Fuzzy Fractional Differential Operators and Equations: Fuzzy Fractional Differential Equations [1st ed.] 9783030512712, 9783030512729

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Studies in Fuzziness and Soft Computing

Tofigh Allahviranloo

Fuzzy Fractional Differential Operators and Equations Fuzzy Fractional Differential Equations

Studies in Fuzziness and Soft Computing Volume 397

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Fuzziness and Soft Computing” contains publications on various topics in the area of soft computing, which include fuzzy sets, rough sets, neural networks, evolutionary computation, probabilistic and evidential reasoning, multi-valued logic, and related fields. The publications within “Studies in Fuzziness and Soft Computing” are primarily monographs and edited volumes. They cover significant recent developments in the field, both of a foundational and applicable character. An important feature of the series is its short publication time and world-wide distribution. This permits a rapid and broad dissemination of research results. Indexed by ISI, DBLP and Ulrichs, SCOPUS, Zentralblatt Math, GeoRef, Current Mathematical Publications, IngentaConnect, MetaPress and Springerlink. The books of the series are submitted for indexing to Web of Science.

More information about this series at http://www.springer.com/series/2941

Tofigh Allahviranloo

Fuzzy Fractional Differential Operators and Equations Fuzzy Fractional Differential Equations

123

Tofigh Allahviranloo Faculty of Engineering and Natural Sciences Bahcesehir University Istanbul, Turkey

ISSN 1434-9922 ISSN 1860-0808 (electronic) Studies in Fuzziness and Soft Computing ISBN 978-3-030-51271-2 ISBN 978-3-030-51272-9 (eBook) https://doi.org/10.1007/978-3-030-51272-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 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, express 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 Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

A thousand hours of my loneliness and suffering in the quarantine period is dedicated to Laala who took the responsibility of Sahand for success and progress and also made my academic life prosperous. I Thank you. Tofigh Allahviranloo

Preface

I tried to use my all knowledge and experiences during 22 years on the fuzzy subjects, to write this book. To this end, I studied many sources that all are in the sources section but the writings and contributions come from myself independently. The present book contains useful information for identifying fuzzy fractional differential equations. By studying this book, you can get acquainted with different types of vague information and learn how to use it efficiently in fractional systems. A detailed study of fuzzy fractional differential equations has been performed from a theoretical and numerical point of view. For this purpose, numerical and semi-analytical methods for solving these differential equations have been investigated. Besides, an analysis of the complete error of the solutions is also provided. Interesting applications of these systems in engineering and biology have also been reported. Istanbul, Turkey

Tofigh Allahviranloo

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Contents

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1 1 4 6

2 Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Fuzzy Sets and Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Membership Function . . . . . . . . . . . . . . . . . . . . . . . 2.3 Fuzzy Numbers and Their Properties . . . . . . . . . . . . . . . . . . 2.3.1 Definition of a Fuzzy Number . . . . . . . . . . . . . . . . 2.3.2 Level-Wise Form of a Fuzzy Number . . . . . . . . . . . 2.3.3 Definition of a Fuzzy Number in Level-Wise Form . 2.3.4 Definition of a Fuzzy Number in Level-Wise Form . 2.3.5 Definition of a Fuzzy Number in Parametric Form . . 2.3.6 Non-linear Fuzzy Number . . . . . . . . . . . . . . . . . . . 2.3.7 Trapezoidal Fuzzy Number . . . . . . . . . . . . . . . . . . . 2.3.8 Triangular Fuzzy Number . . . . . . . . . . . . . . . . . . . . 2.3.9 Operations on Level-Wise Form of Fuzzy Numbers . 2.3.10 Piece Wise Membership Function . . . . . . . . . . . . . . 2.3.11 Some Properties of Addition and Scalar Product on Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Some Operators on Fuzzy Numbers . . . . . . . . . . . . . . . . . . . 2.4.1 Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Limit of Fuzzy Number Valued Functions . . . . . . . . 2.4.3 Fuzzy Riemann Integral Operator . . . . . . . . . . . . . . 2.4.4 Some Additional Properties of gH-Difference . . . . . 2.4.5 Proposition—Minimum and Maximum . . . . . . . . . . 2.4.6 Proposition—Cauchy’s Fuzzy Mean Value Theorem

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1 Introduction to Fuzzy Fractional Operators and Equations 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Introduction to Fuzzy Fractional Differential Equations . . 1.3 Structure of the Book . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.4.7 2.4.8 2.4.9 2.4.10 2.4.11 2.4.12 2.4.13

Corollary—Fuzzy Mean Value Theorem . . . . . . . . . . . Integral Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fuzzy Taylor Expansion of Order One . . . . . . . . . . . . Integration by Part . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition—gH-Partial Differentiability . . . . . . . . . . . . Fuzzy Fubini—Theorem . . . . . . . . . . . . . . . . . . . . . . . First Order Fuzzy Taylor Expansion for Two Variables Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Fuzzy Fractional Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Fuzzy Grunwald-Letnikov Derivative—Fuzzy GL Derivative . . 3.2.1 Definition—Length of Fuzzy Function . . . . . . . . . . . . 3.2.2 Level-Wise Form of Grunwald-Letnikov GL-Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Remark—Fuzzy Fractional Integral Operator . . . . . . . . 3.3 Fuzzy Riemann-Liouville Derivative—Fuzzy RL Derivative . . . 3.3.1 Level-Wise Form of Fuzzy Riemann-Liouville Integral Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 The Fuzzy Riemann-Liouville Derivative Operators . . . 3.3.3 RL—Fractional Derivative for m ¼ 1 . . . . . . . . . . . . . 3.4 Fuzzy Caputo Fractional Derivative . . . . . . . . . . . . . . . . . . . . . 3.4.1 Caputo—Fuzzy Fractional Derivative for m ¼ 1 . . . . . 3.4.2 Caputo gH Differentiability . . . . . . . . . . . . . . . . . . . . . 3.5 Fuzzy Riemann-Liouville Generalized Fractional Derivative . . . 3.5.1 Fuzzy Riemann-Liouville Generalized Fractional Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Riemann Liouville–Katugampola gH—Fractional Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Caputo–Katugampola gH–Fractional Derivative . . . . . . . . . . . . 3.7 Riemann-Liouville gH-Fractional Derivative of a 2 ð1; 2Þ . . . . . 3.7.1 Definition—Fuzzy Caputo Fractional Derivative of Order a 2 ð1; 2Þ . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Generalized Fuzzy ABC Fractional Derivative . . . . . . . . . . . . . 4 Fuzzy Fractional Differential Equations . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Fuzzy Fractional Differential Equations—Caputo-Katugampola Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Existence and Uniqueness of the Solution . . . . . . . . . 4.2.2 Some Properties of Mittag-Leffler Function . . . . . . . . 4.3 Fuzzy Fractional Differential Equations—Laplace Transforms . 4.3.1 Definition—Absolutely Convergence . . . . . . . . . . . . . 4.3.2 Definition—Exponential Order . . . . . . . . . . . . . . . . .

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4.3.3 Some Properties of Laplace . . . . . . . . . . . . . . . . . . . 4.3.4 Convolution Theorem . . . . . . . . . . . . . . . . . . . . . . . . 4.3.5 First Translation Theorem . . . . . . . . . . . . . . . . . . . . . 4.3.6 Second Translation Theorem . . . . . . . . . . . . . . . . . . 4.3.7 Remark—Laplace Forms of Fractional Derivatives . . . 4.3.8 Fuzzy Fourier Transform Operator . . . . . . . . . . . . . . 4.3.9 Existence of Fourier Transform . . . . . . . . . . . . . . . . . 4.4 Fuzzy Solutions of Time-Fractional Problems . . . . . . . . . . . . 4.4.1 Fuzzy Explicit Solution of the Time-Fractional Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Fuzzy Impulsive Fractional Differential Equations . . . . . . . . . 4.6 Concrete Solution of Fractional Differential Equations . . . . . . 4.6.1 Fractional Hyperbolic Functions . . . . . . . . . . . . . . . . 4.6.2 Some Derivation Rules for the Fractional Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Numerical Solution of Fuzzy Fractional Differential Equations . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Fuzzy Mean Value Theorem for Riemann-Liouville Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Fuzzy Fractional Taylor’s Expansion with Caputo gH-Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Fuzzy Fractional Taylor Expansion . . . . . . . . . . . . . . . 5.3.2 Fuzzy Generalized Taylor’s Expansion . . . . . . . . . . . . 5.4 Fuzzy Fractional Euler with Caputo gH-Derivative . . . . . . . . . . 5.5 ABC-PI Numerical Method with ABC gH-Derivative . . . . . . . . 5.5.1 Definition—ABCGH Fractional Derivative in the Sense of Caputo Derivative . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Fuzzy Time Fractional Ordinary Differential Equation . 5.5.3 Remark—Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.4 An Efficient Numerical Method for ABC Fractional Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Numerical Method for Fuzzy Fractional Impulsive Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Remark—Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Reproducing Kernel Hilbert Space Method (RKHSM) . 5.6.3 Numerical Solving Fuzzy Fractional Impulsive Differential Equation in W22 ½0; 1 . . . . . . . . . . . . . . . . . 5.6.4 Combination of RKHM and FDTM . . . . . . . . . . . . . . 5.6.5 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Applications of Fuzzy Fractional Differential Equations . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Fuzzy Fractional Calculus—Preliminaries for Control Problem 6.2.1 Theorem—Interchanging Operators . . . . . . . . . . . . . . 6.2.2 Theorem—Fuzzy Fractional Integration by Part I . . . . 6.2.3 Theorem—Fuzzy Fractional Integration by Part II . . . 6.3 Fuzzy Optimal Control Problem . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Definition—Relative Extremum of Fuzzy Functional Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Theorem—Fuzzy Fundamental Theorem of Calculus of Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Necessary Optimality Conditions . . . . . . . . . . . . . . . 6.3.4 Sufficient Optimality Conditions . . . . . . . . . . . . . . . . 6.4 Fuzzy Fractional Diffusion Equations . . . . . . . . . . . . . . . . . . . 6.4.1 Remark—Laplace Transform of Caputo Derivative with Order 0\a\2 . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Fundamental Solution of Fuzzy Fractional Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Theorem—Fundamental Solution . . . . . . . . . . . . . . . 6.4.4 Application of Fuzzy Fractional Diffusion in Drug Release . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Introduction to Fuzzy Fractional Operators and Equations

