead in the multi-user scenario. In future, one can device the more efficient as well secure efficient scheme in mobility devices.

References 1. The Statistics Portal: Forecast Number of Mobile Users Worldwide from 2019 to 2023. Accessed: 2020. https://www.statista.com/statistics/218984/number-of-globalmobile-userssince-2010 2. Lamport, L.: Password authentication with insecure communication. Commun. ACM 24(11), 770–772 (1981) 3. Wazid, M., Das, A.K., Kumar, N., Rodrigues, J.: Secure three-factor user authentication scheme for renewable-energy-based smart grid environment. IEEE Trans. Indus. Inform. 13(6), 3144– 3153 (2017) 4. Roy, S., Chatterjee, S., Das, A.K., Chattopadhyay, S., Kumari, S., Jo, M.H.: Chaotic mapbased anonymous user authentication scheme with user biometrics and fuzzy extractor for crowdsourcing internet of things. IEEE Internet of Things J. 4, 2884–2895 (2018) 5. Islam, S.H., Vijayakumar, P., Bhuiyan, M.Z.A., Amin, R., Varun Rajeev, M., Balusamy, B.: A provably secure three-factor session initiation protocol for multimedia big data communications. IEEE Internet of Things J. 5(5), 3408–3418 (2018) 6. Patel, C., Doshi, N.: Security challenges in IoT cyber world. In: Hassanien, A., Elhoseny, M., Ahmed, S., Singh A. (eds.) Security in Smart Cities: Models, Applications, and Challenges. Lecture Notes in Intelligent Transportation and Infrastructure. Springer, Cham (2018) 7. Arkko, J., Torvinen, V., Camarillo, G., Niemi, A., Haukka, T.: Security mechanism agreement for SIP sessions. IETF Internet Draft (2002) 8. Patel, C., Doshi, N.: Security challenges in IoT cyber world. In: Hassanien, A., Elhoseny, M., Ahmed, S., Singh, A. (eds.) Security in Smart Cities: Models, Applications, and Challenges. Lecture Notes in Intelligent Transportation and Infrastructure. Springer, Cham. (2018) 9. Wang, D., Cheng, H.B., He, D.B., Wang, P.: On the challenges in designing identity-based privacy-preserving authentication schemes for mobile devices. IEEE Syst. J. 12(1), 916–925 (2018) 10. Chatterjee, S., Roy, S., Das, A.K., Chattopadhyay, S., Kumar, N., Vasilakos, A.V.: Secure biometric-based authentication scheme using Chebyshev Chaotic map for multi-server environment. IEEE Trans. Dependable Sec. Comput. 15(5), 824–839 (2018) 11. Gope, P., Lee, J., Quek, T.Q.S.: Lightweight and practical anonymous authentication protocol for RFID systems using physically unclonable functions. IEEE Trans. Inform. Forensics Secur. 13(11), 2831–2843 (2018) 12. Wang, D., Wang, P.: On the anonymity of two-factor authentication schemes for wireless sensor networks: attacks, principle and solutions. Comput. Netw. 73, 41–57 (2014) 13. Patel, C., Joshi, D., Doshi, N., Veeramuthu, A., Jhaveri, R.: An enhanced approach for three factor remote user authentication in multi-server environment. IOS J. Intell. Fuzzy Syst. 1–12 (2020) (Pre-press) 14. Das, M.L., Saxena, A., Gulati, V.P.: A dynamic ID-based remote user authentication scheme. IEEE Trans. Consum. Electron. 50(2), 629–631 (2004) 15. Kocarev, L., Lian, S.: Chaos-based cryptography: theory, algorithms and applications. Springer, Berlin (2011)

Cryptanalysis of Fuzzy-Based Mobile Lightweight Protocol Scheme

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16. Xiao, D., Liao, X.F., Wong, K.W.: An efficient entire chaos-based scheme for deniable authentication Chaos. Solitons Fractals. 23, 1327–1331 (2005) 17. Patel, C., Doshi, N.: A novel MQTT security framework in generic IoT model. Procedia Comput. Sci. 171, 1399–1408 (2020) 18. Xiao, D., Liao, X.F., Deng, S.J.: A novel key agreement protocol based on chaotic maps. Inf. Sci. 177(4), 1136–1142 (2007) 19. Han, L.D., Xie, Q., Liu, W.H., Wang, S.B.: A new efficient ChaoticMaps based three factor user authentication-and-key-agreement scheme. Wirel. Pers. Commun. 95(3), 3391–3406 (2017) 20. Lee, T.F., Hsiao, C.H., Hwang, S.H., Lin, T.H.: Enhanced smart card-based password authenticated key agreement using extended chaotic maps. PLoS ONE 12(7), e0181744 (2017). https:// doi.org/10.1371/journal.pone.0181744 21. Vaghashiya, R., Thakore, R., Patel, C., Doshi, N.: IoT—principles and paradigms. Int. J. Adv. Trend. Comput. Sci. Eng. 8(6), 153–158 (2019) 22. Zhu, H., Hao, X.: A provable authenticated key agreement protocol with privacy protection using smart-card based on chaoticmaps. Nonlinear Dyn. 81(1–2), 311–321 (2015) 23. Jiang, Q., Wei, F., Fu, S., et al.: Robust extended chaotic maps-based three-factor authentication scheme preserving biometric templateprivacy. Nonlinear Dyn. 83(4), 2085–2101 (2016) 24. Li, X., Wu, F., Khan, M.K., Xu, L.L., Shen, J., Jo, M.H.: A securechaotic map-based remote authentication scheme for telecaremedicine-information-systems. Fut. Gener. Comp. Syst. 84, 149–159 (2018) 25. Wang, D., Wang, N., Wang, P., Qing, S.H.: Preserving privacy for free: efficient and provably secure two-factor authentication scheme with user anonymity. Inf. Sci. 321, 162–178 (2015) 26. Dankhara, F., Patel, K., Doshi, N.: Analysis of robust weed detection techniques based on the Internet of Things (IoT). Procedia Comput. Sci. 160, 696–701 (2019) 27. Yang, G.M., Wong, D.S., Wang, H.X., Deng, X.T.: Two-factormutual authentication based on smart-cards and passwords. J. Comput. Syst. Sci. 74(7), 1160–1172 (2008) 28. Das, M.L.: Two-factor user authentication in wireless sensor networks. IEEE Trans. Wirel. Commun. 8(3), 1086–1090 (2009) 29. Sachdev, S., Macwan, J., Patel, C., Doshi, N.: Voice-controlled autonomous vehicle using IoT. Procedia Comput. Sci. 160, 712–717 (2019) 30. Lin, H.Y.: Improved chaotic maps-based password-authenticated key agreement using smartcards. Commun. Nonlinear Sci. Numer. Simul. 20, 482–488 (2015) 31. Wang, D., Wang, P.: On the implications of zipf’s law in passwords. In: Proceedings ESORICS 2016, ser. LNCS, vol. 9878. Springer, Berlin, pp. 1–21 32. Islam, S.H.: Provably secure dynamic identity-based three-factor password authentication scheme using extended chaotic maps. Nonlinear Dyn. 78(3), 2261–2276 (2014) 33. Lee, T.F.: Enhancing the security of password authenticated keyagreement protocols based on chaotic maps. Inf. Sci. 290, 63–71 (2015) 34. Guo, X., Zhang, J.: Secure group key agreement protocol based onchaotic hash. Inf. Sci. 180(20), 4069–4074 (2010) 35. Chen, C.M., Fang, W.C., Wang, K.H., Wu, T.Y.: Comments on “An improved secure and efficient password and chaos-based two-party key agreement protocol.” Nonlinear Dyn. 87(3), 2073–2075 (2017) 36. Shin, S., Kobara, K.: Security analysis of password-authenticated key retrieval. IEEE Trans. Dependable Sec. Comput. 14(5), 573–576 (2017) 37. Patel, C., Doshi, N.: Cryptanalysis and improvement of Barman et al.’s secure remote user authentication scheme. Int. J. Circuits Syst. Signal Process. 13, 604–610 (2019) 38. Jablon, D.P.: Password authentication using multiple servers. In: Proceedings of Conference Topics Cryptology: The Cryptographer’s Track at RSA, pp. 344–360 (2001) 39. Wang, D., Wang, P.: Two birds with one stone: two-factor authentication with security beyond conventional bound. IEEE Trans. Depend Secur. Comput. 15(4), 708–722 (2018) 40. Kocarev, L., Tasev, Z.: Public-key encryption based on Chebyshev maps. In: Proceedings of IEEE Symposium on Circuits and Systems (ISCAS’03), vol. 3, pp. 28–31 (2003)

448

N. Doshi

41. Zhang, L.: Cryptanalysis of the public key encryption based on multiple chaotic systems. Chaos Solitons Fract. 37(3), 669–674 (2008) 42. Bergamo, P., Arco, P., Santis, A., Kocarev, L.: Security of publickey cryptosystems based on Chebyshev polynomials. IEEE Trans. Circ. Syst. 52, 1382–1393 (2005) 43. Dodis, Y., Reyzin, L., Smith, A.: Fuzzy extractors: how to generate strong keys from biometrics and other noisy data. In: Proceedings of EUROCRYPT, pp. 523–540 (2004) 44. Wang, D., He, D., Wang, P., Chu, C.: Anonymous two-factor authentication in distributed systems: certain goals are beyond attainment. IEEE Trans. Depend. Secur. Comput. 12(4), 428–442 (2015) 45. Shingala, M., Patel, C., Doshi, N.: An improve three factor remote user authentication scheme using smart card. Wirel. Pers. Commun. 99(1), 227–251 (2018) 46. Kocher, P., Jaffe, J., Jun, B.: Differential power analysis. ProcCrypto. 1666, 388–397 (1999) 47. Messerges, T.S., Dabbish, E.A., Sloan, R.H.: Examining smartcard security under the threat of power analysis attacks. IEEE Trans. Comput. 51(5), 541–552 (2002) 48. Wang, D., Zhang, Z., Wang, P.: Targeted online password guessing: an underestimated threat. Proc. ACM CCS. 16, 1242–1254 (2016) 49. Wang, D., Cheng, H.B., Wang, P., Huang, X.Y., Jian, G.P.: Zipf’s law in passwords. IEEE Trans. Inform. Forensics Secur. 12(11), 2776–2791 (2017) 50. Agrawal, S., Das, M.L., Lopez, J.: Detection of node capture attack in wireless sensor networks. IEEE Syst. J. 13(1), 238–247 (2019) 51. He, D.B., Wang, D.: Robust biometrics-based authentication scheme for multiserver environment. IEEE Syst. J. 9(3), 816–823 (2015) 52. Doshi, N., Kumari, S., Mishra, D., Li, X., Raymond Choo, K.-K., Sangaiah, A.K.: A password based authentication scheme for wireless mobile communications. Multimed. Tools Appl. 76(24), 25893–25918 (2016) 53. Wang, D., Li, W.T., Wang, P.: Measuring two-factor authentication schemes for real-time data access in industrial wireless sensor networks. IEEE Trans. Indus. Inform. 14(9), 4081–4092 (2018) 54. Madhusudhan, R., Mittal, R.: Dynamic id-based remote user password authentication schemes using smart-cards: a review. J. Netw. Comput. Appl. 35(4), 1235–1248 (2012) 55. Juels, A., Rivest, R.L., Honeywords: making password cracking detectable. Proc. ACM CCS 145–160 (2013) 56. Wang, D., Cheng, H.B., Wang, P., Yan, J., Huang, X.Y.: A security analysis of honeywords. Proc. NDSS (2018). https://doi.org/10.14722/ndss.2018.23142 57. Bellare, M., Pointcheval, D., Rogaway, P.: Authenticated key exchange secure against dictionary attacks. In: Proceedings of EUROCRYPT2000, ser. LNCS. Springer (2000), vol. 1807, pp. 139–155 58. Bresson, E., Chevassut, O., Pointcheval, D.: Security proofs for an efficient password-based key exchange. In: Proceedings of ACMCCS 2003, ACM, New York, NY, USA, pp. 241–250 (2003) 59. Shoup, V. (2004) Sequences of games: a tool for taming complexity insecurity proofs. IACR Cryptol. ePrint Archive 332 (2004) 60. Canetti, R., Krawczyk, H.: Analysis of key-exchange protocols and their use for building secure channels. In: Proceedings of EUROCRYPT2001, ser. LNCS. Springer, (2001), pp. 453–474 61. Qiu, S., Wang, D., Xu, G., Kumari, S.: Practical and provably secure three-factor authentication protocol based on extended Chaotic-Maps for mobile lightweight devices. In: IEEE Transactions on Dependable and Secure Computing, vol. 01, pp. 1–1, 5555. https://doi.org/10.1109/ TDSC.2020.3022797

Energetic and Exergetic Analyses of Hybrid Wind-Solar Energy Systems Hardik K. Jani , Surendra Singh Kachhwaha , and Garlapati Nagababu

Abstract The energetic analysis of system gives a clear indication of the system performance. However, in order to attain an accurate idea about the actual potential of the energy conversion and the regimes to be focused on performance improvement, the exergetic analysis is an important and necessary tool. The present study encompasses both energy and exergy analyses on the hybrid wind-solar energy systems (HWSES). The study presents that hybridizing the energy systems would reduce the energy efficiency variations over the month and would ensure the mean efficiency above 15% throughout the year. Moreover, the addition of solar energy system to the existing wind energy system enhances the overall exergy efficiency while achieving a peak value of over 10% without falling below 5% over the year. Hence, HWSES has superior performance than the standalone systems in terms of energetic and exergetic efficiencies. Keywords Wind energy · Solar energy · Hybrid energy · Energy analysis · Exergy analysis

1 Introduction According to the recently declare National Wind-Solar Hybrid Policy, India is aiming the 180 GW capacity through renewable energy (RE) by 2022 [1]. For achieving the national goal, continuous growth of renewable power projects is essential [2]. Hybrid renewable energy (HRE) system is the combination of two or more RE systems founded in the same premises utilizing same infrastructures [3]. Solar and wind energy systems are the most commonly used RE resources [4]. The hybridization lowers the overall power generation cost and provided power with fewer fluctuations.

H. K. Jani (B) · S. S. Kachhwaha · G. Nagababu School of Technology, Pandit Deendayal Energy University, Gandhinagar, Gujarat, India S. S. Kachhwaha e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_39

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The energetic and exergetic analyses are used for assessing the operation of these power generation systems. The energetic and exergetic analyses are performed using the first and second law of thermodynamics, respectively. Such wholesome energetic and exergetic analyses have not yet been conducted for Indian locations. The intent of present research is to assess the operation of standalone and hybrid wind-solar energy systems (HWSES) by means of energetic and exergetic analyses.