1.1

Introduction

Uncertainty and hesitation always live with us and this is the reality. All human beings are accustomed to doubting of everything around them and asking themselves or others why? Because they are not sure and their information is not incomplete or accurate. Now imagine that with all this incomplete information, we are in an environment where there is confusion. It is a fact that we do not know how to answer many of our real questions. This attitude of mind-this attitude of uncertainty-is very important for scientists. Instead of trying to combat uncertainty, our job is to figure out how to embrace it and work with it. Because there is uncertainty about the growth, facilities, and life you want. We need to make sure that uncertainty creates an environment for our further growth and learning. Truth cannot be explained or evaluated simply by right or wrong, true or false. Sometimes it’s almost right, or sometimes it’s probably wrong. Most of the time, the answer is given by personal feelings, for example, “I hope it’s true” or “I think”. This is why mathematicians have to consider linguistic propositions in mathematical logic, and this is an entrance for uncertainty in mathematics. In fact, uncertainty has a history of human civilization, and humanity has long been concerned with controlling and exploiting this type of information. For example, probability theory is one of the ambiguities. Tass gambling was the beginning of what is now called probability theory. In the sixteenth century, there was no way to quantify chance. If someone spun two and a half dice during the game, people would think it’s just luck. This means that we can measure an event and know how lucky we are to be working. One of the oldest and most obscure concepts has been the word “lucky”. Gambling and dice have played an important role in the development of probability theory. In the fifteenth century, Gerolamo Cardano was one of the most well-known figures in a formal algebraic activity. In “Game of Chance,” he presented his first © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. Allahviranloo, Fuzzy Fractional Differential Operators and Equations, Studies in Fuzziness and Soft Computing 397, https://doi.org/10.1007/978-3-030-51272-9_1

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1 Introduction to Fuzzy Fractional Operators and Equations

analysis of the rules of chance. In this century, he solved such numerical problems. In 1657, Christine Huygens wrote the first book on probability, “In Calculating Chance.” This book was the real birth of probability. Probability theory was developed mathematically by Blaise Pascal and Pierre in the seventeenth century, which sought to solve mathematical problems in some gambling problems. The concept of “predictive value” is now an essential part of economics and finance. By calculating the expected value of an investment, the value of each party can be understood. Since the seventeenth century, probability theory has been consistently developed and applied in various disciplines. Today it is very important in most fields of engineering and instrument management and even in medicine, ethics, law, and so on. In classical logic, the values of “true” and “false” or “zero” and “one” are the values of a decision in binary logic. But with multi-valued logic, it will not work well with the hierarchy of truth. Another aspect of uncertainty appears in multi-value logic. Multi-value logic is a non-classical logic that is a special case of classical logic because it is the embodiment of the principle of the performance of truth. Multi-value logic uses the degree of its truth. It is very difficult to discuss the nature of “degrees of truth” or “values of truth”. As a practical example, in the literature of artificial intelligence and uncertain modeling, there is a long misconception about the role of fuzzy set theory and multi-value logic. The frequent question is that for the performance of a compound calculation and the validity of the rules there is an exception between mathematical meanings. This confusion, despite some early philosophical warnings, involves early developments in probable logic. Given this fact, three main points can be realized. First, it shows that the root of the disagreement lies in the unpleasant confusion between the hierarchy of beliefs and what logicians call “degrees of truth.” The latter is usually complicity, while the former is not. The growth of nature and the phenomena in it, all often do not follow the fixed law, and in this environment a material transfer model works with inaccurate information such as more or less… How do you think we can model this transfer? To do this, we need knowledge of transmission tools and transmission operators that can work with this inaccurate information. We call these types of operators fuzzy operators. Concerning this modeling, the fractional operators will be more useful and effective. As far as we know, the different materials and processes in many applied sciences like electrical circuits, biology, bio-mechanics, electrochemistry, electromagnetic processes and, others are widely recognized to be well predicted by using fractional differential operators in accordance with their memory and hereditary properties. For the complex phenomena, the modeling and their results in diverse widespread fields of science and engineering, are also so complicated and for achieving the accurate method the only powerful tool is fractional calculus. Indeed the fractional calculus is not only a very important and productive topic, it also represents a new point of view that how to construct and apply a certain type of non-local operators to real-world problems. Since the uncertainty in our real environment and data has an important role, this causes us to discuss the uncertainty in our mentioned topics.

1.1 Introduction

3

In engineering and biological science, many subjects are modeled by fractals and fractional operators like fractional differential operators. The heat, wave, chaos, and other phenomena can be modeled by fractional differential equations and especially if the data and information are uncertain and fuzzy the fuzzy fractional operators are better candidates to model the problems. Fractional differential computations are one of the branches of mathematical analysis that study and research the properties and applications of integrals and derivatives of arbitrary orders. The fractional derivatives and integrals are not only useful for describing many phenomena and properties of materials, but according to the results, it is clear that new modeling based on fractional derivatives is much more appropriate and accurate than modeling using derivatives with integer order. This advantage stems from the fact that fractional derivatives are an excellent tool for describing the memory and hereditary properties of different materials and how they are processed, and this is the most important advantage of fractional derivatives compared to derivatives with integer order. In other words, integer order derivatives depend only on the local behavior of the function, while modeling based on fractional derivatives allows all the information of the function to be condensed in a weighted form. Today, the advantages of fractional derivatives in modeling the electrical and mechanical properties of materials, explaining the properties of rock deformation, nonlinear earthquake oscillations, diffusion equations in mathematics, physics and dynamics of turbulence, quantum mechanics and plasma physics, etc. are well seen. In recent years, due to the frequent emergence of deficit differential equations in flow mechanisms, viscoelastic, biology, electrochemistry, and technical and physical issues, it has attracted the attention of many researchers in engineering and medical sciences to research in this regard. Successful theories can be found in mechanics (Vascular theory Static and viscoelastic) were referred to in biological chemistry (modeling of polymers and proteins), electrical engineering (transmission of ultrasonic waves), and medicine (modeling of human tissue under mechanical loads). The origins of this branch of mathematics came as Leibniz sought to introduce a new symbol for derivation by playing with symbols in 1695. In his journal, he used n the symbol ddtnx to define the derivative, which led to the controversial question of Hopital by Leibniz: What happens if n ¼ 12? In fact, this question was the beginning of a great change in mathematics, which led to the birth of a new branch of mathematics called fractional arithmetic. In this regard, scientists and researchers with a new perspective on mathematics began to investigate this issue. In 1730, Euler formed some questions about the fractional derivative and integral, which was a spark for finding the fractional derivative. He published the first work on derivatives of any order. In 1812, Laplace introduced an integral fractional derivative operator, and in 1819 he published the first work on derivatives of any order. Then in the same year, Lacroix, introduced the Gamma function as an extension of factorial, in the following form,

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1 Introduction to Fuzzy Fractional Operators and Equations

d n xðtÞ Cðm þ 1Þ mn t ; ¼ dtn Cðm  n þ 1Þ

mn

The concepts of fractional differentiation and fractional integration were examined further over the course of the eighteenth and nineteenth centuries. The topic attracted the attention of mathematical giants such as Riemann, Liouville, Abel, Laurent, and Hardy and Littlewood. The paradoxes described by Leibniz were resolved by others later, but this is not to say that the field of fractional calculus is now wholly free of open problems. One recurring issue over the centuries is the existence of numerous contradictory definitions. By the mid-nineteenth century, several different definitions of the deficit account had been proposed: Liouville defined his definition based on the distinction of exponential functions and another based on an integral formula for inverse power functions, while Lacroix defined differently by differentiation. Had created. The power functions of Liouville and Lacroix’s definitions are not equivalent, leading some critics to conclude that one must be right and the other wrong. De Morgan, however, wrote that: Therefore, both systems may be part of a more general system.

His words, like those of Leibniz 145 years ago, were prophetic. Liouville’s and Lacroix’s formulas are, in fact, specific to what is now known as Riemann– Liouville’s definition of fractional calculus. This includes a c-integration integration constant, which, when set to zero, adjusts the Lacroix formula and gives Liouville’s performance when set as infinity.

1.2

Introduction to Fuzzy Fractional Differential Equations

Several scientists in their earliest works introduced fuzzy fractional calculus as an uncertain fractional calculus to consider fractional-order systems with uncertain initial values or uncertain relationships between parameters. The uniqueness and existence of the solution of fractional differential equations with uncertain initial value are utilized as well. The other ones employed the Riemann–Liouville generalized H-differentiability in order to solve the fuzzy fractional differential equations and presented some new results under this notion. Then they applied the technique of fuzzy Laplace transforms to solve some types of fuzzy fractional differential equations based on the Riemann–Liouville fuzzy derivative. As an uncertain set-valued problem and considering the delta-Hukuhara derivative in the fuzzy case for uncertain functions, the stability criteria for hybrid fuzzy systems on time scales in the Lyapunov sense were introduced by others. The other scientists switched to the applied topics of fractional calculus and solved a class of time-dependent fuzzy fractional optimal control problems.

1.2 Introduction to Fuzzy Fractional Differential Equations

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In general, the majority of the fuzzy fractional differential equations as same as fuzzy differential equations do not have exact solutions. This is why approximate and numerical procedures are important to be developed. On the other hand, the complicity of many parameters in mathematical modeling of natural phenomena appears as an uncertain fractional model and they play an important role in various disciplines. Hence, it motivates the researchers to investigate effective numerical methods with error analysis to approximate the fuzzy fractional differential equations. As a result, researchers started to develop numerical techniques for fuzzy fractional differential equations. The first method was introduced as a fuzzy approximate solution using the Euler method to solve fuzzy fractional differential equations. The others adopted the operational Jacobi operational matrix based on the fuzzy Caputo fractional derivative using shifted Jacobi polynomials. The clear advantage of the usage of this method is that the matrix operators have the main role to find the approximate fuzzy solution of fuzzy fractional differential equations instead of considering the methods required the complicated fractional derivatives and their calculations. Then the spectral numerical method for solving fuzzy fractional kinetic equations was introduced. The simplicity, efficiency, and high accuracy of their method are the main advantages. After a while other scientists exploited a cluster of orthogonal functions, named shifted Legendre functions, to solve fuzzy fractional differential equations under Caputo type. The benefit of the shifted Legendre operational matrices method, over other existing orthogonal polynomials, is its simplicity of execution as well as some other advantages. The achieved solutions present satisfactory results, obtained with only a small number of Legendre polynomials. Recently, investigated an analytical method (Eigenvalue–Eigenvector) for solving a system of fuzzy differential equations under fuzzy Caputo’s derivative. To this end, they exploited generalized H-differentiability and derived the solutions based on this concept. Two definitions of differentiability of type-2 fuzzy number valued functions in sense of Riemann–Liouville and Caputo fractional order are introduced and clearly more accuracy will have more complicity cost. Then for solving fuzzy fractional differential equations, the notion of revisited Caputo’s H-differentiability based on the generalized Hukuhara difference and proposed a novel analytical method entitled fuzzy Laplace transforms are developed. Employing Laplace transforms, they proposed a novel efficient technique for the solution of this type of equation that can efficiently make the original problem easier to achieve the numerical solution. The suggested algorithm for the fuzzy fractional differential equations uses the level-wise fuzzy fractional derivative in Caputo sense. Moreover, some researchers have investigated the variation iteration method (VIM) to solve the fuzzy linear fractional differential equation under Caputo generalized Hukuhara differentiability. The VIM method is the semi-analytical method and it was introduced in fuzzy sense and modelled as fuzzy VIM in recent years.

6

1 Introduction to Fuzzy Fractional Operators and Equations

Then Semi-Analytical methods for solving fuzzy impulsive fractional differential equations based on fuzzy differential transform method in the fractional case in Caputo fractional derivative sense have been defined and discussed as well. Then two-dimensional Legendre wavelet was studied to approximate the solution of the fuzzy fractional integro-differential equation under Caputo generalized Hukuhara differentiability. In this research, the existence and uniqueness theorems for a fuzzy fractional integro-differential equation by considering the type of differentiability of solutions were proved. Then the generalized Taylor’s expansion was presented for fuzzy-valued functions. To this end, a fuzzy fractional mean value theorem for integral, and some properties of Caputo generalized Hukuhara derivative were discussed. Also, the authors derived the fractional Euler’s method for solving fuzzy fractional differential equations in the sense of Caputo’s differentiability. Moreover, the fuzzy q-derivative and fuzzy q-fractional derivative in Caputo sense by using generalized Hukuhara difference by means of q-Mittag-Leffler function are provided recently. Moreover, the characterization theorem between the solutions of the fuzzy Caputo q-fractional initial value problem and system of ordinary Caputo q-fractional differential equations was presented. The right and left fuzzy fractional Riemann–Liouville integrals and the right and left fuzzy fractional Caputo derivatives are studied in some researches and then the right and left fuzzy fractional Taylor formula has established for a fuzzy fractional Ostrowski type inequality with applications.

1.3

Structure of the Book

In Chap. 2, we are going to cover the fuzzy sets in several forms and discuss their properties and any other related preliminaries including operations. In Chap. 3, fuzzy fractional differential operators are introduced. Several operators including fuzzy fractional integrals are also considered. Moreover, the relations of fractional integrals and also derivatives are considered. In Chap. 4, we study the fuzzy fractional differential equations with the existence and uniqueness of the solutions. In this chapter, each fractional differential equation is explained with the corresponding fractional derivative in detail. One of the main discussions of this chapter is fuzzy impulsive fractional differential equations that have much application in engineering science. Besides the fuzzy Laplace transforms are expressed to solve the equations theoretically. In Chap. 5 application of these equations is brought. The first application is about the fuzzy optimal control problems and the second one discusses the drug release for Tumor of cancer.

Chapter 2

Fuzzy Sets

2.1

Introduction

In this chapter, we are going to cover the fuzzy sets in several forms and discuss the properties and any other related preliminaries.

2.2

Fuzzy Sets and Variables

Mathematical modeling seeks to describe the phenomenon formally, but it always faces some inconveniences, namely uncertainty, and ambiguity. To be vague, the fuzzy set theory, which is formulated by Prof. Lotfi A. Zadeh in 1965, aims to give the subject matter a mathematical treatment. In addition, it is considered as an important tool for better understanding some of the real situations. Fuzzy variables are tools for modeling data in an uncertain context that cannot be accurately predicted. For example, coin throwing, dice throwing, poker games, stock pricing, marketing and market demand, longevity and more. What is the answer to the question, who is handsome or beautiful? Apparently, it depends on the personality of the individual and differs from different personalities. It is clear that we are composed of a fuzzy set of beautiful people. Membership in this set is called membership grade. The degree of membership can be considered as a real number of [0, 1] intervals. In fact, a fuzzy set is formed by ordered pairs such that the second component is the degree of membership of the first component. Graphically, these sorted pairs are on the function and this function is called the membership function. Sometimes the fuzzy set plays the fuzzy variable role. It depends on the situation. For example, to consider a fuzzy valued function, it must have a fuzzy variable and in fact, is a set of probability measurements and has a membership function. Below, several fuzzy sets and variables are mathematically mapped and modeled. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. Allahviranloo, Fuzzy Fractional Differential Operators and Equations, Studies in Fuzziness and Soft Computing 397, https://doi.org/10.1007/978-3-030-51272-9_2

7

8

2 Fuzzy Sets

• Gaussian form membership function on ½a; b (Fig. 2.1). 8 xa < 0; 2 f ðxÞ ¼ e0:3x ; a  x  b : 0; xb

• Triangular form membership function on ½a; b 8 0; > > > xa1 > < ca1 ; gðxÞ ¼ 1; > a2 x > ; > > : a2 c 0;

xa a  x\c x¼c c\x  b xb

If suppose that a ¼ 1; c ¼ 1; b ¼ 4 the figure is (Fig. 2.2). • Trapezoidal form membership function on ½a1 ; a2  8 0; x  a1 > > > 0 1 > <
axa a1 ; a1  x\a 0 00 hðxÞ ¼ 1; a xa > x > ; a00 \x  a2 > aa22
> a : 0; x  a2 The following figure shows the Trapezoidal fuzzy number for any a1 ; a2 ; a0 ; a00 . It is clear, the functions should define a line between ½a1 ; a0 , ½a0 ; a00  and ½a00 ; a2 . The Trapezoidal fuzzy number is a general form of a Triangular fuzzy number, because it is sufficient that a0 ¼ a00 . For this reason, in approximation methods, the best approximation of a Triangular fuzzy number is a Trapezoidal fuzzy number (Fig. 2.3). Consider the following function, in this figure xðtÞ 2 ½0; 1R þ , the variable x is a fuzzy variable and can be chosen from uncertainty space (Fig. 2.4).