2 Mathematical Modeling The input resource data was obtained from the ERA5 reanalysis dataset which has already been identified as superior with respect to other reanalysis datasets in previous research [5, 6]. The system particulars of the wind and solar energy systems were taken as given in Tables 1 and 2 in the same order. Table 1 Wind turbine technical specifications [7] Parameter

Value

Rated power (PW,r )

2.8 MW

Hub height (h)

105 m

Swept area (AT )

13,070 m2

Rotor diameter

128 m

Rated WS (ur )

9.5 m/s

Cut-out WS (uco )

20 m/s

Cut-in WS (uci )

3 m/s

Table 2 System particulars of thin-film solar PV system [8, 9] Parameter

Value

Rated power (Ps,r )

2.8 MW 16,473 m2

Surface area of PV panel (APV ) Solar irradiation at

STCa

1000 W/m2

(H STC )

Derating coefficient (c1 )

93%

Power temperature coefficient (c2 )

-0.5%

Temperature of PV panel at STC (T STC )

298 K

Temperature of PV panel at

TETCb

(T s,TETC )

320 K

Ambient temperature at TETC (T a,TETC )

293 K

Solar irradiation at TETC (H TETC )

800 W/m2

a STC

Standard test conditions Temperature estimation test conditions

b TETC

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2.1 Wind Energy System (WES) The wind speed (WS) data of 10 m height is obtained from ERA5 reanalysis dataset. For evaluating the power generation from wind turbine, the wind data is extrapolated to the hub height using the power law [8],  u i = u ref

h h ref

β (1)

where uref is the reference WS at height href , u is the extrapolated WS at height h and β is wind shear exponent (taken as 1/7) [10]. Now, the extrapolated WS values are used for the computation of power generated by wind turbine (PW in kW) for available WS (u in m/s) values. It is computed using the power curve provided by the manufacturer of selected wind turbine SUZLON-S128 and can be presented as following, ⎧ 0 ⎪ ⎪ ⎪ 5 4 3 ⎪ ⎨ − 496.9 · u + 11493 · u − 94363 · u 2 + 372336 · u −539386 · u + 3 PW = ⎪ ⎪ ⎪ P ⎪ ⎩ W,r 0

(u < u ci ) (u ci ≤ u ≤ u r )

(2)

(u r ≤ u ≤ u co ) (u > u co )

The capacity factor (CF) for wind turbine (CFW ) is calculated, which is given by dividing the power generation by rated power [10], CFW =

PW,avg PW,r

(3)

The wind turbine energy efficiency (ηw ) is derived by dividing power generation through by power present in wind in the form of kinetic energy and can be given as following [9], ηW =

Pwind PW = 1 Pkinetic ρ AT u 3 2

(4)

where ρ is air density (in kg/m3 ). Overall specific exergy (e in kJ/kg) of a system is summation of kinetic, potential, physics, and chemical specific exergies (eK , eP , ePh , and eCh , respectively) and can be given as, e = eK + eP + ePh + eCh

(5)

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The exergy efficiency is derived by dividing exergy generation by net exergy input [11], ψW =

E˙ gen PW = ˙ m[(e ˙ Ki − eKo ) + (ePhi − ePho ) + (eChi − eCho )] E N i,W

(6)

where m˙ is air mass flow rate at turbine rotor [12].

2.2 Solar Energy System (SES) The power generated through photovoltaic (PV) depends on multiple parameters and can be modeled as given below [8], PS =

Hs · PS,r · c1 · [1 + c2 (Ts − TSTC )] HSTC

(7)

where PS is the power generated by PV panels (W). H s is surface incoming shortwave flux (SISF) (W/m2 ). The temperature of PV panel in working condition (T S ) (in K) can be given by [8], T s = Ta + Hs ·

Ts,TETC − Ta,TETC HTETC

(8)

where T a is surface skin temperature (ambient temperature) (K). The CF of the solar PV system (CFS ) can be stated as follows, CF S =

PS PS,r

(9)

The energy efficiency of solar PV systems (ηS ) is derived using the following expression [9], ηS =

PS HS × APV

(10)

where APV is the area of PV panel. The expression for the exergy available through the solar radiation can be given by [13],      1 4 Ta Ta 4 ˙ + · E N i,S = 1 − · · HS APV 3 TSun 3 TSun

(11)

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where T Sun is the temperature of the surface of the Sun, i.e., 5780 K [13]. The expression for the exergy efficiency (ψ S ) is given as [14]. ψS =

PS =

E˙ N i,S 1−

4 3

·



T0 TSun



PS +

1 3

·



T0 TSun

4

(12) · HS APV

2.3 Hybrid Wind-Solar Energy System (HWSES) Overall energy efficiency of the HWSES (ηH ) is calculated using total energy inflow and outflow for the system as given below [9], ηH =

PW + PS Pkinetic + (HS · APV )

(13)

The CF for the HWSES can be given by, CF H =

PW + PS PW,r + PS,r

(14)

The exergy efficiency of the HWSES can be given by, ψH =

PW + PS m[(e ˙ Ki − eKo ) + (ePhi − ePho ) + (eChi − eCho )]      T0 4 T0 + 0.33 · + 1 − 1.33 · · HS APV TSun TSun

(15)

3 Results Monthly mean wind and solar resource data for 41 years (1979–2019) time duration were obtained from the ERS5 reanalysis dataset for the selected site location in Gujarat, India (20.75° N, 71.25° E) [6]. With the help of the mathematical model explained earlier, the energy and exergy analyses were performed on HWSES of 5.6 MW capacity. Figure 1 shows the monthly variation of system efficiencies of standalone wind and solar compared to hybrid wind-solar energy systems, respectively. Here it can be observed that the variation in CF and exergy efficiencies in wind energy systems is (from zero to 55%) higher than the same in solar energy systems (between 16 and

454

Fig. 1 Month-wise system performance of a WES, b SES, and c HWSES

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27%). Hybridizing energy systems reduces month-to-month energy efficiency fluctuations and maintains a year-round average efficiency of 15% or higher. Furthermore, combining a solar energy system with an installed wind energy system improves total energy efficiency, with a peak value of over 10% and a year-over-year average of less than 5%. Moreover, the cycle of achieving crest and valley in also inverse with respect to each other. Hence, the maximum CF in WES is obtained in the month of July (55%), which is the least CF month (17%) in case of SES. Similarly, the highest CF in SES is obtained in the month April (around 28%), whereas the system CF in WES in below 8% in the same month. Hence, it can be concluded that the solar and wind energy systems complement each other, and the hybridization would contract the overall power generation variations and improve the year-round system performance with the CF and exergy efficiency ranges of 10–36% and 5–11%, respectively. Moreover, in order to identify the long-term variation in the system efficiencies over 41 years, the time series of the system efficiencies have been depicted in Fig. 2, respectively, for WES, SES, and HWSES. It has been found that the system efficiencies go through the similar pattern of cycle over the years and do not have any significant alteration. As a future work, the advance and extended exergy analyses can also be conducted on the hybrid energy systems, in order to get the idea about the avoidable and unavoidable regimes of the exergy destruction.

4 Conclusion The analyses of standalone and hybrid wind-solar energy systems are performed using the actual system specifications and ERA5 reanalysis resource data. According to the findings, combining the energy systems reduces month-to-month energy efficiency fluctuations and guarantees a mean efficiency of 15% or higher over the year. Furthermore, combining a solar energy system with an established wind energy system improves total energy efficiency while maintaining a peak value of over 10% without dipping below 5% over the course of the year. Hence, the hybrid systems have superior performance throughout the year as compared to the standalone systems. Moreover, the monthly means energy efficiencies indicate that the windsolar energy resource distribution complements each other and have more stabilized power generation.

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Fig. 2 System performance of a WES, b SES, and c HWSES (1979–2019)

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References 1. Das, A., Jani, H.K., Nagababu, G., Kachhwaha, S.S.: Wind and solar power deployment in India: economic aspects and policy implications. Afr. J. Sci. Technol. Innov. Dev. (2020). https://doi.org/10.1080/20421338.2020.1762302 2. Das, A., Jani, H.K., Kachhwaha, S.S., Nagababu, G.: Assessment of factors affecting onshore wind power deployment in India. Environ. Clim. Technol. 24, 185–208 (2020). https://doi.org/ 10.2478/rtuect-2020-0012 3. Das, A., Jani, H.K., Nagababu, G., Kachhwaha, S.S.: A comprehensive review of wind–solar hybrid energy policies in India: Barriers and Recommendations. Renew. Energy Focus. (2020). https://doi.org/10.1016/j.ref.2020.09.004 4. Das, A., Jani, H.K., Nagababu, G., Kachhwaha, S.S.: Influence of techno-economic factors on the Levelized cost of electricity (LCOE ) of wind and solar power projects in India. In: ICTEA: International Conference on Thermal Engineering (2019) 5. Jani, H.K., Nagababu, G., Patel, R.P., Kachhwaha, S.S.: Comparative study of meteorological and reanalysis wind data for offshore wind resource assessment. In: ICTEA: International Conference on Thermal Engineering (2019) 6. Jani, H.K., Nagababu, G., Patel, R.P., Kachhwaha, S.S.: A comparative analysis of LiDAR and wind mast measured wind data with the reanalysis datasets for an offshore location of Gujarat. In: Vijayaraghavan, L., Reddy, K., and Jameel Basha, S. (eds.) Emerging Trends in Mechanical Engineering. Lecture Notes in Mechanical Engineering. pp. 627–634. Springer, Singapore (2020). https://doi.org/10.1007/978-981-32-9931-3_61 7. Wind Turbine Manufacturer | Wind and Solar—Products and Solutions | Suzlon Energy LTD. https://www.suzlon.com/in-en/energy-solutions. Last accessed 18 Mar 2020 8. Ren, G., Wan, J., Liu, J., Yu, D.: Spatial and temporal assessments of complementarity for renewable energy resources in China. Energy 177, 262–275 (2019). https://doi.org/10.1016/j. energy.2019.04.023 9. Hasan, A., Dincer, I.: Development of an integrated wind and PV system for ammonia and power production for a sustainable community. J. Clean. Prod. 231, 1515–1525 (2019). https:// doi.org/10.1016/j.jclepro.2019.05.110 10. Manwell, J.F., McGowan, J.G., Rogers, A.L.: Wind Energy: Explained Theory, Design and Applications. Wiley (2009) 11. Allouhi, A.: Energetic, exergetic, economic and environmental (4 E) assessment process of wind power generation. J. Clean. Prod. 235, 123–137 (2019). https://doi.org/10.1016/j.jclepro. 2019.06.299 12. Ehyaei, M.A., Ahmadi, A., Rosen, M.A.: Energy, exergy, economic and advanced and extended exergy analyses of a wind turbine. Energy Convers. Manag. 183, 369–381 (2019). https://doi. org/10.1016/j.enconman.2019.01.008 13. Shaygan, M., Ehyaei, M.A., Ahmadi, A., Assad, M.E.H., Silveira, J.L.: Energy, exergy, advanced exergy and economic analyses of hybrid polymer electrolyte membrane (PEM) fuel cell and photovoltaic cells to produce hydrogen and electricity. J. Clean. Prod. 234, 1082–1093 (2019). https://doi.org/10.1016/j.jclepro.2019.06.298 14. Yanan, W., Jiekang, W., Xiaoming, M.: Exergy analysis of cogeneration system for the wind– solar–gas turbine combined supply. Appl. Sol. Energy (English Transl. Geliotekhnika). 54, 369–375 (2018). https://doi.org/10.3103/S0003701X18050213

Analysis of State of Health Estimation for Lithium-Ion Cell Using Unscented and Extended Kalman Filter Chaitali Mehta, Amit V. Sant, and Paawan Sharma

Abstract There is tremendous potential for the growth of electric vehicles (EVs) in near future. Hence, investigations on different aspects of propulsion system, battery management and charging infrastructure are focused upon by academic and industrial researchers. Battery management systems (BMSs) are responsible to estimate battery parameters and protection. Precise estimation of the State of Charge (SoC) and State of Health (SoH) is crucial functionality of the BMS. Such estimation can inform the driver about the remaining range, replacement of battery pack in their vehicle and the charging requirements. SoC and SoH estimation necessitate a reliable model for battery. This paper provides a comparative study of the SoH and SoC assessment of lithium-ion cells with the help of Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF). Different factors such as temperature and variations in internal resistance are considered in the simulation studies. Important methods for estimation of SoH and SoC are reviewed. An equivalent circuit model with 3-RC branch is used in analyses, to simulate lithium-ion cell. The rise in internal resistance of lithiumion cell as a consequence of charging and discharging is analyzed, and the resulting degradation is studied using UKF and EKF. Keywords Extended Kalman Filter · Unscented Kalman Filter · State of charge · State of health · Lithium-ion battery

1 Introduction Electric vehicles (EVs) are considered as future of transportation segment as they are promising candidates for low-carbon footprint. EVs can be defined as any vehicle which is propelled by electric motor that takes power from some rechargeable energy source, mostly battery energy storage system. From overall perspective of public, there are apprehensions regarding the use of this technology. But with C. Mehta (B) · A. V. Sant · P. Sharma Pandit Deendayal Petroleum University, Gandhinagar, Gujarat, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_40

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recent development in battery technology, more reliable battery storage sources are developed. EVs utilize an electric motor and rechargeable battery for its drivetrain as compared to internal combustion engine, fuelled by petrol/diesel. Currently, there are quite a few low-speed and high-speed EVs which are commercially available, most of which use lithium-ion based batteries. Recent developments in lithium-ion based battery technology have resulted in the battery having higher specific power, higher energy density, higher terminal voltage, better efficiency and long usable life [1]. Estimating SoH and SoC accurately will play crucial role in reducing range anxiety amongst EV users [2]. The real-time status of battery is maintained by BMS. One of the prime tasks of BMS is to indicate battery charge consumption which reveals the remaining driving range and to ensure optimal operation of a battery during overcharge and over-discharge process. The SoC estimation is essential for BMS as it performs cell balancing, protection against overcharge and over-discharge, etc. [3]. All batteries that meet performance criteria at the time of manufacturing eventually degrades with time due to calendar life, charge/discharge cycle, resulting in gradual loss in battery capacity and an increase in internal resistance [4]. By monitoring increase in internal resistance and degrading battery capacity accurate SoH estimation can be achieved. In [5, 6], the effect of various factors that has some effect on the battery life has been examined, and it is observed from the results that with increase in battery age. In [7], an online technique to estimate SoH is presented for LiFePO4 (LFP) cells, the method examines the variation voltage curve in capacity vs. voltage graph in SoH estimation. In [8], the author has developed a dynamic framework with EKF for SoC and usable capacity estimation after every single cycle. Here, the estimated capacity is used to find the SoH of the battery. In [9], an Artificial Neural Networks method is presented, and this method is compared with EKF for SoH estimation. In this method, Artificial Neural Networks use the equivalent circuit model for finding battery parameters to estimate the battery life. From the results presented, it can be said that computational complexity of Artificial Neural Networks is less compared to EKF. In [10], the author proposes a Lyapunov based adaptive SoH estimation technique. The work done in [11] employs change in capacity degradation and increase in internal resistance is modelled through particle filter technique to estimate the available battery life. The accuracy of the proposed method vastly depends on the input data used for the modelling degradation. In [12, 13], a fuzzy logic-based SoH and SoC estimation technique are presented. This technique uses electrolytic impedance spectroscopy based measured data for estimation. Fuzzy logic used in this paper is a powerful technique; however, it requires huge computations and large amount of test data for accurate estimation. This paper presents a comparative study to estimate SoH of a lithium-ion cell of 31 Ah, 3.6 V using EKF and UKF. Generally, battery modelling for SoH prediction is done using 1-RC or 2-RC equivalent circuit. In this work, a 3-RC equivalent circuit is used to model battery parameters. In this battery model, internal resistance represents instantaneous voltage drop, and the RC branch depicts delayed response

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in voltage drop. The nonlinear nature of battery is modelled efficiently by EKF and UKF variants of Kalman Filter. With the increase in number of RC branches, the accuracy of battery model is improved. The SoH estimation for the battery cell with EKF and UKF is implemented in MATLAB/SIMULINK® . The degradation in SoC with battery ageing is also estimated in this work.

1.1 State of Charge SoC of a battery is an indicator of the remaining capacity that a battery can deliver to a load at any given time. Coulomb counting method is widely employed methods for the determination of SoC of a battery [14]. This method banks on the integration of the current flowing in and out of the battery terminals during the charging and discharging states, respectively [14]. In Coulomb counting method, SoC is determined as SoCt = SoCt−1 +

1 1 ∫ I dt C 0

(1)

where, SoCt is existing SoC, SoCt−1 is initial SoC value, C is battery capacity, and I is current drawn from the battery.