Fig. 2.1 Gaussian membership function on ½a; b ¼ ½4; 4

2.2 Fuzzy Sets and Variables

Fig. 2.2 Triangular membership function on ½1; 4

Fig. 2.3 Trapezoidal membership function

Fig. 2.4 Fuzzy variable

9

10

2 Fuzzy Sets

Some examples x1 ðtÞ ¼ t;

x2 ðtÞ ¼

n X

ai t i ;

n 2 Rþ ;

x3 ðtÞ ¼ exp t; . . .

i¼1

The Zigzag form of an uncertain variable This example is the shape of zigzag uncertain variables. Suppose we are talking about old people, not everyone who is less than a1 years old, but all people aged between a1 to a2 years old can be called old. The people with the age of a2 to a3 years old are almost old and the rest more than a3 years old is completely old. Suppose that xðtÞ is the function of oldness of aged people, the following membership function shows us the membership of the people in the function.

xðtÞ ¼

8 0; > > > < ata1 ; a2 a1 bta

2 > a3 a2 ; > > : ata1 þ

a2 a1

bta2 a3 a2

þ ðt  a3 Þ;

t  a1 a1 \t  a2 a2 \t  a3 a3 \t

In the following figure some type of a zigzag membership function is shown. (Fig. 2.5) Experimental uncertain variables The following membership function is for an experimental uncertain variable with the following figure.

Fig. 2.5 The set of tall kids as an uncertain variable

2.2 Fuzzy Sets and Variables

11

8 t  t1 < 0; xðtÞ ¼ f ðhÞ; t ¼ ti þ hðti þ 1  ti Þ : 1; tn \t where 1  i  n  1; 0\h  1: The function f ðhÞ is any function between the points ti and ti þ 1 as a jump function. In the following figure the relation between two arbitrary points is not restricted to the linear function. It can be any function even a curve (Fig. 2.6). Or another case of experimental uncertain variable (Fig. 2.7).

Fig. 2.6 An experimental (piecewise) uncertain variable

Fig. 2.7 An experimental (non-linear) uncertain variable

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2 Fuzzy Sets

xðtÞ ¼

1  ; ðetÞ ffiffi 1 þ exp pp 3d

e; d 2 R þ x 2 R

Note. Now we have enough information about the uncertainty and uncertain sets and variables. One of the uncertainties cases is fuzzy sets. Like as other uncertain sets, the fuzzy set is a set of ordered pairs, (member, its membership). It is a membership function as well and their structures and forms are the same as the structures and forms of uncertain sets. See the figures.

2.2.1

Membership Function

For a fuzzy set (uncertain set), M, the membership of its elements, x 2 M, has a membership degree, MðxÞ 2 ½0; 1. If the degree is 0, the member does not belong to the set and if the degree is 1, the member belongs to the set completely. This is the reason that, any real set is a special case of an uncertain set and has the characteristic degree and function. In this case, the range of the membership degree is f0; 1g. It is obvious than the set M is a fuzzy set and M ¼ fðx; MðxÞÞjMðxÞ 2 ½0; 1; x 2 Mg This set is called membership function. In this concept, if x belongs to the uncertain set with the membership degree MðxÞ, at the same time it does not belong to the set with 1  MðxÞ membership degree. In the following section, we discuss fuzzy numbers and their properties.

2.3

Fuzzy Numbers and Their Properties

First of all, we should know that why we need to fuzzy number and it is important. Because, we need to computations and ranking the fuzzy sets. This the reason that fuzzy sets must have some additional properties. Now we are going to explain their graphical and mathematical forms and computations. First of all, we should define a fuzzy set as a fuzzy number.

2.3.1

Definition of a Fuzzy Number

A fuzzy membership function M: R ! ½0; 1; is called a fuzzy number if it has the following conditions:

2.3 Fuzzy Numbers and Their Properties

13

1. M, is normal. It means there is at least a real member x0 such that M ðx0 Þ ¼ 1: 2. M, is fuzzy convex. It means, For two arbitrary real points x1 ; x2 and k 2 ½0; 1 we have M ðkx1 þ ð1  kÞx2 Þ  minfM ðx1 Þ; M ðx2 Þg 3. M, is upper semi-continuous. It means, if we increase its value at a certain point x0 to f ðx0 Þ þ  (for some positive constant ), then the result is upper-semicontinuous; if we decrease its value to f ðx0 Þ   then the result is lower-semi-continuous. 4. The closure of the set SuppðMÞ ¼ fx 2 RjMðxÞ [ 0g; as a support set, is a compact set. As an example; the triangular, trapezoidal fuzzy sets are fuzzy numbers. The definition of a fuzzy number can be defined as other forms, parametric and level-wise form.

2.3.2

Level-Wise Form of a Fuzzy Number

The level-wise membership function is, in fact, an inverse function of membership function that proposes an interval-valued function. In fact, any level in the vertical axis gives us an interval in the horizontal axis. For example, consider one of the following triangular membership functions (Fig. 2.8). In this figure, all real numbers in the interval [a, b] have degree of membership greater than or equal to the value of “r  level” in the fuzzy set M, i.e.

Fig. 2.8 Level-wise form of a fuzzy set

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2 Fuzzy Sets

8x 2 ½a; b; MðxÞ  r;

0r1

Then the r—cut or r—level set of the membership function can be defined as, M 1 ðrÞ ¼ fx 2 RjMðxÞ  r g ¼ ½a; b ¼ ½Ml ðrÞ; Mu ðrÞ := M½r; 0  r  1 Based on the inequality property in the set, M½r, MðxÞ  r, it is clear that the membership function, MðxÞ, can be obtained by, MðxÞ ¼ supf0  r  1jx 2 M 1 ðrÞg; x 2 R It means that there is one to one map between two functions, membership function M 1 ðrÞ; and level wise membership function MðxÞ (Fig. 2.9). The figure shows that for each interval there is a degree or level and vice versa. In fact, it can be claimed that, [ Domain of MðxÞ ¼ ½Ml ðrÞ; Mu ðrÞ 0r1

Now, in general, a fuzzy set M, in level-wise form can be shown as follows, M½r ¼ ½Ml ðrÞ; Mu ðrÞ; 0  r  1 Another form of the definition can be defined as follows.

2.3.3

Definition of a Fuzzy Number in Level-Wise Form

A fuzzy membership function M : R ! ½0; 1; is called a fuzzy number if its level-wise form M½r ¼ ½Ml ðrÞ; Mu ðrÞ; is a compact interval for any 0  r  1:

Fig. 2.9 One-to-one corresponding

2.3 Fuzzy Numbers and Their Properties

15

The following definition shows, when the stacking of levels or cuts in a fuzzy set can establish a fuzzy number.

2.3.4

Definition of a Fuzzy Number in Level-Wise Form

The sufficient and necessary conditions for M 1 ðrÞ ¼ M½r to be a level-wise membership function of a fuzzy number are: (i) (Nesting property) For any two r  levels; r1 ; r2 If r1  r2 then M ½r1   M ½r2  (ii) For any monotone increasing sequence of levels, 0\r1 \r2 \    \rn \1, If frn gn % r; then M ½rn  ! M½r; for any 0  r  1 (Fig. 2.10). A singleton fuzzy number A real number ‘a’ is called a singleton fuzzy number, if a½r ¼ ½al ðrÞ; au ðrÞ ¼ ½a; a It means in the membership function the membership degree at ‘a’ is 1 and at other values is zero. al ðrÞ ¼ au ðrÞ ¼ a.

Fig. 2.10 The levels stacking

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2 Fuzzy Sets

Based on this definition we have the same for fuzzy zero number or origin and al ðrÞ ¼ au ðrÞ ¼ 0:

2.3.5

Definition of a Fuzzy Number in Parametric Form

Any fuzzy number, M, has the parametric form MðrÞ ¼ ðMl ðrÞ; Mu ðrÞÞ for any 0  r  1, if and only if, (i) Ml ðrÞ  Mu ðrÞ (ii) Ml ðrÞ is an increasing and left continuous function on ð0; 1 and right continuous at 0 with respect to r. (iii) Mu ðrÞ is a decreasing and left continuous function on ð0; 1 and right continuous at 0 with respect to r. Note that, in items (ii) and (iii), both functions can be bounded. In both forms of fuzzy numbers—level-wise form and parametric form—both functions, lower, Ml ðrÞ and upper, Mu ðrÞ, are the same. But the differences can be listed as follow: 1. In level-wise form, the values of both functions for any arbitrary but fixed r, are real numbers. But in parametric form, they act the role of function with respect to r. 2. In level-wise form, the level is an interval for any arbitrary but fixed r. But in parametric form, it is a couple of functions with respect to r. Now we are going to introduce other forms of a fuzzy number in linear, non-linear cases. The following figure shows that it doesn’t matter which fuzzy number we do we consider for analyzing. Indeed, a membership function which corresponds to a fuzzy number is a piece-wise function. For examples, 8 0x1 < x; M1 ðxÞ ¼ 2  x; 1  x  2 : 0; otherwise  M2 ðxÞ ¼

1  j xj; 0; 

1  x  1 otherwise

j xj; 0;

1  x  1 otherwise

 pffiffiffi x; M4 ðxÞ ¼ 1;

1  x  1 otherwise

M3 ðxÞ ¼

2.3 Fuzzy Numbers and Their Properties

17

In general form, the function can be shown as follow, 8   L xa ; a  x  b > > < ba 1;  b  x  c MðxÞ ¼ R dx ; c  x  d > > : dc 0; otherwise where L; R: ½0; 1 ! ½0; 1 are two non-decreasing shape functions and Rð0Þ ¼ Lð0Þ ¼ 0; Rð1Þ ¼ Lð1Þ ¼ 1: To obtain the level-wise or parametric forms of the fuzzy number in the general form, the relations are, Ml ðrÞ ¼ a þ ðb  aÞL1 ðrÞ; Mu ðrÞ ¼ d  ðd  cÞR1 ðrÞ;

r 2 ½0; 1

Clearly, if the functions L and R are linear then we will have Trapezoidal and Triangular membership functions as fuzzy numbers. If they are non-linear then they will appear as curves look like Trapezoidal or Triangular. The general form of a fuzzy number ın linear or non-linear cases can be shown as the following figure (Fig. 2.11).