1.2 State of Health The change in battery parameters such as capacity fade, increase in internal resistance are indicators of degradation in battery health. Looking at the change in battery capacity the SoH is defined as SoHc =

Cact × 100% Ccap

(2)

where SoHc is the value of SoH, Cact is the actual battery capacity and Ccap is the rated battery capacity. The BMS will give a warning to change the battery if the battery capacity degrades below 80% [14]. Internal resistance of a battery changes during lifespan of a battery due to charge/discharge cycles. The SoH of a battery can be modelled as a function of internal resistance (R O ) of a battery. A battery that has SoH equal to 100%, and its internal resistance R O will be equal to internal resistance of new battery Ri . When SoH is 0%, R O will be twice of Ri [14]. SoH can be framed as 

SoH Ri

Ri − R O = 1+ Ri

 × 100%

(3)

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where SoH Ri shows the current battery health, R O is the current internal resistance of a battery, and Ri is the internal resistance of new battery.

2 State of Health Estimation Algorithm 2.1 Kalman Filter Kalman Filter is optimal estimator, meaning it infers required parameters from inaccurate, uncertain and indirect data. It is recurrent in nature, which means that data can be processed as and when it arrives. Unlike any algorithm that works on batch data, where all data needs to be present beforehand. The Kalman Filter reduces the mean squared error between estimated and measured values. The advantages of Kalman filtering are, accurate estimation on practical data due to optimality and structure, the filter is convenient for online real-time processing, ease of formulation and implementation, noise attenuation and projection of the measurements onto the next state estimate. There are two steps in Kalman Filter, (i) State Prediction and (ii) Measurement Prediction [1]. The filter processes the data ensure its convergence to predefined real value. Since the measured values fit relatively well to the predicted values, the filter performs correct estimation in spite of presence of noise in the data. Figure 1 shows flow chart for Kalman Filter. The Kalman Filter is best suited for problems having linear assumptions. Battery behaviour is highly nonlinear. It is advantageous to use derivations of Kalman Filter, EKF and UKF to tackle the nonlinearity aspect. These variants are more suitable for nonlinear assumptions.

Fig. 1 Flow chart for Kalman Filter

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Fig. 2 Algorithm for Kalman Filter

Kalman Filter gives optimal results under varying conditions. To start with, a nonlinear state space model can be assumed as xk = Axk−1 + Bu k−1 + wk

(4)

yk = C xk + vk

(5)

where xk is the state matrix for the system at time k, u k is the control variable matrix, wk is the state noise matrix that represents some unmeasured inputs, yk is the output matrix for the system, vk is the sensor noise that effects the measurement of the output, A, B, C are used to describe dynamic properties of the system. The twoinput variable xk and u k computed with Eq. (4) affects the state of the system. The measurement matrix is computed with Eq. (5) as a linear combination of measured input and state. Figure 2 puts the Kalman Filter in a nutshell. The variables used in Kalman Filter are defined in Table 1. The state equations can be estimated iteratively using Kalman Filter equations as shown in Fig. 2.

2.2 Extended Kalman Filter Nonlinear state estimation is a challenging problem. The EKF is regarded as an improvement to the traditional Kalman Filter algorithm that can work with nonlinear

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Table 1 Variables of Kalman Filter algorithm Variable

Description

X

State matrix

P

Process covariance matrix (represents error in estimate of process)

U

Control variable matrix

W

State noise matrix

A, B, C, H

Adaptation matrix (used to convert one form of matrix to another)

Q

Process noise

Y

Measurement state

Z

Noise in measurement (uncertainty)

K

Kalman gain

R

Sensor noise

I

Identity matrix

state estimation. This improved filter provides the tools to efficiently deal with nonlinearity of the system. In order to implement the EKF, obtaining Jacobian Matrix is necessary. For every time step, the Jacobian matrix (JM) is calculated with current predicted state. The calculated Jacobian matrices are used in estimation using EKF. This filtering method linearizes the nonlinear function around the current estimate [15]. Figure 3 shows block diagram of the EKF. A nonlinear system used for EKF can be represented in the form of state space equations as given in (3) and (4) X kp = f (X k−1 , Uk−1 ) + Wk

(6)

Yk = h(X km ) + Z k

(7)

where f (X k−1 , Uk−1 ) and h(X km ) are nonlinear functions. The nonlinearities of these state functions are approximated in the EKF with first order Taylor Expansion. From the equations in (5) and (6), Ak−1 is the JM of partial derivatives of function f with

Fig. 3 Extended Kalman Filter block diagram

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respect to state functions xk−1 and u k−1 , and C is defined as JM of partial derivatives of nonlinear function h with respect to state functions xk and u k [16]. Ak−1 =

∂ f (X k−1 , Uk−1 ) ∂ X k−1

Ck =

∂ f (h k , u k ) ∂ Xk

(8) (9)

The steps to be followed to implement the EKF are 1. 2. 3. 4. 5.

Predicted State Estimation Predicted Error Covariance Calculating Kalman Gain State Estimate Measurement Update Error Covariance Measurement Update.

2.3 Unscented Kalman Filter If the state estimates and measurements are highly nonlinear, then EKF can might give erroneous results [17]. This is because the covariance is transmitted through linearization of original nonlinear model [17]. The UKF makes use of a sampling technique which is known as unscented transformation to select a marginal number of sigma points around the mean [17]. These sigma points are then passed from the nonlinear functions, and then a new mean and covariance are calculated. The filter depends on how efficiently the transformed statistics of the unscented transform are calculated and how accurately the sigma points are selected. For some systems, the UKF can accurately estimate mean and covariance [17]. Figure 4 shows pictorial representation of the UKF. The mathematical equations used for UKF algorithm can be expressed as Prediction Step   Predicted state estimate X k−1 , Pk−1 = U T X k|k−1 , Pk|k−1 , f (X k−1 , Uk−1 ) + Wk (10) Fig. 4 Pictorial representation of unscented Kalman Filter

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Predicted error covariance Pkp = Pk|k−1 + Q

(11)

Measurement Step   Measurement Residual Yk|k−1 , Sk , Ck = U T X kp|kp−1 , Pk|k−1 + h(X km ) + Z k (12) Pkp H +R H Pkp H T   Update State Estimate X k = X kp + K Y − H X kp

(14)

Update Error Covariance Pk = (I − H K )Pkp

(15)

Kalman Gain K =

(13)

where UT is unscented transform, X is state space, Y is measurement state, P is covariance matrix, Q is noise covariance matrix, and K is Kalman gain.

3 Electrical Equivalent Battery Model An electrical equivalent circuit battery model is represented using a voltage source, series resistance and one or more resistance–capacitance pair in series/parallel. Figure 5 represents a 3-RC battery model with equivalent circuit. This model is employed in this paper for modelling a lithium-ion cell. The equivalent circuit of considering ageing effect for lithium-ion cell have been simulated using MATLAB and SIMULINK tool. This simulation is done to investigate the runtime and capacity degradation characteristics for battery.

Fig. 5 3-RC equivalent circuit model

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The voltage source is used to replicate the open circuit voltage of battery whereas the other components model the internal resistance and time dependent behaviour of battery cell. The internal resistance replicates instantaneous voltage drop response, and the RC branch models the delayed voltage drop response. A 31 Ah, 3.6 V lithium-ion cell was used in testing. All parameters of equivalent circuit as shown in Fig. 5 are determined based their relationship with SoC and temperature. The delayed voltage response in the battery is driven by RC pairs. In [4], the initial parameters were derived assuming that the current during pulse discharge was constant. As a result, the contribution of parameters can be linearly scaled from starting SoC value to ending SoC value during each pulse. The linear problem statement was posed using Matrix Equation [16]. Vt = VEm + VRO + VRC1 + VRC2 + VRC3

(16)

where Vt is the terminal voltage of battery, VEm is the open circuit voltage of battery, VRO is the voltage drop at resistance RO and VRCn are voltage drops at three RC pairs. The voltage response driven by series resistance and RC component pairs is given by V (t) = I (t) × RO + τ1

dVC1 dVC2 dVC3 + τ2 + τ3 dt dt dt

(17)

where V (t) is the terminal voltage, I (t) is the current passing through battery, RO is battery’s internal resistance, and τn dVdtCn are the voltage drops due to RC component pair. The battery parameters are modelled using lookup tables presented in Table 2. Table 2 Lookup table values for determination of voltage, resistance and capacitance State E m of charge

R0

R1

R2

R3

C1

9.71544

C2

C3

1

3.49 0.009913 0.0002936 0.0009794 0.000510

0.9

3.55 0.008750 0.0009064 0.0009984 0.000143 12.1200

50.465935 649.8549

0.8

3.61 0.008322 0.0012650 0.0010142 0.000570 12.0370

84.585061 938.7252

0.7

3.64 0.008720 0.0003146 0.0010868 0.000208 10.9054

54.864791 349.4190

0.6

3.68 0.007959 0.0011525 0.0009806 0.000651

55.660370 983.4442

0.5

3.74 0.008565 0.0004844 0.0008780 0.000327 13.4646

66.795537 958.5700

0.4

3.83 0.008433 0.0007226 0.0008938 9.253943 10.2523

92.003741 962.0885

0.3

3.94 0.008503 0.0007885 0.0009926 0.000107 10.0421

81.040401 402.2209

0.2

4.06 0.008466 0.0008089 0.0008680 9.602820

0.1

4.10 0.008465 0.0007946 0.0009133 0.000476 10.0848

9.25131

9.65077

61.445390 650.0073

90.833521 405.7126 104.58288

568.2598

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4 Results and Discussions The results discussed in this section are based on the simulated data. Both the EKF and UKF are implemented and compared for estimation of SoH of the battery. Simulations are carried out on the same dataset at 20 °C. The conventional Coulomb counting based SoC estimation technique requires 100% depth of discharge and constant current discharge rate to estimate battery degradation [1]. This may not be practically possible in each case. In applications like EVs, the discharge current and depth of discharge level varies based on load profile. Figures 6 and 7 show the SoH estimation after 10 cycles based on the EKF and UKF, respectively. Here, battery’s initial conditions are 50% depth of discharge and 20 °C temperature. There is degradation of battery health from 0.5 to 0.454. Figure 8 indicates increase in the internal resistance of the cell occurring due to rise in temperature. This rise indicates the change in SoH of battery. The internal resistance of the battery will increase in accordance with the age of the battery and time of usage. SoH estimations can be used to determine if the battery is still harmless to use or it needs replacement. The simulations show an increase in internal resistance is from 0.008 to 0.012. Figure 9 indicates rise in cell temperature. The test was conducted at 20 °C. From figure, it can be seen that there is slight rise in the temperature of cell from 20 to 21.65 °C. This rise in temperature also contributes to the rise in internal resistance of the cell. Figures 10 and 11 show the cell current and voltage profiles used in the

Fig. 6 Real versus estimated SoC degradation using EKF

Fig. 7 Real versus estimated SoC degradation using UKF

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Fig. 8 Estimated increase in internal resistance

Fig. 9 Cell temperature

Fig. 10 Cell current

charge–discharge cycle. The current and voltage profiles used are added with some noise to simulate real-time scenarios.

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Fig. 11 Cell voltage

5 Conclusion Estimating SoH of a battery is more subjective than estimating the SoC, as there is no universal agreement on how state of health is to be defined. The paper compares EKF and UKF for SoH estimation. The work extends upon equivalent circuit model structure presented in the prior research. The filters are employed for state of health estimation for a lithium-ion cell of 31Ah, 3.6 V capacity. The method presented here continuously monitors the battery discharge current and SoC to estimate the state of health. Both EKF and UKF are employed for SoH estimation considering the rise in internal resistance. The rise in internal resistance and reduction in battery capacity indicates the degradation of battery state of health. The paper uses the data obtained by carrying out the tests on a single cell for simulation studies. The electric vehicle batteries are a combination of several such cells connected in series and parallel, and this work can be further extended with to the battery pack, whilst considering the ageing effect of different cells in a battery pack is different.

References 1. Saikrishna, G., Kalpana, R., Singh, B.: An online method of estimating state of health of a Li Li-Ion battery. IEEE Trans. Energy Convers. (Early Access) 8583–8592 (2020). https://doi. org/10.1109/TEC.2020.3008937 2. Huria, T., Ceraolo, M., Gazzarri, J. Jackey, R.: High fidelity electrical model with thermal dependencies for characterization and simulation of high-power lithium battery cell. In: IEEE International Electric Vehicle Conference, IEEE Press, Greenville, SC, USA (2012). https:// doi.org/10.1109/IEVC.2012.6183271 3. Cheng, K.W.E., Divakar, B.P., Wu, H., Ding, K., Ho, F.H.: Battery-management system (BMS) and SOC development for electrical vehicles. IEEE Trans. Veh. Technol. 60, 76–88 (2010). https://doi.org/10.1109/TVT.2010.2089647 4. Ahmed, R., Gazzarri, J., Onori, S., Habibi, S.: Model-based parameter identification of healthy and aged Li-ion batteries for electric vehicle applications. SAE Int. J. Altern. Power. 4, 233–247 (2015). https://doi.org/10.4271/2015-01-0252

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5. Lam, L., Bauer, P.: Practical capacity fading model for Li-Ion battery cells in electric vehicles. IEEE Trans. Power Electron. 28, 5910–5918 (2013). https://doi.org/10.1109/TPEL.2012.223 5083 6. de Vries, H., Nguyen, T.T., Veld, B.O.H.: Increasing the cycle life of lithium. Elsevier J. Microelectron. Reliab. 55, 247–2253 (2015). https://doi.org/10.1016/j.microrel.2015.08.014 7. Berecibar, M., Garmendia, M., Gandiaga, I., Crego, J., Villarreal, I.: State of health estimation algorithm of LiFePO4 battery packs based on differential voltage curves for battery management system application. J. Energy. 103, 784–796 (2016). https://doi.org/10.1016/j.energy. 2016.02.163 8. Chao, H., Youn, B.D., Chung, J.: A multiscale framework with extended Kalman filter for lithium-ion battery SOC and capacity estimation. Appl. Energy 92, 694–704 (2012). https:// doi.org/10.1016/j.apenergy.2011.08.002 9. Andre, D., Nuhic, A., Soczka-Guth, T., Sauer, D.U.: Comparative study of a structured neural network and an extended Kalman filter for state of health determination of lithium-ion batteries in hybrid electric vehicles. Eng. Appl. Artif. Intell. 26, 951–961 (2013). https://doi.org/10.1016/ j.engappai.2012.09.013 10. Chaoui, H., Golbon, N., Hmouz, I., Souissi, R., Tahar, S.: Lyapunov-based adaptive state of charge and state of health estimation for lithium-ion batteries. IEEE Trans. Indus. Electron. 62, 1610–1618 (2015). https://doi.org/10.1109/TIE.2014.2341576 11. Guha, A., Patra, A.: State of Health estimation of lithium-ion batteries using capacity fade and internal resistance growth models. IEEE Trans. Transport. Electrification 4, 135–146 (2017). https://doi.org/10.1109/TTE.2017.2776558 12. Zenati, A., Desprez, P., Razik, H.: Estimation of the SOC and the SOH of li-ion batteries, by combining impedance measurements with the fuzzy logic inference. In: IECON 2010— 36th Annual Conference, IEEE Industrial Electronics Society, IEEE Press, Glendale, AZ, pp. 1773–1778 (2010). https://doi.org/10.1109/IECON.2010.5675408 13. Singh, P., Vinjamuri, R., Wang, X., Reisner, D.: Fuzzy logic modeling of EIS measurements on lithium-ion batteries. Electrochim. Acta 51, 1673–1679 (2006). https://doi.org/10.1016/j. electacta.2005.02.143 14. Topan, P.A., Ramadan, M.N., Fathoni, G., Cahyadi, A.I., Wahyunggoro, O.: State of Charge (SOC) and State of Health (SOH) estimation on lithium polymer battery via Kalman Filter. In: 2nd International Conference on Science and Technology-Computer, IEEE Press, Yogyakarta, Indonesia (2016). https://doi.org/10.1109/ICSTC.2016.7877354 15. Zhao, L., Xu, G., Li, W., Taimoor, Z.: LiFePO4 battery pack capacity estimation for electric vehicles based on unscented Kalman filter. In: IEEE International Conference on Information and Automation (2013). https://doi.org/10.1109/ICInfA.2013.6720314 16. Aung, H., Soon, J.J., Goh, S.T., Lew, J.M., Low, K.S.: Battery management system with stateof-charge and opportunistic state-of-health for a miniaturized satellite. IEEE Trans. Aerosp. Electron. Syst. 56, 2978–2898 (2020). https://doi.org/10.1109/TAES.2019.2958161 17. Marelli, S., Corno, M.: Model-based estimation of lithium concentrations and temperature in batteries using soft-constrained dual unscented Kalman Filtering. IEEE Trans. Control Syst. Technol. (Early Access). 1–8 (2020). https://doi.org/10.1109/TCST.2020.2974176 18. Zou, Y., Hu, X., Ma, H., Li, S.E.: Combined state of charge and state of health estimation over lithium-ion battery cell cycle lifespan for electric vehicles. J. Power Sources 273, 793–803 (2015). https://doi.org/10.1016/j.jpowsour.2014.09.146 19. Lam, L., Bauer, P., Kelder, E.: A practical circuit-based model for Li-ion battery cells in electric vehicle applications. In: IEEE 33rd International Telecommunications Energy Conference (INTELEC), IEEE Press, Amsterdam, Netherlands (2011). https://doi.org/10.1109/INTLEC. 2011.6099803

Control of 7-Level Simplified Generalized Multilevel Inverter Topology for Grid Integration of Photovoltaic System Nirav R. Joshi and Amit V. Sant

Abstract The present scenario of environmental crisis and energy crisis has led to most of the countries concentrating on increasing the grid penetration of renewable energy sources. Clean, free, and abundant availability of renewable energy sources, such as solar and wind energy are becoming the prime choice for energy generation. Solar photovoltaic (PV) energy system when integrated with the grid through dc-dc converter and dc-ac converter known as the dual stage conversion system. Maximum power transfer is assured by the dc-dc converter. Grid synchronization, dc-link voltage, and total harmonic distortion (THD) content in injected currents are controlled by dc-ac converter. Generally, 2-level dc-ac converters are employed for integration of small and medium power PV to the grid. However, multilevel inverters (MLI) offer numerous advantages for integration of high-power PV to the grid. The current MLI topologies suffer from the higher switch and capacitor count. In this paper, 7-level simplified generalized MLI topology has been employed for integration of PV arrays with the single-phase distribution network. The developed control algorithm comprises of dc-link voltage control, phase locked loop, maximum power point tracking (MPPT), hysteresis current controller (HCC), and switch selector algorithm to ensure that adherences to the standards set by the IEEE 1547. The proposed grid-tied 7-level simplified generalized MLI-based PV energy system is simulated on MATLAB/SIMULINK and its operation is analyzed considering the case where in the PV array supplies surplus power as compared to the load demand. Keywords Grid-tied inverter · Multilevel inverter · PV energy system · Hysteresis current control

N. R. Joshi (B) Electrical Engineering Department, Saffrony Institute of Technology, Mehsana, India e-mail: [email protected] A. V. Sant Electrical Engineering Department, Pandit Deendayal Petroleum University, Gandhinagar, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_41

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1 Introduction The current ecological state of affairs along with the continuously mounting demand of power generation necessitates grid integration of energy sources which are free from gaseous emissions and free to use [1–3]. In many other tropical countries, due to its abundant availability over the year, the solar photovoltaic (PV)-based generation is finding prominence [4]. The growth in solid-state technology has further improved the energy utilization from PV array [5]. Standalone PV energy system feds the power to the local loads, and grid-tied PV energy system feds the power to the grid and/or local loads. In dual stage connection scheme, PV module is interfaced with the grid through a dc-dc converter followed by dc-ac converter. Maximum power transfer is assured by the dc-dc converter. Grid synchronization, dc-link voltage, and total harmonic distortion (THD) content in injected currents are controlled by dc-ac converter. In spite of the increased component count, the dual stage conversion system is preferred due to the improved quality in the supplied voltage and current [6, 7]. PV being integrated to the grid via dc-ac converter hence its control strategy decides the quality of the power being injected to the grid. As per the grid codes, the control of grid-tied inverter includes control over the harmonic content in injected currents, islanding detection and electromagnetic compatibility. Conventional 2-level voltage source inverters (VSIs) are widely employed for integration of PV modules with the grid in low/medium power range. However, for higher power range, 2level VSI is not preferable due to the higher voltage stress involved. Multilevel inverters (MLIs), generate stepped voltage using string of small dc voltage at the input side, are recommended for the high-power applications [8]. MLI has the merits of enhanced power quality at the output, lower current, and voltage distortions, reduced voltage stress on switches, lower dv/dt requirement and reduced switching losses [9]. Reference [10] reports the application of MLI for the grid integration of high-power PV arrays. Similarly, [11, 12] have reported the grid integration of renewable energy sources using MLI. Reference [13] has presented a comprehensive survey regarding the different power circuit configuration of MLI. Peng has reported a generalized MLI topology with self-voltage balancing irrespective of inverter control and load characteristics [14]. Reference [15–17] has also reported MLI topologies with reduced switch count and voltage balancing facility. Reference [15] reports a simplified generalized MLI topology which has merits of reduced component count. This topology has been employed for medium voltage drives. However, this topology has not been explored for PV applications. This paper proposes the application of 7-level simplified generalized MLI topology for grid integration of PV system. The operation of the reported grid-tied PV system, comprising of 7-level simplified generalized MLI topology, is analyzed for a single-phase system. For single-phase system, reported 7-level simplified generalized MLI has the advantage of reduced switch and capacitor counts by seven and three units, respectively. This results in a marked reduction in system complexity,

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gating circuitry requirements and overall converter cost. Per phase, the PV arrays are dived into three groups, with each group corresponding to a voltage level of the MLI. The control architecture of this grid-tied system involves MPPT, dc-link voltage control, and hysteresis current control (HCC). The performance and operation of this single-phase grid-tied PV system are analyzed for the case in which PV generates surplus power than demanded.

2 General Architecture of Grid-Tied Photovoltaic System PV system being integrated with the single-phase network is shown in Fig. 1, where V pv , V dc , and V grid are the voltages available across the ends of PV array, dc link capacitor and grid, respectively, I pv , I inv , I load , and I grid are the currents available across the ends of PV array, VSI, load, and grid, respectively, C pv and C dc are the capacitors connected across the ends of PV array and dc-dc converter, L f , C f , and Rf serves filtering operation at the VSI end, and Z grid is the grid side impedance. PV array supplies the power Ppv , and grid supplies the power Pgrid . In this grid-tied energy system, PV serves as the energy source. PV converts the solar irradiation to generate electrical energy. The maximum efficiency of the PV panel during the day time is usually 15% [18]. With high system cost and low efficiency, the MPPT control ensures the delivery of maximum power from PV module under given operating condition. Control over duty cycle of dc-dc converter ensures that the PV module operates at the voltage corresponding to maximum power point. A dc-dc converter is interfaced by VSI with the grid. VSI provides the valuable tasks in the structure such as control over the dc-link voltage, grid synchronization, and power flow control. This is achieved through dc-link control, inverter control, and phase locked loop (PLL). In grid-tied PV energy system, grid supplies the local load with the PV power cannot meet the demand and conversely acts to absorb the excess power coming from PV array. In this system, PV array is the primary power source, and grid operates as

Fig. 1 Architecture of single-phase grid-tied PV system

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the secondary source. Thus, the grid-tied architecture results in increased reliability of power supply and independence from reliance on costly energy storage devices. With the inclusion of an MLI in place of 2-level VSI, the architecture can work at higher voltage and power levels while providing better power quality.

3 Simplified Generalized 7-Level Multilevel Inverter MLI converts the different dc voltage sources, at the input, to the stepped ac voltage at the output. The three basic MLI topologies are (i) Cascaded H-bridge (CHB) MLI. (ii) Neutral point clamped (NPC) MLI, and (iii) Flying capacitor (FC) MLI [13]. These topologies suffer from increased switch and capacitor count which further initiates requirements of increased gate drivers, protection units, losses, and system complexity. The proposed grid-tied MLI-based PV system uses 7-level simplified generalized MLI which comprises of seven switches, six diodes, and string of three

Fig. 2 Power structure of 7-level simplified generalized MLI topology

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Table 1 Switching groups Group

Switches

Purpose

Group-I

S p1 , S p2 , S p3

Generates unidirectional stepped voltage at terminals A–B

Group-II

S b1 , S b2 , S b3 , S b4

Generates bidirectional stepped voltage at terminals X–Y

dc voltage sources. Figure 2 shows the power circuit of 7-level simplified generalized MLI. The three dc voltage sources have equal magnitudes of V dc , resulting in symmetric MLI topology. As shown in Table 1, switches of the MLI topology can be categorized in two groups. The group-I switching action is providing the generation of stepped dc voltage available at the input of H-bridge inverter. For different dc voltage levels, appropriate switching state needs to be employed as shown in Table 2. The group-II switching action results in generation of stepped ac voltage from stepped dc voltage. In groupII, the activation of switches S b1 and S b2 will generate the positive stepped voltage, whereas the activation of switches S b3 and S b4 will generate the negative stepped voltage. Switching states of the reported 7-level simplified generalized MLI topology is shown in Table 2. To obtain V AB = V dc3 , S p3 need to be turned ON. The direction of flow of current in this state is: V dc3 -D4 -S p3 -A-S b1 -X-load-Y-S b2 -B-V dc3 . Similarly, activation of S p2 along with S p3 results in V AB = V dc2 + V dc3 = 2V dc . On the other hand, the activation of switch S p1 along with S p2 and S p3 will result in V AB = V dc1 + V dc2 + V dc3 = 3V dc . For each switching state, whether the individual switches that are ON or OFF, the resulting dc-link voltage, ac output voltage, and the direction of flow of current are shown in Table 2.

4 Proposed Architecture of Grid-Tied PV System The proposed 7-level simplified generalized MLI-based grid integration of PV system is given in Fig. 3. The proposed architecture consists of three separate PV arrays. Each of these arrays serves as a source for the individual boost dc-dc converter, which implements MPPT control. The output of boost dc-dc converter acts as a separate dc voltage source for the 7-level simplified generalized MLI topology. This results in elimination of need of voltage balancing as the dc-dc converters ensure the control over the output voltages V dc1 -V dc2 -V dc3 . The maximum available power has been harvested by employing Perturb and observe (P & O) technique. Each dc-dc converter’s output is controlled to V dc1 , V dc2 and V dc3 , respectively. These voltages are basically the input voltages for 7-level simplified generalized MLI topology which interfaces the system with the grid. The developed control algorithm comprises of PLL, MPPT controller, dc-link voltage controller, hysteresis current controller, switch selector algorithm for switches in group-I.

1

1

1

0

1

1

1

2

3

4

5

6

7

1

1

0

0

0

1

1

1

0

0

0

0

0

1

0

0

0

1

1

1

1

Sb1 Sb2

Sp1

Sp3

Sp2

Switches in group-II

Switches in group-I

1

State

1

1

1

0

0

0

0

Sb3 Sb4

V dc3 +V dc2 +V dc1

V dc3 +V dc2

V dc3

0

V dc3

V dc3 +V dc2

V dc3 +V dc2 +V dc1

V AB

D6 -A-S b1 -X-load-Y-S b2 -B V dc3 -D4 -S p3 -A-S b3 -X-load-Y-S b4 -B-V dc3 V dc3 -V dc2 -D1 -S p2 -S p3 -A-S b3 -X-load-Y-S b4 -B-V dc3 V dc3 -V dc2 -V dc1 -S p1 -S p2 -S p3 -A-S b1 -X-load-Y-S b2 -B-V dc3

−V dc3 −V dc3 −V dc2 −V dc3 −V dc2 −V dc1

V dc3 -D4 -S p3 -A-S b1 -X-load-Y-S b2 -B-V dc3

V dc3 -V dc2 -D1 -S p2 -S p3 -A-S b1 -X-load-Y-S b2 -B-V dc3

V dc3 -V dc2 -V dc1 -S p1 -S p2 -S p3 -A-S b1 -X-load-Y-S b2 -B-V dc3

Current direction

0

V dc3

V dc3 +V dc2

V dc3 +V dc2 +V dc1

V XY

Table 2 States of switches available in 7-level simplified generalized MLI topology

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479

Fig. 3 Proposed 7-level simplified generalized MLI-based grid-tied PV system

Mathematical modeling of PV array and 7-level simplified generalized MLI is discussed in Sect. 4.1 and 4.2. Moreover, Sect. 4.3 discusses the developed control system for the grid-tied PV system with 7-level simplified generalized MLI.

4.1 Mathematical Modeling of PV Cell The series–parallel connection of PV panels results in formation of PV array. The equivalent circuit of a PV panel is illustrated in Fig. 4. In this figure, G stands for the level of irradiance, photon current is represented by I ph , the recombination of charge carriers is represented in the form of current flowing through the diode, I D , and the voltage across the terminals of the PV panel, and the current flowing out of the positive terminal of PV panel is, respectively, represented as V PV and I PV . Furthermore, the series resistance, Rs , and shunt resistor, Rsh , are also incorporated in the model. From the equivalent circuit, PV current can be given as [18]. Ipv = Iph − Ish − ID

(1)

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Fig. 4 Equivalent circuit of PV cell [18]

I pv can be mathematically expressed as [19].   V× G = Npa (Iscref + μsc (T − Tref )) − G ref    V   + I × NRpas Nsa − Npa × I0 exp −1 A × Vt 

Ipv



Npa Nsa

+ I × Rs



Rsh (2)

where G is the irradiance level in W/m2 , Gref is the irradiance as per STC in W/m2 , N sa , and N pa are the connected PV panels in series and parallel, respectively, I scref is the short circuit current at STC, μsc is the temperature coefficient of short circuit current, T is the cell temperature in K, T ref is the cell temperature at STC in K, I o is the saturation current in A, V t is the thermal voltage in V, k is Boltzmann’s constant (1.3805 × 10−23 J/K), N s is the connected PV cells in series, and q is the electron charge (1.6 × 10−19 C). I o and V t can further be expressed as [19].  I0 = Irs ×

T Tref

3

Vt =

 exp

  q × Eg 1 1 − A×k T Tref

k × T × Ns q

(3) (4)

where E g is the band gap energy of semiconductor in eV, A is the ideality factor, and I rs is the reverse saturation current in A, which is expressed in (5). Iscref

Irs = exp

q Voc Ns k AT



(5)

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4.2 Modeling of 7-level Simplified Generalized MLI Topology The proposed 7-level simplified generalized MLI produces seven different voltage levels, namely 0, ± V dc , ±2V dc , and ±3V dc . Switching states and its relationship with output voltage for this MLI topology is shown in Table 2. Mathematical modeling for this MLI topology can be derived from this table, and the output voltage of MLI is expressed as, T × [Si ]7×7 × [U ]7×1 [Vout ]1×1 = [xi ]1×7

(6)

where V out is the output voltage matrix of size 1 × 1, x i is the switching state matrix of size 7 × 1, S i is the switching sequence matrix of size 7 × 7, and U is the 7 × 1 input matrix that includes input dc voltage sources for 7-level simplified generalized MLI. Equation (6) can further be expanded and written as, ⎡