Fig. 2.11 General from of a fuzzy number

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2 Fuzzy Sets

2.3.6

Non-linear Fuzzy Number

For instance, in the following cases, the fuzzy numbers do not have any linear lower and upper functions. LðxÞ ¼

1 ; 1 þ x2

RðxÞ ¼

1 ; 1 þ 2j x j

a ¼ 3;

b ¼ 2;

m¼4

Then the membership function is, 8 4x 1 L 3 ¼ ; > 4x 2 < x4 1 þ ð1 3 Þ MðxÞ ¼ R ¼ 1 þ 2 x4 ; 2 > j2j : 0;

x4 4x otherwise

The figure of this membership function is (Fig. 2.12). Generally, the r-level set of a fuzzy set M is, M½r ¼ ½Ml ðrÞ; Mu ðrÞ ¼ ½a þ ðb  aÞL1 ðrÞ; d  ðd  cÞR1 ðrÞ Note that, members of an r-level set as an interval is included in the membership function with membership degree or uncertain measure, r.

2.3.7

Trapezoidal Fuzzy Number

The general level-wise or parametric forms are,

Fig. 2.12 Non-linear fuzzy number

2.3 Fuzzy Numbers and Their Properties

19

Ml ðrÞ ¼ b  ðb  aÞL1 ðrÞ; Mu ðrÞ ¼ c þ ðd  cÞR1 ðrÞ;

r 2 ½0; 1

where in linear case the left and right functions can be replaced by linear functions like RðrÞ ¼ LðrÞ ¼ r or others. Now suppose that b  a ¼ a; d  c ¼ b; b ¼ m1 ; c ¼ m2 then the membership function will be as, 8 m x L 1 ; > > >  a  < xm2 MðxÞ ¼ R b ; > > 1; > : 0;

m1  a  x  m1 m2  x  m2 þ b m1  x  m2 otherwise

where a; b are left and right spreads, and m1 ; m2 are cores of a trapezoidal fuzzy number. Ml ðrÞ ¼ m1  aL1 ðrÞ; Mu ðrÞ ¼ m2 þ bR1 ðrÞ;

r 2 ½0; 1

The formal formats to show this number can be written as, M ¼ ða; b; c; d Þ := M ¼ ðm1 ; m2 ; a; bÞ := M½r ¼ ½Ml ðrÞ; Mu ðrÞ := MðrÞ ¼ ðMl ðrÞ; Mu ðrÞÞ Please note that these formats are only for presenting the fuzzy number and each one has own properties and calculations. The calculations on them will be explained in the next sections. The r-level set of a trapezoidal fuzzy set M ¼ ða; b; c; d Þ is, M½r ¼ ½Ml ðrÞ; Mu ðrÞ ¼ ½b þ ðb  aÞðr  1Þ; c þ ðd  cÞð1  r Þ

2.3.8

Triangular Fuzzy Number

The only difference of triangular fuzzy number from trapezoidal is the number of the cores in membership functions. In the trapezoidal, if the cores are the same, i.e. m1 ¼ m2 ¼ m then it will be triangular fuzzy number. The general level-wise or parametric forms are, Ml ðrÞ ¼ b  ðb  aÞL1 ðrÞ; Mu ðrÞ ¼ b þ ðc  bÞR1 ðrÞ;

r 2 ½0; 1

20

2 Fuzzy Sets

where in linear case the left and right functions can be replaced by linear functions like RðrÞ ¼ LðrÞ ¼ r or others. Now suppose that b  a ¼ a; c  b ¼ b; b ¼ m then the membership function will be as, 8   xm þ a > < L a ; m  a  x  m MðxÞ ¼ R m þbbx ; m  x  m þ b > : 0; otherwise where a; b are left and right spreads, and m is the core of a triangular fuzzy number. Ml ðrÞ ¼ m  aL1 ðrÞ; Mu ðrÞ ¼ m þ bR1 ðrÞ;

r 2 ½0; 1

The formal formats to show this number can be written as, M ¼ ða; b; cÞ := M ¼ ðm; a; bÞ := M½r ¼ ½Ml ðrÞ; Mu ðrÞ := MðrÞ ¼ ðMl ðrÞ; Mu ðrÞÞ The r-level set of a triangular fuzzy set M ¼ ða; b; cÞ is, M½r ¼ ½Ml ðrÞ; Mu ðrÞ ¼ ½b þ ðb  aÞðr  1Þ; b þ ðc  bÞð1  r Þ For both format of the r-level sets of triangular and trapezoidal fuzzy numbers the lower and upper functions can be obtained practically as follow, Consider the following figure of fuzzy set M. First, the line segment between two points A and B is defined as, y  yA yB  yA 1 ¼ ) y ¼ ð x  m þ aÞ a x  xA xB  xA Then after finding the equation of line, we have this system, 1 y ¼ ð x  m þ aÞ a

& y¼r

So, 1 r ¼ ð x  m þ aÞ a

2.3 Fuzzy Numbers and Their Properties

21

The inverse is reflective function on x ¼ r and it is as, Ml ðrÞ ¼ x ¼ m þ aðr  1Þ The same procedure will be true for the points B and C. Then it will be obtained as, y  yB yC  yB 1 ¼ ) y ¼ ð m  xÞ þ 1 b x  xB xC  xB Again, we will have 1 r ¼ ð m  xÞ þ 1 b And the inverse function will be Mu ðrÞ (Fig. 2.13). Mu ðrÞ ¼ m þ bð1  r Þ

2.3.9

Operations on Level-Wise Form of Fuzzy Numbers

Here, the main calculations on fuzzy sets in parametric or level-wise form are going to be defined and discussed. Since the concept of the difference is different and needs more attention, so this operation will be discussed more. Because here the subtraction has the meaning of difference between two sets or two functions. Before the discussion on the operations, we need some explanations. Please note that the fuzziness will be growing under these operators. It means, for two arbitrary fuzzy numbers M and N, and a continuous measurable function like f 2 f ; ; øg, the fuzziness of M and fuzziness of N is less than or equal to the fuzziness of f ðM; N Þ. One of the concepts of the fuzziness is diameter of a fuzzy number. The diameter of an interval in level-wise form of a fuzzy number can be called as fuzziness of M in any level of r. FuzzðMr Þ ¼ diamð½Ml ðrÞ; Mu ðrÞÞ ¼ Mu ðrÞ  Ml ðrÞ;

0  r  1:

Now it is proven that     FuzzðMr Þ  Fuzz f ðM; N Þr & FuzzðNr Þ  Fuzz f ðM; N Þr also, K ¼ f ðM; N Þ ) K½r ¼ f ðM½r; N½rÞ

22

2 Fuzzy Sets

Fig. 2.13 Fuzzy number in triangular form

This means that if the function f is continuous and measurable then if M N ¼ K then M½r þ N½r ¼ K½r; if M N ¼ K then M½r N½r ¼ K½r; if MøN ¼ K then M½røN½r ¼ K½r: Now we can discuss on the operators separately. For any two arbitrary fuzzy numbers M and N and any arbitrary but fixed 0  r  1, if M N ¼ K; we have, K½r ¼ ½Kl ðrÞ; Ku ðrÞ ¼ M½r þ N½r ¼ ½Ml ðrÞ; Mu ðrÞ þ ½Nl ðrÞ; Nu ðrÞ Then Kl ðrÞ ¼ Ml ðrÞ þ Nl ðrÞ;

Ku ðrÞ ¼ Mu ðrÞ þ Nu ðrÞ

An example Suppose that M ¼ ½r  1; 1  r ;

 N ¼ r; 2  r 2 ;

To compute the summation M N ¼ K we need to, Ml ðrÞ ¼ r  1;

Mu ðrÞ ¼ 1  r;

Nl ðrÞ ¼ r;

Nu ðrÞ ¼ 2  r 2

2.3 Fuzzy Numbers and Their Properties

23

So, Kl ðrÞ ¼ 2r  1;

Ku ðrÞ ¼ 3  r  r 2 :

The following figure shows the summation of two fuzzy numbers in the example (Fig. 2.14). As it is seen we have Mu ðrÞ  Ml ðrÞ ¼ FuzzðMr Þ  FuzzðM N Þr ¼ Ku ðrÞ  Kl ðrÞ & Nu ðrÞ  Nl ðrÞ ¼ FuzzðNr Þ  FuzzðM N Þr ¼ Ku ðrÞ  Kl ðrÞ For all levels.