⎤T ⎡ x1 S p1 ⎢x ⎥ ⎢ S ⎢ 2 ⎥ ⎢ p1 ⎢x ⎥ ⎢ S ⎢ 3 ⎥ ⎢ p1 ⎢ ⎥ ⎢ [Vout ] = ⎢ x4 ⎥ ⎢ S p1 ⎢ ⎥ ⎢ ⎢ x5 ⎥ ⎢ −S p1 ⎢ ⎥ ⎢ ⎣ x6 ⎦ ⎣ −S p1 x7 −S p1

S p2 S p2 S p2 S p2 −S p2 −S p2 −S p2

S p3 S p3 S p3 S p3 −S p3 −S p3 −S p3

Sb1 Sb1 Sb1 Sb1 Sb1 Sb1 Sb1

Sb2 Sb2 Sb2 Sb2 Sb2 Sb2 Sb2

Sb3 Sb3 Sb3 Sb3 Sb3 Sb3 Sb3

⎤⎡ ⎤ Vdc1 Sb4 ⎢ ⎥ Sb4 ⎥ ⎥⎢ Vdc2 ⎥ ⎢ ⎥ Sb4 ⎥⎢ Vdc3 ⎥ ⎥ ⎥⎢ ⎥ Sb4 ⎥⎢ 0 ⎥ ⎥⎢ ⎥ Sb4 ⎥⎢ 0 ⎥ ⎥⎢ ⎥ Sb4 ⎦⎣ 0 ⎦ Sb4 0

(7)

where x 1 –x 7 represents the switching states shown in Table 2. Here, x i = 1 to get ith state. Based on i, states of switches S p1 –S p3 and S b1 –S b4 gets decided from Table 2. For instance, for i = 1, x i = 1 and corresponding to that S p1 , S p2 , S p3 , S b1 and S b2 are 1 and x 2 –x 7 , S b3 and S b4 are 0. Considering V dc1 = V dc2 = V dc3 = V dc , Eq. (7) can further be simplified and written as, ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ [Vout ] = ⎢ ⎢ ⎢ ⎢ ⎣

x1 x2 x3 x4 x5 x6 x7

⎤⎡

⎤ 3Vdc ⎥⎢ 2V ⎥ dc ⎥ ⎥⎢ ⎥⎢ V ⎥ ⎥⎢ dc ⎥ ⎥⎢ ⎥ ⎥⎢ 0 ⎥ ⎥⎢ ⎥ ⎥⎢ −Vdc ⎥ ⎥⎢ ⎥ ⎦⎣ −2Vdc ⎦ −3Vdc

(8)

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4.3 Control Algorithm Figure 3 shows the developed control algorithm for the proposed grid-tied PV system, comprising of three dc-dc converters and a 7-level simplified generalized MLI for interfacing the PV arrays with the grid. The MPPT compares the PV array voltage with the reference value determined by the P&O algorithm. PI controller takes resulting voltage error and processes it to get the duty cycle for each boost converter to harvest maximum power from each PV modules under the given atmospheric conditions. The three dc-link voltages available across each dc-dc converters are measured by separate voltage sensors. The sum of the three dc-link voltages, (V dc1 + V dc2 + V dc3 ), and the reference value, V dcref , of the dc-link voltage is compared to determine the dc-link voltage error, which is then processed by a PI controller to get the peak magnitude of the current, I gm * , to be injected in the grid for ensuring that the dc-link voltage at the MLI input matches V dcref . The grid codes as well as the IEEE standard 1547, mandates that the distributed energy resource  only inject active power into the grid. Hence, the product of I gm * , and sin θgrid ) serves as the reference value of the instantaneous current, I grid * , to be injected by the MLI into the grid. θgrid , the phase angle of the grid voltage, is determined by PLL. A HCC processes I grid * , generated by the control algorithm, and I inv , sensed by the current sensor, to generate the signal x. This signal along with absolute value of I grid * is fed to the Switch Selector Algorithm for switches of group-I. The algorithm determines which of the group-I switches are to be turned ON and OFF. Thus, the gating signals for S p1 −S p2 −S p3 is controlled, thereby controlling the stepped dc voltage available at the input of the H-bridge inverter. Gate pulses are provided to S p1 if it is required to have V AB = (V dc1 + V dc2 + V dc3 ) across the A–B terminals. During this time, S p2 and S p3 will be kept ON. Similarly, application of gate pulse to S p2 and S p3 , with S p1 OFF, results in V AB = (V dc2 + V dc3 ). While application of gate pulse to S p3 with S p1 −S p2 OFF, leads to V AB = V dc3 . A comparator process I grid * and generates logic High if the current is positive or Logic Low if it is negative. This logic is given to the switches of group-II. Thus, group-I switches ensure control over amplitude of the current, whereas the group-II switches control the polarity of the current being injected in the grid. These switching actions ensure in phase operation of injected current with the V grid . This implies that the current is injected by the MLI at unity power factor, and thus, only active power is injected into the grid. Moreover, the switching action also ensures that the THD of the injected current is well below 5%. The passive filter along with the control algorithm and switching action ensures the grid code compliant operation of the proposed PV system.

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5 Results Simulation studies on the proposed grid-tied PV system are carried out on MATLAB/SIMULINK software. Steady-state operation of the proposed grid-tied architecture has been analyzed at STC considering PV generation surplus than that demanded by the load. The system parameters considered have been shown in Table 3. Figure 5 shows the performance of 7-level simplified generalized MLI-based proposed grid-tied PV system. The 7-level simplified generalized MLI integrates PV array 1, 2, and 3 with the single phase 230 V, 50 Hz distribution network. Figure 5a shows the grid voltage of 230 V. Figure 5b, c and d illustrates three currents at PCC such as the grid current, load current, and the current supplied by the MLI which are sinusoidal with the THD of 2.06%, 0.15%, and 1.16%, respectively. The THD is within the 5% limit imposed by standards. The peak amplitudes of the grid current, load current, and the current injected by the MLI are 19.57 A, 15.29 A, and 34.86 A, respectively. It is to be also noted that the phase difference between grid voltages and injected currents is almost negligible which results in the power factor of 0.9995. V AB and V XY of the 7-level simplified generalized MLI are shown in Fig. 5e, f. V AB is unipolar stepped voltage at the input of H-bridge inverter. V AB has 0, V dc , 2V dc , and 3V dc levels with V dc being 125 V. The output of H-bridge inverter, made up of group-II switches, consists of 7-levels, namely 0, ±V dc , ±2V dc , and ±3V dc with RMS value of 230 V. From the waveforms, it can be clearly observed that the developed control algorithm of the 7-level simplified generalized MLI-based PV system ensures in phase operation of injected currents and grid voltages. Also, the distortion in current is below the permissible limits thereby eliminating the chances of power quality degradation and grid pollution. Figure 5g, h, and i shows the active power and reactive power provided by the grid, consumed by the load and supplied by the PV systems, respectively. As resistive load is considered, the grid is not required to supply any reactive power. The active power supplied by the PV array is 5667 W which is greater than the active power Table 3 System parameters PV array [Quantity: 3]

Parallel connection of 3 PV string, each comprising of series connection of 2 PV panels V mpp = 95.82 V, I mpp = 24.42 A

Boost dc-dc converter [Quantity: 3]

Prated = 2.5 kW (each) 95.82–125 V (Boost type)

7-level simplified generalized MLI [Quantity: 1]

Prated = 3 × 2.5 kW = 7.5 kW, f r = 50 Hz, V out = 345 V (peak to peak)

LC filter

L f = 1.5 mH, C f = 5 µF and Rf = 3 

Grid

1-phase, 230 V, 50 Hz ac supply

Load

1-phase balanced linear load 2500 W

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Fig. 5 a Grid voltage, b grid current, c load current, d current injected by 7-level simplified generalized MLI, e voltage across terminals X and Y of MLI, V XY , f voltage across terminals A and B of MLI, V AB , g power supplied by the grid, h power consumed by the load and i total power supplied by PV arrays

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485

need of 2500 W posed by the load. The surplus PV power is injected to the grid. Therefore, the active power supplied by the grid is −3179 W. The negative sign that indicates that the surplus power is given to the grid. Thus, it is evident from the results that the proposed grid-tied PV system operates to supply only the active power and grid while maintaining the THD of injected currents within the specified limits. The developed control system ensures adherence to the grid code, extraction of maximum power from PV arrays, and reduction in fossil fuel dependencies. This can culminate in development of distribution system with high penetrations of renewable energy sources such as PV.

6 Conclusion In this work, grid integration of 7-level simplified generalized MLI-based PV energy system has been simulated and its operation is analyzed at STC considering case where PV generates surplus power. This enables injection of the surplus power to the grid. Inclusion of 7-level simplified generalized MLI benefits in the reduced voltage stresses and passive filter requirement. With the reduced switch and capacitor count, 7-level simplified generalized MLI makes the design of control system less complex. This further aids in terms of reduced gate circuitry and protection requirements. The developed control algorithm of this grid-tied 7-level simplified generalized MLI ensures adherence to the grid code by injecting currents with THD well below 5%. Also, the injected currents are in phase with the PCC voltage, thereby ensuring that only active power is injected into the grid. Additionally, the developed control system ensures extraction of maximum power from PV arrays. This system enables high penetrations of renewable energy sources such as PV to the distribution network irrespective of voltage levels.

References 1. Bae, Y., Trung-Kien, V., Kim, R.-Y.: Implemental control strategy for grid stabilization of grid-connected PV system based on German grid code in symmetrical low-to-medium voltage network. IEEE Trans. Energy Convers. 28(3), 619–631 (2013) 2. Carrasco, J.M., et al.: Power-electronics system for the grid integration of renewable energy source: a survey. IEEE Trans. Ind. Electron. 53(4), 1002–1016 (2006) 3. Kjaer, S., Pedersen, J., Blaabjerg, F.: A review of single-phase grid connected inverters for photovoltaic modules. IEEE Trans. Ind. Appl. 41(5), 1292–1306 (2005) 4. Quan, L., Wolfs, P.: IEEE Trans. Power Electron. 23(3), 1320–1333 (2008) 5. Cramer, G., Ibrahim, M., Kleinkauf, W.: PV system technologies: state-of-the-art and trends in decentralised electrification. Refocus 38–42 (2004) 6. Errouissi, R., Al-Durra, A.: Disturbance observer-based control for dual-stage grid-tied photovoltaic system under unbalanced grid voltages. IEEE Trans. Ind. Electron. (2018) 7. Errouissi, R., Al-Durra, A., Muyeen, S.M.: Design and implementation of a nonlinear pi predictive controller for a grid-tied photovoltaic inverter. IEEE Trans. Ind. Electron. 64(2), 1241–1250 (2017)

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8. Babaei, E.: A cascade multilevel converter topology with reduced number of switches. IEEE Trans. Power Electron. 23(6), 2657–2664 (2008) 9. Ali, J.S.M., Vijayakumar, K.: An assessment of recent multilevel inverter topologies with reduced power electronics components for renewable applications. Renew. Sustain. Energy Rev. (2017) 10. González, R., Gubía, E., López, J., Marroyo, L.: Transformerless single-phase multilevel-based photovoltaic inverter. IEEE Trans. Ind. Electron. 55(7), 2694–2702 (2008) 11. Selvaraj, J., Rahim, N.A.: Multilevel inverter for grid-connected PV system employing digital pi controller. IEEE Trans. Ind. Electron. 56(1), 149–158 (2009) 12. Calais M., Agelidis V.G.: Multilevel converters for single-phase grid connected photovoltaic systems—an overview. In: Proceedings IEEE International Symposium on Industrial Electronics, pp. 224–229 (1998) 13. Rodriguez, J., Lai, J.-S., Peng, F.Z.: Multilevel inverters: a survey of topologies, controls, and applications. IEEE Trans. Ind. Electron. 49(4), 724–738 (2002) 14. Peng, F., Qian, W., Cao, D.: Recent advances in multilevel converter/inverter topologies and applications. In: IEEE International Power Electronics Conference IPEC, pp. 492–501 (2010) 15. Joshi, N.R., Sant, A.V.: Analysis of a new symmetic multilevel inverter topology with reduced component count. In: International Conference on Emerging Trends in Information Technology and Engineering (IC-ETITE), pp. 1–6. IEEE, Vellore (2020) 16. Ceglia, G., et al.: A new simplified multilevel inverter topology for DC–AC conversion. IEEE Trans. Power Electron. 21, 1311–1319 (2006) 17. Gonzalez, S.A., Valla, M.I., Christiansen, C.F.: Five-level cascade asymmetric multilevel converter. IET Power Electron. 3, 120–128 (2010) 18. Sant, A.V., Khadkikar, V., Xiao, W., Zeineldin, H, Al-Hinai, A.: Adaptive control of grid connected photovoltaic inverter for maximum VA utilization. In: IECON 2013—39th Annual Conference of the IEEE Industrial Electronics Society, pp. 388–393. Vienna (2013) 19. MathWorks: Simscape/Electrical/Specialized Power Systems/Renewables/Solar: PV Array (R2015a). From https://in.mathworks.com/help/physmod/sps/powersys/ref/pvarray.html (2015)

Implementation of Constraint Handling Techniques for Photovoltaic Parameter Extraction Using Soft Computing Techniques P. Ashwini Kumari

and P. Geethanjali

Abstract Parameterized photovoltaic (PV) model accurately predicts the performance characteristics and output of PV module. These parameters can be obtained from experimentally measured performance curves using technical data sheet. Extrapolating the PV and IV curves obtained using experimental values yield appreciable results with highest degree of accuracy. Due to unpredictable climatic conditions, static modelling fails to extrapolate exact performance curves. This paper unfolds the new method of parameter extraction of single and double-diode model with respect to photovoltaic system. A new methodology using Ensemble of Constraint Handling Techniques (ECHTE) is developed using MATLAB platform. In spite of various extraction methods presented in the recent survey, there is scope for new estimation method which can address convergence issues and achieve optimized solution under varying environmental condition. The proposed ECHTE-based approach has been evaluated with different PV technologies under varying irradiation condition. The legitimacy of ECHTE was compared with existing state-of-art. On interpreting the results obtained by estimated and experimental values, it is clear that ECHTE achieves better results with reduced root mean square error (RMSE). Keywords Energy conversion ratio (ECR) · Ensemble constraint handling techniques (ECHTE) · Differential evolution (DE) · PV parameter extraction

1 Introduction The interest for electrical energy has raised exponentially because of expanded worldwide populace and industrialization. This elevated energy consumption has been demanding more efficient, reliable and eco-friendly source of energy. Harnessing P. A. Kumari · P. Geethanjali (B) VIT University, Vellore 632014, India e-mail: [email protected] P. Geethanjali REVA University, Bangalore 560064, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_42

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solar power to maximum extent can be of paramount significance to resolve global energy crisis. However, high installation cost and efficiency improvement have always been an open research domain for academia-industries. Enabling higher energy conversion ratio can be of paramount significance to provide low cost and reliable energy solution. In recent years, various researchers are trying their best to achieve optimal results in extracting the photovoltaic (PV) parameters. Estimation approaches including analytical and numerical methods have made effort to estimate accurate PV design parameters [1–5]. However, their respective limitations such as error proneness due to environmental parameter changes and computational complexity confine their applicability. A generic module datasheet merely gives the information at standard test condition (STC). Nevertheless, the perturbation of PV parameters due to dynamic variation such as temperature changes, irradiation, partial shading, soiling, etc., cannot be ignored [6]. To assess optimal parameters of solar cell, various evolutionary computing (EC) algorithms have been derived. The authors have used genetic algorithm (GA) and adaptive genetic algorithm (AGA) as multi-objective optimization problems [7, 8]. Particle swarm optimization (PSO) [9], hybrid patter search (HPS) [10], simulated annealing (SA) [11], chaotic whale optimization algorithm (CWOA) [12], bacterial forging optimization (BFO) [13], cuckoo search [14], wind driven optimization (WDO) [15] and bacterial foraging optimization approach (BFOA) [16]. However, many of the existing methods are still suffering due to local minima and convergence. Enabling an optimal EC-based approach while alleviating issue of the local minima, and convergence is the major motivation of the presented work. Unlike traditional EC-based parameter estimation models, in this paper, advanced evolutionary computing algorithm-based ECHTE has been applied to perform PV design parameter estimation. This paper employs ECHTE [17] algorithm to determine optimized values of one and double-diode model (DDM). Three sets of data are considered in this paper to validate the preciseness and compliance of the algorithm proposed which are as follows: 1. 2. 3.