2.3.9.1

Multiplication

For any two arbitrary fuzzy numbers M and N and any arbitrary but fixed 0  r  1, if M N ¼ K; we have, K½r ¼ ½Kl ðrÞ; Ku ðrÞ ¼ M½r  N½r ¼ ½Ml ðrÞ; Mu ðrÞ  ½Nl ðrÞ; Nu ðrÞ Then Kl ðrÞ ¼ minfMl ðrÞ  Nl ðrÞ; Ml ðrÞ  Nu ðrÞ; Mu ðrÞ  Nl ðrÞ; Mu ðrÞ  Nu ðrÞg; Ku ðrÞ ¼ maxfMl ðrÞ  Nl ðrÞ; Ml ðrÞ  Nu ðrÞ; Mu ðrÞ  Nl ðrÞ; Mu ðrÞ  Nu ðrÞg In the previous example

    Kl ðrÞ ¼ min ðr  1Þ  r; ðr  1Þ  2  r 2 ; ð1  r Þ  r; ð1  r Þ  2  r 2

    Ku ðrÞ ¼ max ðr  1Þ  r; ðr  1Þ  2  r 2 ; ð1  r Þ  r; ð1  r Þ  2  r 2

Fig. 2.14 Summation

24

2 Fuzzy Sets

We will see that   Kl ðrÞ ¼ ðr  1Þ 2  r 2 ;

  Ku ðrÞ ¼ ð1  r Þ 2  r 2

The following figure shows the multiplication of two fuzzy numbers in the example (Fig. 2.15).

2.3.9.2

Scalar Multiplication

The difference of two fuzzy numbers is indeed the difference of two membership functions. Two different differences in the sense of standard and non-standard cases will be discussed in level-wise form. First of all, we should consider the multiplying a membership function by a scalar in level-wise form. Suppose that k 2 R is a scalar. Then In triple form of fuzzy number:  k M ¼ k  ða; b; cÞ ¼

ðk a; k b; k cÞ; k  0 ðk c; k b; k aÞ; k\0

In level-wise form of fuzzy number:  kM½r ¼

½kMl ðrÞ; kMu ðrÞ; ½kMu ðrÞ; kMl ðrÞ;

k0 k\0

For any 0  r  1: The concept of scalar multiplication is the same as the multiplication of the scalar to each member of the interval. It means, M½r ¼ ½Ml ðrÞ; Mu ðrÞ ¼ fzt jzt ¼ Ml ðrÞ þ tðMu ðrÞ  Ml ðrÞÞ; 0  t  1g For any 0  r  1: So, in r  level;

Fig. 2.15 Multiplication

2.3 Fuzzy Numbers and Their Properties

25

 kM½r ¼ fkzt j0  t  1g ¼

½kMl ðrÞ; kMu ðrÞ; k  0 ½kMu ðrÞ; kMl ðrÞ; k\0

For instance, if consider one of the previous fuzzy numbers like,  M½r ¼ ½Ml ðrÞ; Mu ðrÞ ¼ r; 2  r 2 And k ¼ 1, then  ð1ÞM½r ¼ ½Mu ðrÞ; Ml ðrÞ ¼ r 2  2; r For more illustration see the following figure. (Figure 2.16) For using this scalar multiplication in the definition of difference of two fuzzy numbers, let M½r ¼ ½Ml ðrÞ; Mu ðrÞ and B½r ¼ ½Nl ðrÞ; Nu ðrÞ are two fuzzy numbers in level-wise form. In this case the difference is defined as follow, M½r  N½r ¼ M½r þ ð1ÞN½r ¼ ½Ml ðrÞ; Mu ðrÞ þ ð1Þ½Nl ðrÞ; Nu ðrÞ ¼ ½Ml ðrÞ; Mu ðrÞ þ ½Nu ðrÞ; Nl ðrÞ ¼ ½Ml ðrÞ  Nu ðrÞ; Nu ðrÞ  Ml ðrÞ Or in the format of convex combination, for any arbitrary but fixed 0  r  1, M½r ¼ fz0t jz0t ¼ Ml ðrÞ þ tðMu ðrÞ  Ml ðrÞÞ; 0  t  1g ð1ÞN½r ¼ fz00t jz00t ¼ Nu ðrÞ þ tðNu ðrÞ  Nl ðrÞÞ; 0  t  1g M½r þ ð1ÞN½r ¼ fz0t  z00t j0  t  1g ¼ ½Ml ðrÞ  Nu ðrÞ; Nu ðrÞ  Ml ðrÞ In this definition, M ð1Þ M 6¼ 0

Fig. 2.16 Multiplication of ð1Þ

26

2 Fuzzy Sets

Because, based on the definition the result is a symmetric interval centred at zero and it is always non-zero interval. M½r  M½r ¼ ½Ml ðrÞ  Mu ðrÞ; Mu ðrÞ  Ml ðrÞ 6¼ 0 ¼ ½0; 0 Please note that the symmetric interval centered at zero is called as a zero interval. It came from the concept of equivalency class. If we consider an equivalency class of zero as a set of all symmetric intervals centered at zero, then all of the members of the class are called as a zero interval. Moreover,   FuzzðMr Þ  Fuzz ðk M Þr ;

2.3.9.3

0  r  1:

Hukuhara Difference

Suppose that M and N are two fuzzy numbers in level-wise form. The Hukuhara difference of M H N is defined as, 9K; M H N ¼ K , M ¼ N K It is clear, the existence of the difference is conditional and depends on the existence of fuzzy number K. Note. For the existence of H-difference, all M, N and K must be fuzzy numbers. It means that if the fuzzy set B can be transformed by C then it will be fallen into A. Now consider M ¼ N N, and level-wise form of both side of the equality, we have, M½r ¼ N½r þ K½r ½Ml ðrÞ; Mu ðrÞ ¼ ½Nl ðrÞ; Nu ðrÞ þ ½Kl ðrÞ; Ku ðrÞ ¼ ½Nl ðrÞ þ Kl ðrÞ; Nu ðrÞ þ Ku ðrÞ Ml ðrÞ ¼ Nl ðrÞ þ Kl ðrÞ; Mu ðrÞ ¼ Nu ðrÞ þ Ku ðrÞ Finally, Kl ðrÞ ¼ Ml ðrÞ  Nl ðrÞ; Ku ðrÞ ¼ Mu ðrÞ  Nu ðrÞ The level-wise form of the Hukuhara difference or H-difference is defined as subtractions of two endpoints of two intervals respectively. 

ðM H N Þl ðrÞ; ðM H N Þu ðrÞ ¼ ½Kl ðrÞ; Ku ðrÞ ¼ ½Ml ðrÞ  Nl ðrÞ; Mu ðrÞ  Nu ðrÞ

2.3 Fuzzy Numbers and Their Properties

27

Please note that the difference M H N 6¼ M ð1Þ N Because in level-wise form the differences between intervals in both sides are not the same. ðM H N Þ½r ¼ ½Ml ðrÞ  Nl ðrÞ; Mu ðrÞ  Nu ðrÞ 6¼ ½Ml ðrÞ  Nu ðrÞ; Nu ðrÞ  Ml ðrÞ ¼ ðM ð1Þ N Þ½r An example Consider the following two fuzzy numbers in parametric forms, M½r ¼ ½Ml ðrÞ; Mu ðrÞ ¼ ½2r; 4  2r ; N½r ¼ ½Nl ðrÞ; Nu ðrÞ ¼ ½r  1; 1  r ; Now to obtain M H N ¼ K, Kl ðrÞ ¼ r þ 1;

Ku ðrÞ ¼ 3  r

So, the Hukuhara difference in parametric form is, K½r ¼ ½r þ 1; 3  r  In the following figure the H-difference is shown (Fig. 2.17). As it is seen in the figure and based on the definition of H-difference, Mu ðrÞ  Ml ðrÞ ¼ FuzzðMr Þ  FuzzðM H N Þr ¼ Ku ðrÞ  Kl ðrÞ & Nu ðrÞ  Nl ðrÞ ¼ FuzzðNr Þ  Mu ðrÞ  Ml ðrÞ ¼ FuzzðMr Þ Because the difference K is a shift for extending of N to be into M. Now we are going to find some sufficient conditions for existence of H-difference. To this purpose let us consider,       Mr ¼ M1;r ; M2;r ; M3;r ; Nr ¼ N1;r ; N2;r ; N3;r ; Kr ¼ K1;r ; K2;r ; K3;r

Fig. 2.17 M H N

28

2 Fuzzy Sets

where M1;r  M2;r  M3;r ; N1;r  N2;r  N3;r ; K1;r  K2;r  K3;r are representation of fuzzy numbers in triple forms for each level r. It means in each level the intervals are satisfied the conditions of a real interval. Lemma The sufficient condition for the existence of the H-difference M H N ¼ K is

 FuzzðNr Þ ¼ N3;r  N1;r  min M2;r  M1;r ; M3;r  M2;r To show M ¼ N K with condition K1;r  K2;r  K3;r , our assertion for the existence is only proving K1;r  K2;r  K3;r or M1;r  N1;r  M2;r  N2;r  M3;r  N3;r . Because, M ¼ N K , M1;r ¼ N1;r þ K1;r ; M2;r ¼ N2;r þ K2;r ; M3;r ¼ N3;r þ K3;r To show M1;r  N1;r  M3;r  N3;r it is enough to show N3;r  N1;r  M3;r 

 M1;r . In the first case suppose that, min M2;r  M1;r ; M3;r  M2;r ¼ M2;r  M1;r [ 0. Now we have N3;r  N1;r  M2;r  M1;r  M3;r  M1;r and the proof is completed.