The data obtained from modelling PV cell using MATLAB at various irradiation levels; PV cell data provided by different manufacturer’s and Realistic experimental values of voltage and current recorded by National Renewable Energy Laboratory (NREL) at the different climatic condition.

In Addition, the comparative study of various optimization techniques with the proposed approach is analysed. The results obtained prove that ECHTE outperforms existing algorithms in the literature with respect to reduced RMSE. We recommend that the ECHTE can be unhesitatingly prescribed as the best and reliable alternative for estimating PV parameter.

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2 Mathematical Modelling Modelling of PV cell is highly nonlinear due to dependency on various climatic condition. In this paper, both single and double-diode modelling is done using the mathematical equations. Problem formulation is carried out including the constraints with objective of reducing the RMSE using differential evolutionary algorithms.

2.1 Single-Diode and Double-Diode PV Model Identifying the optimal PV design parameter is quite a gruelling process. The PV design parameters for one and two diode are extracted to derive the performance characteristics. The current output of PV can be estimated using Eqs. (1) and (2).       vpv + i pv Rs vpv + i pv Rs −1 − i pv = Iph − I01 exp a1 Vt Rsh

(1)

The functional behaviour of the double-diode PV cell can be obtained by connecting current source in parallel with diode. The equivalent circuit of two-diode model is given in Fig. 1.         vpv + i pv Rs vpv + i pv Rs − 1 − I02 exp −1 i pv = Iph − I01 exp a1 Vt a2 Vt   vpv + i pv Rs (2) − Rsh

Rs Iph

Id1

Id2

Ish

Rsh D1

Fig. 1 Double-diode PV model

D2

Ipv

Vpv

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3 Problem Formulation Optimization techniques are extensively used to obtain the parameters of the pv cell. The fitness function here is taken as a minimization problem which intends to minimize the RMSE as represented in Eq. (3). Here, ε represents the RMS difference of actual measured and simulated values of current, N is the total count of I–V data obtained experimentally and decision vector is represented by ∅.   N 1

2 f k vpv , i pv , ∅ ε= N k=1

(3)

The fitness function for DDM is given by,     

vpv + i pv Rs f vpv , i pv , ∅ = Iph − I01 exp −1 a1 Vt     vpv + i pv Rs − I02 exp −1 a2 Vt vpv + i pv Rs − i pv − Rsh

∅ = Iph , I01 , I02, Rs , Rsh , a1 , a2

(4) (5)

In many PV applications, it is important to estimate maximum power under various climatic conditions. It is very essential to validate the PV curve accuracy with respect to experimental and measured values. The objective of this work is to minimize the difference in the error with respect to decision vector. Lower the value of difference indicates reduction in error between the actual and estimated values. The error between computed and estimated power must be evaluated, and the mean value of the power is calculated using Eqs. (6) and (7). Perror = Pm − Pe N MAE =

Perror N

i=1

(6)

(7)

where error in power is Perror, Pm is the measured power, Pe is the estimated power and N is total experimental data.

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4 Ensemble Constraint Handling Techniques and Its Implementation 4.1 ECHTE Algorithm Differential evolution (DE)-based constraint optimization problems is grabbing researcher’s attention due to randomization of initial population by asexual reproduction to generate offspring’s. This method generally depends on phenotypic behavioural evolvement than genetic transformation. Various forms of DE are used to explore and exploit the entire search space [18–21]. It is very hard for one single CHT to perform better for a given problem. This method is preferred where each population has unique constraint managing technique to solve the constrained-based optimization problems. The general formulation of the optimization problem subjected to constraints is given in Eqs. (8) and (9). Min/Max : f (Y ), Y = (yi , y2 . . . yn )

(8)

Constraints Gi (y) ≤ 0 i = 1 . . . . . . m

(9)

Hi (y) = 0 J = m + 1 . . . . . . . . . . . . ..n where m being the inequality constraint, (n–m) is the equality constraint and Y ∈ V. V is taken as complete search space. Here ‘f ’ should be bounded else it may not be continuous. Inequality constraints which satisfies Gi (y) = 0 are referred as active constraints because it achieves global optima, converting equality to inequality and ensembling with other inequality constraints as given below. P(Y ) =

w

k i (G i (Y )) w i=1 k i

i=1

(10)

  1 , where k i is weighing factor, Gmaxi indicates the parameter which where k i = G maxi violates maximum constraints.

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4.2 ECHTE Implemented for Extracting Solar PV Parameters The flowchart present in Fig. 2 represents ensembling of various constraints to solve EC-based optimization problem.

Fig. 2 Flowchart of ECHTE algorithm

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5 Experimental Analysis of the Estimated Parameters In order to accomplish higher ECR, this work presents two distinct advanced EC algorithms named EVPS and ECHTE. The proposed EC models explore and exploit the parameters of two-diode model. The double-diode eight parameters are obtained for different PV models. The performances of the proposed models have been extracted for both single as well as DDM. To exhibit relative performance assessment, both algorithms are applied distinctly. The proposed model is validated with different PV systems such as Kyocera KS20T [22] which is m-crystalline, sun module poly-crystalline SW245 [23], mono crystalline cell Shell 140-PC [24]. The limits have been assigned to the parameters as Rs ∈ [0.01, 3] , Rsh ∈ [100, 3000] , a1 , a2 ∈ [0.5, 3], I ph ∈ [0, 10] A, I o2 , I o2 ∈ [1e−5, 1e−15] µA. The experimental data is obtained from the datasheet. From Fig. 3a and b, it is evident that the estimated values DDM agrees completely with the experimental values for different irradiations. Further NRMSE is calculated mathematically by dividing RMSE by short circuit current [25]. The estimated values of NRMSE of different PV technologies presented in Table 1 validate the accuracy of the ECHTE algorithm at STC which further helps in accurate assessment of PV and IV characteristics at any given climatic conditions.

5.1 Experimental Data Analysis of RTC Cell at the Controlled Climatic Condition To access the effectiveness of modelling using ECHTE approach, the results obtained using experimental values of RTC France cell and the one estimated by optimization techniques are compared and plotted as shown in Fig. 4a and b. Various algorithms with various CH methods are simulated in order check the performance of ECHTE in comparison with the literature [29–37]. The realistic data of 57 mm diameter RTC France silicon solar cell at standard irradiance of 1000 W/m2 irradiance, 33 °C is used for comparison [26]. The contrast of results evaluated by ECHT-DE and other techniques such as EVPS, SF and SP has been enumerated in Table 2. From Table 2, it is remarkable that the RMSE of ECHTE algorithm stands at the lowest point among all the methods considered for comparison. The experimental and calculated values of current available at the output of RTC France cell are plotted as given in Fig. 4a and b. With just a small inappreciable error, the experimental data trails the estimated data with a close precision. Figure 4c shows the contrast in root mean square error of EHCTE, EVPS [27], SPDE and SFDE. The results manifested by the EHCT-DE are superior to the other three techniques.

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Fig. 3 a and b depicts I–V and P–V characteristics of Kyocera KC200GT-215 PV module at 25 °C and various irradiation conditions

5.2 Comparison of Different Constraint Handling Techniques Concerning Performance Parameters and Literature The assessment of real measurements and datasheet values has been done in [28] considering PSO, GA and Newton Raphson method to promote the adherence of PV in power system studies. Considering the RMSE as the performance index, the parameters estimated using ECHTE are compared with existing outperforming algorithms such as general algebraic modelling (GAMS) [29], improved JADE IJADE [30], cat

Implementation of Constraint Handling Techniques …

495

Table 1 Parameter estimation of different solar PV modules under various irradiation Parameters

Kyocera, KS20T

Sun module, SW245

Shell, SP140-PC

a1

2.1264

4.1710

1.9618

a2

1.2656

1.518

1.3609

RS (m )

90

5.8

440

RP ()

1329.1

799.7526

1148.6

I ph (A)

1.263

8.4901

4.6994

I o1 (µA)

4.226

0.035459

1.3858

I o2 (µA)

0.01

0.92614

1.3334

NRMSE (%)

0.442

0.040765

0.1264

swarm optimization (CSO) [31], improved Jaya algorithm IJAYA [32], self-adaptive teaching learning-based optimization SATLBO [33], improved whale optimization algorithm IWAO [34], improved shuffled complex evolution algorithm ISCE [35], GOFPANM [36] and improved teaching learning-based optimization ITLBO [37]. With reference to all above obtained results, a comparative study of distinct differential evolutionary algorithms is carried out with reference to execution time for SDM. From Table 3, it is evident that this approach has reduced NRMSE which greatly contributes in improving the accuracy of the parameter extracted. Similar to the single-diode PV parameter estimation, EVPS and ECHTE algorithms are applied to estimate design parameters of two-diode PV cells under STC conditions which are charted in Table 4. The obtained results affirm EHCTE to take more time for its execution but in contrary, it rewards the least error inferring its effectiveness in achieving higher ECR. A reference work [17] containing different evolutionary algorithms has been considered to perform parameter estimation. Thus, the parameters estimated using the above EC models are given as follows (Table 5).

5.3 Uncontrolled Climatic Data Validation Conditions with Measured Sets These algorithms are used extensively in case of stochastic climatic conditions like changing temperature and irradiation. In order to substantiate the obtained PV and IV characteristic curves, data obtained from NREL for different time of the day under varying irradiation and temperature are considered. The variations in temperature and irradiance with respect to time are presented in Fig. 4. The graph clearly reveals that the parameterized model using proposed approach exhibit close characteristic curves to that of the experimental model. The performance indices obtained using parameterized model validates the effectiveness of the proposed method (Fig. 5). The real-time experimental data provided by NREL of Golden City at Colorado State was considered for Si-Tandem 90–31 solar cell at 1012.9 W/m2 irradiance

496

P. Kumari and P. Geethanjali EsƟmated Data Experimental Data

Current (A)

1

0.5

0

Voltage (V)

-0.5

(a) 0.4

EsƟmated Data Experimental Data

Power (W)

0.2

0

-0.2

Voltage (V) (b)

(c) Fig. 4 a I–V characteristics. b P–V characteristics. c RMSE computation for different methods

Implementation of Constraint Handling Techniques …

497

Table 2 Double-diode parameter estimation for RTC France PV cell Parameter I ph (A)

SF-DE

SP-DE

0.76081244

*EVPS

0.760811879

**ECHT-DE

0.7608118

0.7608131

Rsh ()

58.3646509

58.3867470

58.3646509

58.37134386

I o1 (µA)

2.1724074

2.17536

1.96706

0.7608131

I o2 (µA)

0.0872713

0.0871260

0.07760

0.0869527

a1

1.37149073

1.37136571

1.363412

1.371206

a2

1.999996

1.99999913

1.9590989

1.999999

Rs ()

0.0380289

0.0380294

0.03808612

0.03803330

RMSE × 10–4

7.325520662

7.32552092

7.3406838415

7.325513

Table 3 Diode parameter estimation for RTC France PV cell and comparison of ECHT-DE parameter estimation with literature Algorithm

Iph (A)

Io (µA)

Rsh ()

n

Rs ()

RMSE (10–4 )

ECHT-DE

0.760788

0.310684

52.889787

1.475258

0.036546

7.730062

GAMS [29] Gnetchejo (2019)

0.760776

0.323020

53.718524

1.481184

0.036377

9.860218

ITLBO [37] Li (2019)

0.7608

0.3230

53.7185

1.4812

0.0364

9.8602

IWAO [34] Xiong (2018)

0.760877519 0.3232

53.73168644

1.48122913 0.03637529 9.8602

ISCE [35] Xiankum (2018)

0.76077553

0.32302083 53.71852771

1.48118360 0.03637709 9.860219

SATLBO [33] Kunjie (2017)

0.7608

0.32315

53.7256

1.48123

0.03638

9.8602

GOFP-ANM 0.7607755 [36] Shuhui (2017)

0.3230208

53.7185203

1.4811836

0.0363771

9.8602

IJAYA [32] Kunjie (2017)

0.7608

0.3228

53.7595

1.4811

0.0364

9.8603

CSO [31] Guo (2016)

0.76078

0.3230

53.7185

1.48118

0.03638

9.8602

Rcr- IJADE [30] Gong (2013)

0.760776

0.323021

53.718526

1.481184

0.036377

9.8602

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P. Kumari and P. Geethanjali

Table 4 Comparative analysis of different constraint handling techniques for one diode model Algorithm

Mean

ECHT-DE

7.730062

SF-DE

8.8244371

SP-DE

8.633495

EVPS

7.762812

Worst

Standard deviation

CPU time (s)

7.730062

4.53e−12

3.6

11.292247

8.53e−5

1.75

10.863042

8.20e−5

1.78

7.785691

1.52e−6

1024

Table 5 Performance parameter comparison of various constraint handling methods for two-diode model Algorithm

Mean

ECHT-DE

7.388335

SF-DE

0.00079319

SP-DE

7.99556

EVPS [27]

7.53386

Worst

Standard deviation

CPU time (s)

7.507677

5.04e−6

3.76

0.0019911

8.53e−5

1.83

1.219e−6

1.84

8.5519e−6

2.556

12.384429 7.6765868

Fig. 5 Experimental data considering environmental factors

and 57.7 °C temperature. The cell had short circuit current of 1.664 A, maximum current I mp equal to 0.9506 V, maximum voltage V mp = 39.8593 V, open circuit voltage V oc of 53.7476 V and maximum power Pmp = 37.8917 W. Total number of cells in series was 38. Considering this varying uncontrolled climatic conditions, the parameters were estimated using ECHTE. The curves exhibit good agreement with the estimated values proving its accuracy. This underwrites that the applied

Implementation of Constraint Handling Techniques …

499

calculations can precisely be utilized to build a virtual PV simulator which can anticipate the behaviour under any ecological conditions.

6 Conclusion Optimal design parameter selection is very essential for achieving higher energy conversion ratio and maximum power point tracking (MPPT). In proposed work, a novel and robust PV design parameter extraction techniques are developed. Two distinct and advanced evolutionary computing algorithms named enhanced vibrating particles system algorithm (EVPS) and (ECHTE) algorithm were applied to perform PV design parameter estimation. Unlike classical evolutionary computing methods, ECHTE algorithms avoid higher iterations and provide optimal solution without imposing issues like local minima and convergence. Simulations with different PV models affirm that the proposed evolutionary approaches ensures optimal design parameter estimation under different insolation. Comparative results reveal that ECHTE outperforms other state-of-art techniques for PV parameter estimation. This process of accurate estimation helps in designing most efficient inverter with precise control techniques. In future, the performance could be assessed in terms of MPPT under varying irradiation condition.

References 1. Silva, E.A., Bradaschia, F., Cavalcanti, M.C., Nascimento, A.J.: Parameter estimation method to improve the accuracy of the photovoltaic electrical model. IEEE J. Photovoltaics 6(1), 278–285 (2016) 2. Chan, D.S., Phang, J.C.: Analytical methods for the extraction of solar-cell single-and doublediode model parameters from IV characteristics. IEEE Trans. Electron Devices 34(2), 286–293 (1987) 3. Farivar, G., Asaei, B., Mehrnami, S.: An analytical solution for tracking photovoltaic module MPP. IEEE J. Photovoltaics 3(3), 1053–1061 (2013) 4. Bharadwaj, P., Chaudhury, K.N., John, V.: Sequential optimization for PV panel parameter estimation. IEEE J. Photovoltaics 6(5), 1261–1268 (2016) 5. Kumari, P.A., Geethanjali, P.: Parameter estimation for photovoltaic system under normal and partial shading conditions: a survey. Renew. Sustain. Energy Rev. 84, 1–11 (2018) 6. Silva, E.A., et al.: An eight-parameter adaptive model for the single diode equivalent circuit based on the photovoltaic module’s physics. IEEE J. Photovoltaics 7(4), 1115–1123 (2017) 7. Ismail, M.S., Moghavvemi, M., Mahlia, T.M.: Characterization of PV panel and global optimization of its model parameters using genetic algorithm. Energy Convers. Manage. 1(73), 10–25 (2013) 8. Kumari, P.A., Geethanjali, P.: Adaptive genetic algorithm based multi-objective optimization for photovoltaic cell design parameter extraction.Energy Proc. 117 (2017). https://doi.org/10. 1016/j.egypro.2017.05.165 9. Ye, M., Wang, X., Xu, Y.: Parameter estimation of solar cells using particle swarm optimization. J. Appl. Phys. 105(9) (2009)

500

P. Kumari and P. Geethanjali

10. Bradaschia, F., Cavalcanti, M.C., Nascimento, A.J., Silva, E.A., Azevedo, G.M.S.: Parameter identification for PV modules based on an environment-dependent double-diode model. IEEE J. Photovoltaics 9(5), 1388–1397 (2019) 11. El-Naggar, K.M., Al Rashidi, M.R., Al Hajri, M.F. Al-Othman, A.K.: Simulated annealing algorithm for photovoltaic parameter identification. Solar Energy 86(1), 266–274 (2012) 12. Oliva, D., El Aziz, M.A., Hassanien, A.E.: Parameter estimation of photovoltaic cells using an improved chaotic whale optimization algorithm. Appl. Energy 200, 141–54 (2017) 13. Rajesekar, N., Krishna Kumar, N., Venugopalan, R.: Bacterial for-aging algorithm based PV parameter estimation. Solar Energy 97, 255–265 (2013) 14. Ma, J., Ting, T.O., Man, K.L., Zhang, N., Guan, S.U., Wong, P.W.: Parameter estimation of photovoltaic models via cuckoo search. J. appl. math. 2013 (2013) 15. Derick, M., Rani, C., Rajesh, M., Busawon, K., Binns, R.: Estimation of solar photovoltaic parameters using pattern search algorithm. In: International Conference on Emerging Trends in Electrical, Electronic and Communications Engineering, pp. 184–191. Springer, Cham (2016) 16. Subudhi, B., Pradhan, R.: Bacterial foraging optimization approach to parameter extraction of a photovoltaic module. IEEE Trans. Sustain. Energy 9(1), 381–389 (2017) 17. Mallipeddi, R., Suganthan, P.N.: Ensemble of constraint handling techniques. IEEE Trans. Evol. Comput. 14(4), 561–579 (2010) 18. Ishaque, K., I., Salam, Z., Taheri, H., Shamsudin, A.: Parameter extraction of photovoltaic cell using differential evolution method. In: 2011 IEEE Applied Power Electronics Colloquium (IAPEC), pp. 10–15. IEEE (2011) 19. Yang, X., Gong, W., Wang, L.: Comparative study on parameter extraction of photovoltaic models via differential evolution. Energy Convers. Manage. 201, 112113 (2019) 20. Abido, M.A., Khalid, M.S.: Seven-parameter PV model estimation using differential evolution. Electr. Eng. 100(2), 971–981 (2018) 21. Das, S., Abraham, A., Chakraborty, U.K., Konar, A.: Differential evolution using a neighborhood-based mutation operator. IEEE Trans. Evol. Comput. 13(3), 526–553 (2009) 22. Kyocera-KS20T multi-crystal PV module. [Online]. Available: https://www.kyocerasolar.com/ dealers/product-center/archives/spec-sheets/Solartec-KS20.pdf. Accessed on 3 July 2017 23. Sunmodule SW245 poly PV module. [Online]. Available: http://la.solarworld.com/∼/media/ www/files/datasheets/sunmodule-poly/sunmodule-solar-panel-245-poly-ds.pdf. Accessed on 3 July 2017 24. Shell solar SP140-PC monocrystalline PV module, M55 photovoltaic solar module. [Online]. Available: http://abcsolar.com/pdf/sp140pc.pdf. Accessed on 3 July 2017 25. Jadli, U., Thakur, P., Shukla, R.D.: A new parameter estimation method of solar photovoltaic. IEEE J. Photovoltaics 8(1), 239–247 (2017) 26. Easwarakhanthan, T., Bottin, J., Bouhouch, I., Boutrit, C.: Nonlinear minimization algorithm for determining the solar cell parameters with microcomputers. Int. J. Solar Energy 4(1), 1–12 (1986) 27. Gnetchejo, P.J., Essiane, S.N., Ele, P., Wamkeue, R., Wapet, D.M., Ngoffe, S.P.: Enhanced vibrating particles system algorithm for parameters estimation of photovoltaic system. J. Power Energy Eng. 7(08), 1 (2019) 28. Waly, H.M., Azazi, H.Z., Osheba, D.S.M., El-Sabbe, A.E.: Parameters extraction of photovoltaic sources based on experimental data. IET Renew. Power Gener. 13(9), 1466–1473 (2019) 29. Gnetchejo, P.J., Essiane, S.N., Ele, p., Wamkeue, R., Wapet, D.M., Ngoffe, S.P.: Important notes on parameter estimation of solar photovoltaic cell. Energy Convers. Manage. 197, 111870 (2019) 30. Gong, W., Cai, Z.: Parameter extraction of solar cell models using repaired adaptive differential evolution. Sol. Energy 94, 209–220 (2013) 31. Lei, G., Zhuo, M., Yize, S., Libiao, W.: Parameter identification and sensitivity analysis of solar cell models with cat swarm optimization algorithm. Energy Convers. Manage. 108, 520–528 (2016)

Implementation of Constraint Handling Techniques …

501

32. Yu, K., Liang, J.J., Qu, B.Y., Xu, C., Wang, H.: Parameters identification of photovoltaic models using an improved JAYA optimization algorithm. Energy Convers. Manage. 150, 742–53 (2017) 33. Kunjie, Y., Chen, X., Xin, W., Zhenlei, W.: Parameters identification of photovoltaic models using self-adaptive teaching-learning-based optimization. Energy Convers. Manage. 145, 233– 246 (2017) 34. Guojiang, X., Jing, Z., Dongyuan, Y., Shi, H.: Parameter extraction of solar photovoltaic models using an improved whale optimization algorithm. Energy Convers. Manage. 174, 388–405 (2018) 35. Gao, X., Cui, Y., Hu, J., Xu, G., Wang, Z., Qu, J., Wang, H.: Parameter extraction of solar cell models using improved shuffled complex evolution algorithm. Energy Convers. Manage. 157, 460–479 (2018) 36. Xu, S., Wang, Y.: Parameter estimation of photovoltaic modules using a hybrid flower pollination algorithm. Energy Convers. Manage. 53–68. https://doi.org/10.1016/j.enconman.2017. 04.042 37. Shuijia, L., Wenyin, G., Xuesong, Y., Chengyu, H., Danyu, B., Ling, W., Liang, G.: Parameter extraction of photovoltaic models using an improved teaching learning-based optimization. Energy Convers. Manage. 186, 293–305 (2019)

Single Zonal Building Energy Modelling and Simulation Nayan Kumar Singh

and V. S. K. V. Harish

Abstract Building energy control strategies involve development of building energy models with appreciable accuracy. Several methods are available for model development and simulation. Present paper adopts resistance–capacitance network to develop a building energy model which is to be conditioned by a single-zonal heating, ventilation and air-conditioning (HVAC) system. A white box mathematical model is developed, based on the fundamentals of energy physics, in MATLAB/Simulink. Differential equations are formulated and modelled in state space form for a multilayered building construction element. The element is configured as a three resistance and two capacitance model (pi-network) for a single-zonal room by considering the thermal resistance and thermal capacitance of the external walls, window glass, internal walls, ceiling and floor. Energy balance equations for each node of the 3R2C model are formulated as differential equations and solved when excited by step inputs. The input parameters for the developed model involve weather parameters of wind velocity, outdoor air temperature; thermos-physical properties of the building construction elements such as thermal resistance and thermal capacitance. The output parameter is the dry-bulb indoor air temperature for an input response of dry-bulb outdoor temperature and relative humidity. Developed modelling routine can act as benchmark for developing energy control strategies and their implementation. Keywords Building energy · State space model · Single-zonal model · 3R2C network

1 Introduction As per the information from International Energy Agency (IEA), the building sector is accountable for one-third total energy consumption globally, and it is also accountable for 40% of total indirect and direct CO2 emissions [1]. In recent years, the CO2 N. K. Singh · V. S. K. V. Harish (B) Department of Electrical Engineering, School of Technology, Pandit Deendayal Energy University (PDEU), Gandhinagar, Gujarat 382007, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2_43

503

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N. K. Singh and V. S. K. V. Harish

emission from building has risen. In 2019, the direct and indirect CO2 emission from buildings has risen to 10GtCO2 (Ten gigatonnes of carbon dioxide), which is the highest ever recorded according to IEA report [1]. According to the International Energy Outlook 2017 report of IEA states that, the up to 2040, India will be the country, which has the fastest growth in energy consumption in building. International Energy Outlook 2017 report also states that, it is expected that in India, there will be rise of an average of 2.7% energy consumption per year for commercial and residential buildings, between the year 2015 and 2040, which is much more than the twice the global average rise [2]. The energy demand in the building sector continues to grow due to increase in energy requirement for heating and cooling purpose in residential building by using the energy consuming devices [3]. However, building sector despite being the one of the largest contributor to emission and energy usage, the building sector holds the potential to tackle these problems [4]. Several buildings and project already exists that can show the incredible feast which can be achieved with energy efficient practice today, with technology and clever use of design playing a key role [5]. Several studies have been done on the heat transfer building modelling by researchers, and through these studies, many mathematical models [6] have been developed. Mosaico et al. [7] have developed an energy model for non-residential building by using state space equations. This model can accurately predict the energy consumption in HVAC system. In this model, occupant is estimated through deep transfer leering process which uses an image from thermal camera. Perera and Skeie [8] have developed different physics-based building heating model for ingle zone, multi-floor and multi-room building. These models are compared to each other to find out the good heating model, which can be used by the building energy management system for better control purpose and control energy consumption in building. These models are simulated on the MATLAB on multi-story residential building having several rooms. Fateh et al. [9] have developed the dynamic model for the single zone building with considering the internal radiation, solar radiation and PCM effects. The model than tested on the building, which is located in Genoa. The model was simulate in MATLAB/Simulink, and detailed analysis was taken on the model. In this paper, the mathematical model is developed for the single-zonal building, and state space analysis is done on it. The mathematical equation is derived for the building envelop, and then the model is developed in the MATLAB for further analysis.

2 Building Energy Model The single-zonal model of a building in this paper is formed on the basis of the following assumptions [10]:

Single Zonal Building Energy Modelling and Simulation

505

• Heat transfer is considered, only in one direction and that is across the thickness of each slab or wall. • Heat transfer in the building materials is isotropic. • Properties of material are independent of temperature. • There is no internal heat source or sink. • Air inside the room is well mixed. • The temperature of walls inner surface and the temperature of the zone space air are taken same. By considering the above assumption, 3R2C thermal network model is developed for a room. Figure 1 shows complete 3R2C thermal network model of a room, in which there are 2 external walls, 2 internal walls, a window glass, a ceiling and a floor are represented in there corresponding thermal resistance and thermal capacitance. T out , QHVAC and T bs1 are the inputs of the model. Based on Fig. 1, the energy balance equation for each nodes is developed. Energy balance equation for external wall 1 At node Tw,o       Aw1 ∂ Tw,o 1 Aw1 1 Tw,o + Tw,i + = − ∂t Rw1 Rw2 Cw1 Cw1 Rw2   Aw1 Tout + Rw1 Cw1

(1)

At node Tw,i      Aw1 Aw1 1 1 Tw,o + − Tw,i + Rw2 Cw2 Rw2 Rw3 Cw2     Aw1 Aw1 Tbs + Q HVAC + Rw3 Cw2 Cw2

∂ Tw,i = ∂t



(2)

Energy balance equation for external wall 2 At node Tw2,o       Aw2 1 ∂ Tw2,o 1 Aw2 Tw2,o + Tw2,i = − + ∂t Rw21 Rw22 Cw21 Cw21 Rw22   Aw2 Tout + Rw21 Cw21 At intermediate node Tw2       Aw Aw2 ∂ Tw2 1 1 = Tw2,o + − Tw2,i + ∂t Rw22 Cw22 Rw22 Rw23 Cw22     Aw2 Aw2 Tbs + Q HVAC + Rw23 Cw22 Cw22

(3)

(4)

506

N. K. Singh and V. S. K. V. Harish Rc1

Rc2

Tc,o

Ceiling

Cc1

Rw21 External wall - 2

Rc3

Tc,i

Cc2

Tw2,o

Rw22

Cw21

Tw2,i

Rw23

Cw22

Rg1

Rg2

Tg

Window on external wall 1

Cg

QHVAC Rw1

Tw,o

Rw2

Tw,i

Rw3 Tbs

Tout External wall - 1 Cw2

Cw1

Rp1

Tp,o

Rp2

Tp,i

Rp3

Tp2,i

Rp23

Tf,i

Rf3

Tbs1 Partition wall - 1

Cp2

Cp1

Rp21

Partition wall - 2

Tp2,o

Cp21

Rf1 Floor

Rp22

Cp22

Tf,o

Cf1

Fig. 1 3R2C thermal network model of a room

Rf2

Cf2

Single Zonal Building Energy Modelling and Simulation

507

Energy balance equation for window glass At node Tg       ∂ Tg Ag 1 1 Ag Tg + Tout = − + ∂t Rg1 Rg2 Cg Rg1 Cg     Ag Ag Tbs + Q HVAC + Rg2 Cg Cg

(5)

Energy balance equation for ceiling At node Tc,o         Ac 1 ∂ Tc,o 1 Ac Ac Tc,o + Tc,i + Tout = − + ∂t Rc1 Rc2 Cc1 Cc1 Rc2 Rc1 Cc1

(6)

At node Tc,i      Ac Ac 1 1 Tc,o + − Tc,i + Rc2 Cc2 Rc2 Rc3 Cc2     Ac Ac Tbs + Q HVAC + Rc3 Cc2 Cc2

∂ Tc,i = ∂t



(7)

Energy balance equation for partition wall 1 At node TP,o       AP 1 ∂ TP,o 1 AP TP,o + TP,i = − + ∂t R P1 R P2 C P1 C P1 R P2   AP Tbs1 + R P1 C P1

(8)

At node TP,i      AP AP 1 1 TP,o + − TP,i + R P2 C P2 R P2 R P3 C P2     AP AP Tbs + Q HVAC + R P3 C P2 C P2

∂ TP,i = ∂t



(9)

Energy balance equation for partition wall 2 At node TP2,o       A P2 1 ∂ TP2,o 1 A P2 TP2,o + TP2,i = − + ∂t R P21 R P22 C P21 C P21 R P22   A P2 Tbs1 + R P21 C P21