 In the second case suppose that, min M2;r  M1;r ; M3;r  M2;r ¼ M3;r  M2;r [ 0. Now we have N3;r  N1;r  M3;r  M2;r  M3;r  M1;r and the proof is also completed. To show M1;r  N1;r  M2;r  N2;r it is enough to show N2;r  N1;r  M2;r 

 M1;r . In the first case suppose that, min M2;r  M1;r ; M3;r  M2;r ¼ M2;r  M1;r [ 0. Now we have N2;r  N1;r  N3;r  N1;r  M2;r  M1;r and the

proof is completed. In the second case suppose that, min M2;r  M1;r ; M3;r  M2;r g ¼ M3;r  M2;r [ 0. Now we have N2;r  N1;r  N3;r  N1;r  M3;r  M2;r  M2;r  M1;r and the proof is also completed. To show M2;r  N2;r  M3;r  N3;r it is enough to show N3;r  N2;r  M3;r 

 M2;r . In the first case suppose that, min M2;r  M1;r ; M3;r  M2;r ¼ M2;r  M1;r [ 0. Now we have N3;r  N2;r  N3;r  N1;r  M2;r  M1;r  M3;r 

M2;r and the proof is completed. In the second case suppose that, min M2;r  M1;r ; M3;r  M2;r g ¼ M3;r  M2;r [ 0. Now we have N3;r  N2;r  N3;r  N1;r  M3;r  M2;r and the proof is also completed. An example As we mentioned the existence of the difference is conditional. Now in this example we will see that it does not always exist. Suppose that in M H N

 FuzzðNr Þ ¼ N3;r  N1;r [ min M2;r  M1;r ; M3;r  M2;r

2.3 Fuzzy Numbers and Their Properties

29

For instance, M½r ¼ ½1  r; r  1 and N½r ¼ ½2  r; r  2, now ðM H N Þ½r ¼ ½Ml ðrÞ  Nl ðrÞ; Mu ðrÞ  Nu ðrÞ ¼ ½1; 3 As you see it is not an interval and for any r. So, the difference does not exist.

2.3.9.4

Generalized Hukuhara Difference

In this case, we can define the difference in another way. Suppose that we want to try N H M ¼ ð1ÞK and may the difference K does exist. Considering the level-wise forms of two sides, ðN H M Þ½r ¼ ½Nl ðrÞ  Ml ðrÞ; Nu ðrÞ  Mu ðrÞ ¼ ðð1ÞK Þ½r ¼ ½Ku ðrÞ; Kl ðrÞ We have, Nl ðrÞ  Ml ðrÞ ¼ Ku ðrÞ;

Nu ðrÞ  Mu ðrÞ ¼ Kl ðrÞ

Or Ml ðrÞ  Nl ðrÞ ¼ Ku ðrÞ;

Mu ðrÞ  Nu ðrÞ ¼ Kl ðrÞ

In general,  ðN H M Þl ðrÞ; ðN H M Þu ðrÞ ¼ ½Kl ðrÞ; Ku ðrÞ ¼ ½Mu ðrÞ  Nu ðrÞ; Ml ðrÞ  Nl ðrÞ  ¼  ð1Þ ðM H N Þl ðrÞ; ðM H N Þu ðrÞ Now to define an almost right definition for the difference, we have two cases to consider. M gH N ¼ K ,

8 < ðiÞ

M ¼N K or : ðiiÞ N ¼ M ð1ÞK

The generalized Hukuhara difference is defined in two cases. If the case (i) exist so there is no need to consider the case (ii). Otherwise we will need the second case. The relation between two cases can be explained as follow,

30

2 Fuzzy Sets

• In case (i), M gH N ¼ K • In case (ii), N gH M ¼ ð1ÞK The relationship is, 

    M gH N i ½r :¼ 0 H ð1Þ M gH N ii ½r

In case both exist then, K ¼ ð1ÞK and it is concluded that both types of the difference are the same and equal. The level-wise form of generalized difference As we found the level-wise form in case (i) is as, h

i    M gH N l ðrÞ; M gH N u ðrÞ ¼ ½Kl ðrÞ; Ku ðrÞ ¼ ½Ml ðrÞ  Nl ðrÞ; Mu ðrÞ  Nu ðrÞ

And in case (ii) it is as follow, h

i    N gH M l ðrÞ; N gH M u ðrÞ ¼ ½Kl ðrÞ; Ku ðrÞ ¼ ½Mu ðrÞ  Nu ðrÞ; Ml ðrÞ  Nl ðrÞ

So to define the endpoints of the difference, Kl ðrÞ ¼ minfMl ðrÞ  Nl ðrÞ; Mu ðrÞ  Nu ðrÞg Ku ðrÞ ¼ maxfMl ðrÞ  Nl ðrÞ; Mu ðrÞ  Nu ðrÞg To show two cases at the same time, we use gH-difference notation and define it in the following form, 

 M gH N ½r ¼ ½minfMl ðrÞ  Nl ðrÞ; Mu ðrÞ  Nu ðrÞg; maxfMl ðrÞ  Nl ðrÞ; Mu ðrÞ  Nu ðrÞg

Some properties of gH-difference Please note that all of the following properties can be proved in level-wise form easily.

2.3 Fuzzy Numbers and Their Properties

1. 2. 3. 4. 5. 6.

31

If the gH-difference exist, it is unique. M gH M ¼ 0 If M gH N exists in case (i) then N gH M exists in case (ii) and vise-versa. In both cases ðM N Þ gH N ¼ M: (It is easy to show in level-wise form.)   If M gH N and N gH M exist, then 0gH M gH N ¼ N gH M If M gH N ¼ N gH M ¼ K if and only if K ¼ K and M ¼ N.

The difference even in the gH-difference case may not exist. It can be say that the gH-difference of two fuzzy numbers are not always a fuzzy number. An example This example shows that the generalized Hukuhara difference does not exist for each arbitrary level of difference. Suppose that one of the numbers is triangular and another one is in trapezoidal forms. M ¼ ð0; 2; 4Þ or in parametric form M½r ¼ ½2r; 4  2r  and N ¼ ð0; 1; 2; 3Þ or in parametric form N½r ¼ ½r; 3  r . In case (i), 

 M gH N ½r ¼ ½r; 1  r 

If r ¼ 1, the difference is as ½1; 0 that is not an interval. In case (ii), 

 M gH N ½r ¼ ½1  r; r 

If r ¼ 0, the difference is as ½1; 0 that is not an interval. So as we see the gH-difference does not exist for all r 2 ½0; 1. Note. In all methods of this book, we will suppose that the gH-difference always exists.

2.3.9.5

Partial Ordering

For two fuzzy numbers M; N 2 FR , we call 4 as a partial order notation and M 4 N if and only if Ml ðrÞ  Nl ðrÞ and Mu ðrÞ  Nu ðrÞ also we have the same definition for the strict inequality, M N if and only if Ml ðrÞ\Nl ðrÞ and Mu ðrÞ\Nu ðrÞ For any r 2 ½0; 1:

32

2 Fuzzy Sets

Some properties of partial ordering • If M 4 N then N 4  M • If M 4 N and N 4 M then M ¼ N. To prove the properties, we use the level-wise form and they are very clear. For instance, we prove the first property, M 4 N if and only if Ml ðrÞ  Nl ðrÞ and Mu ðrÞ  Nu ðrÞ N 4  M if and only if  Nl ðrÞ   Ml ðrÞ and  Nu ðrÞ   Mu ðrÞ So the proof is completed. The second on is obtained in similar way. Absolute value of a fuzzy number The absolute value of a fuzzy number M is defined as follow,  jM j ¼

M; M < 1 ðs  tÞma1 xl ðt; r Þdt; m  1\a\m CðmaÞ ds a DRL xl ðs; r Þ ¼ t0 > :  d m1 x ðs; r Þ; a¼m1 l ds

3.3 Fuzzy Riemann-Liouville Derivative—Fuzzy RL Derivative

95

And

DaRL xu ðs; r Þ

¼

8 >
: d m1 x ðs; r Þ; u ds

tÞma1 xu ðt; r Þdt; m  1\a\m a¼m1

Example Assume a fuzzy number a 2 FR , with the membership function, aðr Þ ¼ ðal ðr Þ; au ðr ÞÞ; 0  r  1 In this example, the RL derivative is always i  gH differentiable. Because the function ðs  tÞma1 0 and nothing threat the conditions of fuzzy numbers. DaRLigH xðsÞ ¼

8  d m Rs > 1 < Cðma ðs  tÞma1 adt; m  1\a\m Þ  ds t0

> :  d m1 a; 8 ds  d m h aðstÞma is 1 > > ; < CðmaÞ ds  ma a DaRLigH xðsÞ ¼  Rs > d m > adt ¼ 0; : ds

a¼m1 m  1\a\m a¼m1

a

  m 1 d a  ðs  aÞma  ma Cðm  aÞ ds   m  m1 a d ðs  aÞma a d ¼ ¼   ðs  aÞma1 ma Cðm  aÞ ds Cðm  aÞ ds a ¼  ðs  aÞa Cðm  aÞ

DaRLigH xðsÞ ¼

The RL gH-derivative of this fuzzy number in level-wise form is,
 DaRLigH xðs; r Þ ¼ DaRL xl ðs; r Þ; DaRL xu ðs; r Þ ; s 2 ½t0 ; T  Where DaRL xl ðs; r Þ ¼

al ð r Þ au ð r Þ ðs  aÞa ; DaRL xu ðs; r Þ ¼ ðs  aÞa C ð m  aÞ Cðm  aÞ

Thus, the RL gH-derivative of a fuzzy number is not a zero fuzzy number and it is indeed another fuzzy number. Since Cðm  aÞ [ 0; ðs  aÞa [ 0 then the derivative satisfies the conditions of a fuzzy number. Remark Please note that the RL gH-derivative of a fuzzy number valued function is always a fuzzy number valued function and then it exists.