(10)

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N. K. Singh and V. S. K. V. Harish

At node TP2,i      A P2 1 1 A P2 TP2,o + − TP2,i + R P22 C P22 R P22 R P23 C P22     A P2 A P2 Tbs + Q HVAC + R P23 C P22 C P22

∂ TP2,i = ∂t



(11)

Energy balance equation for floor At node TF,o       AF ∂ TF,o 1 AF 1 TF,o + TF,i + = − ∂t R F1 R F2 C F1 C F1 R F2   AF Tbs1 + R F1 C F1

(12)

At node TF,i      AF AF 1 1 TF,o + − TF,i + R F2 C F2 R F2 R F3 C F2     AF AF Tbs + Q HVAC + R F3 C F2 C F2

∂ TF,i = ∂t



(13)

Energy balance equation for building space air      Ag Aw1 Aw2 Tw,i + Tw2,i + Tg Rw3 ρa V ca Rw23 ρa V ca Rg2 ρa V ca     Ap Ac + Tc,i + TP,i Rc3 ρa V ca R P3 ρa V ca     A P2 AF TP2,i + TF,i + R P23 ρa V ca R F3 ρa V ca  Ag Aw1 Aw2 Ac AP A P2 − + + + + + Rw3 Rw23 Rg2 Rc3 R P3 R P23  Tbs AF Q HVAC + + (14) R F3 ρa V ca ρa V ca

∂ Tbs = ∂t



The state-space representation of the above equation from Eqs. 1–14, can be done by representing the model in the following two equations. X˙ = AX + BU Y = C X + DU

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where state vector (X) is representing the state variables, which are representing the room construction element temperatures, input vector (U) is representing the system input and the output of the system is the inner room temperature. Equations 15 and 16 show the state-space equations of the 3R2C network model, which is obtained by using the above equations from Eqs. 1–14. The building energy system model study is a dynamic system with the nodal temperatures of Fig. 1, representing the state variables. And the inputs to the system are outdoor temperature (Tout ), HVAC heat gain (Q HVAC ) and adjacent room temperature (Tbs1 ). ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

T˙w,o T˙w,i ˙ Tw2,o T˙w2,i T˙g T˙c,o T˙c,i T˙P,o T˙P,i T˙P2,o T˙P2,i T˙F,o T˙F,i T˙bs

⎤

⎡

⎤ Tw,o ⎢ Tw,i ⎥ ⎥ ⎢ ⎥ ⎥ ⎢T ⎥ ⎥ ⎢ w2,o ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ Tw2,i ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ Tg ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ Tc,o ⎥ ⎥ ⎡ ⎤ ⎢ ⎥ ⎥ Tout ⎢ Tc,i ⎥ ⎥ ⎥ = A⎢ ⎥ ⎣ ⎦ ⎢ TP,o ⎥ + B Q HVAC ⎥ ⎢ ⎥ ⎥ Tbs1 ⎢ T ⎥ ⎥ ⎢ P,i ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ TP2,o ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ TP2,i ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ TF,o ⎥ ⎥ ⎢ ⎥ ⎥ ⎣ TF,i ⎦ ⎦ Tbs

Matrix A can be written as: ⎡ A11 A12 0 0 0 0 0 ⎢ A21 A22 0 0 0 0 0 ⎢ ⎢ 0 0 A A 0 0 0 33 34 ⎢ ⎢ A 0 0 0 0 0 A ⎢ 43 44 ⎢ ⎢ 0 0 0 0 A55 0 0 ⎢ ⎢ 0 0 0 0 0 A66 A67 ⎢ ⎢ 0 0 0 0 0 A76 A77 ⎢ ⎢ 0 0 0 0 0 0 0 ⎢ ⎢ 0 0 0 0 0 0 0 ⎢ ⎢ 0 0 0 ⎢ 0 0 0 0 ⎢ ⎢ 0 0 0 0 0 0 0 ⎢ ⎢ 0 0 0 0 0 0 0 ⎢ ⎣ 0 0 0 0 0 0 0 0 A142 0 A144 A145 0 A147 Matrix B can be written as:

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 A88 A89 0 0 0 0 0 0 0 A98 A99 0 0 0 A1010 A1011 0 0 0 0 0 A1110 A1111 0 0 0 0 0 A1212 A1213 0 0 0 0 A1312 A1313 0 A149 0 A1411 0 A1413

(15)

⎤ 0 A214 ⎥ ⎥ 0 ⎥ ⎥ ⎥ A414 ⎥ ⎥ A514 ⎥ ⎥ 0 ⎥ ⎥ A714 ⎥ ⎥ 0 ⎥ ⎥ A914 ⎥ ⎥ ⎥ 0 ⎥ ⎥ A1114 ⎥ ⎥ 0 ⎥ ⎥ A1314 ⎦ A1414

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⎡

B11 ⎢ 0 ⎢ ⎢B ⎢ 31 ⎢ ⎢ 0 ⎢ ⎢ B51 ⎢ ⎢ B61 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎣ 0 0

0 B21 0 B42 B52 0 B72 0 B92 0 B112 0 B132 B142

⎤ 0 0 ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ B83 ⎥ ⎥ 0 ⎥ ⎥ ⎥ B103 ⎥ ⎥ 0 ⎥ ⎥ B123 ⎥ ⎥ 0 ⎦ 0

And the output equation in the matrix form is ⎡

⎤ Tw,o ⎢ Tw,i ⎥ ⎢ ⎥ ⎢T ⎥ ⎢ w2,o ⎥ ⎢ ⎥ ⎢ Tw2,i ⎥ ⎢ ⎥ ⎢ Tg ⎥ ⎢ ⎥ ⎢ Tc,o ⎥ ⎢ ⎥

⎢ Tc,i ⎥ ⎢ ⎥ Y = 00000000000001 ⎢ ⎥ ⎢ TP,o ⎥ ⎢ T ⎥ ⎢ P,i ⎥ ⎢ ⎥ ⎢ TP2,o ⎥ ⎢ ⎥ ⎢ TP2,i ⎥ ⎢ ⎥ ⎢ TF,o ⎥ ⎢ ⎥ ⎣ TF,i ⎦ Tbs ⎡ ⎤ Tout + [0]⎣ Q HVAC ⎦ Tbs1

(16)

All the elements of the D matrix are zero. The state-space model for 3R2C thermal network model of room is developed by using the MATLAB software. The programme is written in MATLAB, which solves Eqs. 15 and 16 by using the MATLAB command. The values for the variables which are used in the MATLAB programme are given in appendix.

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3 Simulation Results In MATLAB, the unit step input is given to the 3R2C thermal network model of the single-zonal building space under study, and following results were found. Figure 2 shows the step response of the inside room temperature (T BS ), when the step outdoor temperature (T out ) as input (1) is given to the model, the reaming two inputs considers as zero. Figure 2 also shows that the inside room temperature rises in approximately 3470 s and settles at final value of 0.541 with approximate settling time of 8300 s.

Fig. 2 Response of building space air temperature for a step input of outdoor air temperature

Fig. 3 Response of building space air temperature for a step input of HVAC system heat rate

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Fig. 4 Response of building space air temperature for a step input of air temperature of an adjacent room

Figure 3 shows the step response of the inside room temperature (T BS ), when the step HVAC heat gain (QHVAC ) as input (2) is given to the model, the reaming two inputs considers as zero. In Fig. 3, the inside room temperature rises in approximately 5960 s and settles at final value of 0.312 with approximate settling time of 1270 s. Figure 4, shows the step response of the inside room temperature (T BS ), when the step building space air temperature 1 (T BS1 , i.e. temperature of adjacent room) as input (3) is given to the model, the reaming two inputs considers as zero. In Fig. 4, the inside room temperature rises in approximately 4520 s and settles at final value of 0.476 with approximate settling time of 1120 s. From the simulation result, it is shown that the rise time is very high, when step input is given to the model. And in Fig. 2, when input as a step outdoor temperature is given to the model, then the settling time is severely high.

4 Conclusion A fully integrated building energy system model is significant to develop and deploy energy control and management strategies. In this paper, a second order 3R2C thermal network model of a single-zonal building room has been developed, and the model is simulated in the MATLAB. The input parameters for the developed model are outdoor air temperature, heat gain of HVAC system and the adjacent room temperature. Thermo-physical properties of the building construction elements such as thermal resistance and thermal capacitance values are given in the appendix. The output parameter is building space air temperature. When the step input is given to the model one at a time, we get the step response from that and the rise time and

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settling time noted from the step response graph. The developed modelling routine can act as benchmark for developing and testing any energy control strategy for attaining building energy efficiency and can also be used in energy control strategy implementation. The study shall be extended further by modelling the power grid in terms of swing equations to analyse the interactions of building energy control and power grid.

Appendix

Building envelope External wall

Parameters External wall area

Values (m2 )

60

Thermal resistance (m2 °C/W) Rw1 = Rw21

0.36

Rw2 = Rw12

1.65

Rw3 = Rw13

1.25

Thermal capacitance

(J/m2

°C)

Cw1 = Cw21

24,010

Cw2 = Cw22 Window glass

Window glass area

105,070 (m2 )

16

Thermal resistance (m2 °C/W) Rg1 = Rg2

0.1785

Thermal capacitance Ceiling and floor

(J/m2

°C)

Ceiling area and floor area (m2 )

33,810 120

Thermal resistance (m2 °C/W) Rc1 = RF1

0.04

Rc2 = RF2

0.11

Rc3 = RF3

0.27

Thermal capacitance

(J/m2

°C)

Cc1 = CF1

24,010

Cc2 = CF2 Partition wall

Partition wall area

278,910 (m2 )

40

Thermal resistance (m2 °C/W) RP1 = RP21

0.36

RP2 = RP22

1.65

RP3 = RP23

1.25

Thermal capacitance (J/m2 °C) CP1 = CP21

24,010 (continued)

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(continued) Building envelope Other parameters

Parameters

Values

CP2 = CP22

105,070

Volume of room (m3) Density of indoor air kg

500 (m−3 )

Specific heat capacity of air (J/(kg °C))

1184 1

References 1. IEA Homepage. https://www.iea.org/topics/buildings. Last accessed 2020/11/23 2. US EIA. Available Online: https://www.eia.gov/todayinenergy/dtail.php?id=33252#:~:text= The%20United%20Nations%20projects%20India%E2%80%99s,consumption%20in%20I ndia%20in%202015. Last accessed 2020/12/11 3. Harish, V.S.K.V., Parab, D., Doshi, K.: 11 building-to-grid integration for smart grids. In. Green Innovation, Sustainable Development, and Circular Economy, p. 157. Taylor and Francis, ISBN: 9780367441746 (2020). https://doi.org/10.1201/9781003011255 4. Harish, V.S.K.V., Kumar, A.: 10 smart energy control and comfort management in buildings. In. Green Innovation, Sustainable Development, and Circular Economy, p. 141, Taylor and Francis, ISBN: 9780367441746 (2020). https://doi.org/10.1201/9781003011255 5. Harish, V.S.K.V., Kumar, A.: Simulation based energy control and comfort management in buildings using multi-objective optimization routine. Int. J. Math. Eng. Manage. Sci. 5(6), 1324–1332 (2020). https://doi.org/10.33889/IJMEMS.2020.5.6.098 6. Harish, V.S.K.V., Kumar, A.: Reduced order modeling and parameter identification of a building energy system model through an optimization routine. Appl. Energy 162, 1010–1023, ISSN: 0306-2619 (2016). https://doi.org/10.1016/j.apenergy.2015.10.137 7. Mosaico, G., Saviozzi, M., Silvestro, F., Bagnasco, A., Vinci, A.: Simplified state space building energy model and transfer learning based occupancy estimation for HVAC optimal control. In: 2019 IEEE 5th International forum on Research and Technology for Society and Industry (RTSI), pp. 353–358. IEEE (2019) 8. Perera, D.W.U., Skeie, N.O.: Modeling and simulation of multi-room buildings (2016) 9. Fateh, A., Borelli, D., Spoladore, A., Devia, F.: A state-space analysis of a single zone building considering solar radiation, internal radiation, and PCM effects. Appl. Sci. 9(5), 832 (2019) 10. Underwood, C., Yik, F.: Modelling Methods for Energy in Buildings. John Wiley & Sons (2008)

Author Index

A Ahmad, Nesar, 369 Ahmed, Sarfraz, 169 Anukiruthika, K., 93 Asraful Haque, Md., 369 B Bansal, Ritik, 393 Bebarta, Shubhashree, 107 Bhatt, Jignesh, 335 Bhatt, Vishva, 381 Bhawaria, Sushil, 229 Bhuju, Gauri, 423 Bisht, Soni, 361 C Chamola, Bhagwati Prasad, 77 Chandaragi, Praveen I., 241 Chavaraddi, Krishna B., 241 D Dasgupta, Srirupa, 143 Desai, Rupande, 299 Dey, Prianka, 157 Dhar, Joydip, 255 Dhiman, S. D., 313 Dhodiya, Jayesh, 285 Doshi, Nishant, 439 Dutta, Biju Kumar, 169 F Faruqi, Shahab, 31

G Geethanjali, P., 487 Ghosh, Medhashree, 143 Gode, Ruchi Telang, 31 Gouder, Priya M., 241 Goudru, N. G., 321 Gourisaria, Mahendra Kumar, 351 Goyal, Nupur, 361 Gupta, Jyoti, 255 Gurung, Dil Bahadur, 423

H Harish, V.S.K.V., 335, 503

J Jahan, Akbari, 121 Jani, Hardik K., 449 Jani, Omkar, 335 Jena, Mahendra Kumar, 107 John, J. Catherine Grace, 3 Joshi, Nirav R., 473

K Kachhwaha, Surendra Singh, 271, 449 Khan, Abhinandan, 143, 157 Khandelwal, Utkarsh, 393 Kiwne, S. B., 129 Kodgire, Pravin, 271 Kumar, Akshay, 361 Kumari, P. Ashwini, 487 Kumar, Saurabh, 393 Kumar, Vijay, 361

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1405, https://doi.org/10.1007/978-981-16-5952-2

515

516 M Marali, G. B., 241 Mehta, Chaitali, 459 Mishra, Ashutosh, 31 Mishra, Sarita, 351 Muthukumar, P., 93

N Nagababu, Garlapati, 449 Narula, Pankaj, 181

P Pal, Rajat Kumar, 143, 157 Pandey, Manjusha, 351 Pandya, Divyang H., 203 Parvat, B. J., 215 Patel, Narendra, 299 Patel, Purvak K., 415 Patil, N. S., 215 Phaijoo, Ganga Ram, 423 Prajapati, Nirav, 271

R Ram, Mangey, 361 Rathore, Himanshu, 229 Rautaray, Siddharth Swarup, 351 Rokad, Vipulkumar, 203

S Sabharwal, Anuradha, 13 Saha, Goutam, 143, 157 Sahni, Manoj, 77 Sahni, Ritu, 77

Author Index Sant, Amit V., 459, 473 Saraswat, Ram Naresh, 57, 191 Shah, Hetvi, 381 Shah, Jigarkumar, 381 Sharma, Anulika, 57 Sharma, Arindam, 181 Sharma, Paawan, 459 Sharma, Sandeep, 181 Singh, Nayan Kumar, 503 Singh, Prabhdeep, 181 Sinha, Poonam, 255 Sonawane, Sandip M., 129 Sonker, Smita, 45

T Tilva, Surbhi, 285

U Udar, Dinesh, 25 Umar, Adeeba, 191

V Veninstine Vivik, J., 407 Vivik., J. Veninstine, 67 Vyas, Rajendra G., 415

X Xavier, P., 67, 407

Y Yadav, Pooja, 13