96

3 Fuzzy Fractional Operators

Remark There is a close relationship between the RL derivative and GL derivative. Indeed, it is possible to see that whenever x 2 Cm ½t0 ; T , with m ¼ a, then DaRLgH xðtÞ ¼ DaGL xðtÞ; t 2 ðt0 ; T  Example—Fuzzy exponential function Consider the fuzzy function as xðtÞ ¼ ect ; t 2 ðt0 ; T ; c 2 FR , the RL gH-derivative is obtained as,

DaRLgH ecs

¼

8 >
:  d m1 ecs ;

ðs  tÞma1 ect dt; m  1\a\m

t0

a¼m1

ds

And cðr Þ ¼ ðcl ðr Þ; cu ðr ÞÞ; 0  r  1: Using the remark, whenever x 2 C m ½t0 ; T ; t0 0, with m ¼ a, then DaRLgH ecs ¼ DaGL ecs ¼ ca  ecs ; s 2 ðt0 ; T  Otherwise, it can be handled as the following when m  1\a\m, DaRLgH ecs

 m Z s 1 d  ¼ ðs  tÞma1 ect dt; s 2 ½t0 ; T  Cðm  aÞ ds t0

let v ¼ s  t, and then dt ¼ dv and ect ¼ ecðsvÞ ; s  v 0. DaRLgH ecs

 m Z0 1 d  ¼ vma1  ð1ÞecðsvÞ dv Cðm  aÞ ds st0

Since, Zst0

Z0 v

I:¼

ma1

 ð1Þe

st0

cðsvÞ

dv ¼

vma1  ecðsvÞ dv 0

Because in level-wise form, I ½r  ¼ ½Il ðr Þ; Iu ðr Þ

3.3 Fuzzy Riemann-Liouville Derivative—Fuzzy RL Derivative

97

where Z0 Il ðr Þ ¼

v

ma1

ð1Þe

cl ðr ÞðsvÞ

Zst0 dv ¼

st0

vma1 ecl ðrÞðsvÞ dv

0

Z0 Iu ðr Þ ¼

v

ma1

ð1Þe

cu ðrÞðsvÞ

Zst0 dv ¼

st0

vma1 ecu ðrÞðsvÞ dv

0

For any r 2 ½0; 1: Finally, we have, DaRLgH ecs

 m Zst0 1 d  ¼ vma1  ecðsvÞ dv Cðm  aÞ ds 0

The level-wise form of the derivative is, DaRLgH ecl ðrÞs

 m Zst0 1 d ¼ vma1 ecl ðrÞðsvÞ dv Cðm  aÞ ds 0

 m Zst0 1 d cl ðrÞs e vma1 ecl ðrÞv dv ¼ Cðm  aÞ ds 0

 m Zst0 d 1 cl ðrÞs e vma1 ecl ðrÞv dv ¼ ds Cðm  aÞ 0

The integral Then

R st0 0

vma1 ecl ðrÞv dv is an incomplete Gamma density function.

DaRLgH ecl ðrÞs ¼

 m d 1 cðv; tÞ ecl ðrÞs ds Cðm  aÞ

The same process for the upper function, DaRLgH ecu ðrÞs ¼

 m d 1 cðv; tÞ ecu ðrÞs ds C ð m  aÞ

It is not clear that, DaRLgH ecl ðrÞs  DaRLgH ecu ðrÞs

98

3 Fuzzy Fractional Operators

3.3.3

RL—Fractional Derivative for m ¼ 1 1 d  ¼ Cð1  aÞ ds

DaRLgH xðtÞ or

0 DaRLgH xðtÞ ¼

1 @ Cð1  aÞ

Zs

Zs t0

t0

xð t Þ dt; 0\a\1 ðs  tÞa

10 xð t Þ dtA ; 0\a\1 ðs  tÞa gH

Assume, Zs t0

So, d ds

xðtÞ dt ¼ f ðsÞ 2 FR ; xðtÞ 2 FR ðs  t Þa

Zs t0

xð t Þ 0 dt ¼ fgH ðsÞ; s 2 ½t0 ; T  ðs  t Þa 0

0

Remark The function f ðtÞ is gH-differentiable if and only if fl ðt; r Þ and fu ðt; r Þ are differentiable with respect to t for all 0  r  1 and h n 0 o n 0 oi 0 0 0 fgH ðtÞ ¼ min fl ðt; r Þ; fu ðt; r Þ ; max fl ðt; r Þ; fu ðt; r Þ Theorem The necessary and sufficient condition for RL gH differentiability of xðtÞ is gH differentiability of f ðsÞ: It is,
    DaRLgH xðt; r Þ ¼ min DaRL xl ðt; r Þ; DaRL xu ðt; r Þ ; max DaRL xl ðt; r Þ; DaRL xu ðt; r Þ The proof is very easy, since s  t [ 0; Zs fl ðs; r Þ ¼ t0

n min

xl ðt; r Þ dt; fu ðs; r Þ ¼ ðs  t Þa

Zs t0

1 0 1 0 Cð1aÞ fl ðt; r Þ; Cð1aÞ fu ðt; r Þ

 max

n

xu ðt; r Þ dt ðs  t Þa o o

1 0 1 0 Cð1aÞ fl ðt; r Þ; Cð1aÞ fu ðt; r Þ

3.3 Fuzzy Riemann-Liouville Derivative—Fuzzy RL Derivative

99

Then     min DaRL xl ðt; r Þ; DaRL xu ðt; r Þ  max DaRL xl ðt; r Þ; DaRL xu ðt; r Þ It means that the following interval defines an interval for all r 2 ½0; 1,
 DaRLgH xðt; r Þ ¼ DaRL xl ðt; r Þ; DaRL xu ðt; r Þ

3.4

Fuzzy Caputo Fractional Derivative

In this section another form of the fractional derivative is expressed. First the order m  1\a\m and then 0\a\1 are displayed. Definition—Caputo fractional derivative of order a In the definition of RL fractional derivative, suppose the integer order of the derivative is an operator inside of the integral and operating on operand function xðtÞ 2 FR ; t 2 ½t0 ; T : DaCgH xðsÞ

¼

8 >
:  d m1 xðsÞ;

ds

a¼m1

Note The Caputo derivative of a fuzzy number is zero. But in case RL derivative it wasn’t. In the Caputo differential operator, there is a derivative of order m on fuzzy number valued function, xðmÞ ðtÞ. To existence of this derivative for any m, it should be a fuzzy number at any point t. To this end, the gH-difference in the definition of the derivative should be well defined. xðm1Þ ðt þ hÞgH xðm1Þ ðtÞ h!0 h

xðmÞ ðtÞ ¼ lim

The gH-difference is defined in two cases and • i  gH differentiable ðmÞ

xðm1Þ ðt þ hÞH xðm1Þ ðtÞ h!0 h

xigH ðtÞ ¼ lim Its level-wise form,

100

3 Fuzzy Fractional Operators


 DaCigH xðs; r Þ ¼ DaC xl ðs; r Þ; DaC xu ðs; r Þ

• ii  gH differentiable xðm1Þ ðtÞH xðm1Þ ðt þ hÞ h!0 h

ðmÞ

xiigH ðtÞ ¼ lim Its level-wise form,


 DaCiigH xðs; r Þ ¼ DaC xu ðs; r Þ; DaC xl ðs; r Þ Which are defined as the following form for m  1\a\m, Rs ma1 ðmÞ 1 DaC xl ðs; r Þ ¼ Cðma xl ðt; r Þdt; Þ ðs  t Þ DaC xu ðs; r Þ

¼

1 CðmaÞ

t0 Rs

ðs  tÞma1 xðumÞ ðt; r Þdt;

t0

For s 2 ½t0 ; T :

3.4.1

Caputo—Fuzzy Fractional Derivative for m ¼ 1

Suppose the order of fractional derivative is in 0\a\1; m ¼ 1. Now we are going to cover some properties of Caputo derivative with this fraction order. The definition is as, Zs 1 a  ðs  tÞa x0gH ðtÞdt DCgH xðsÞ ¼ Cð1  aÞ t0

1  :¼ Cð1  aÞ

Zs t0

x0gH ðtÞ dt ðs  tÞa

Remark The function xðtÞ is gH-differentiable if and only if x0l ðt; r Þ and x0y ðt; r Þ are differentiable with respect to t for all 0  r  1 and
    x0gH ðt; r Þ ¼ min x0l ðt; r Þ; x0u ðt; r Þ ; max x0l ðt; r Þ; x0u ðt; r Þ

3.4 Fuzzy Caputo Fractional Derivative

3.4.2

101

Caputo gH Differentiability

The fuzzy number valued function xðtÞ is Caputo gH differentiable if and only if x0l ðt; r Þ and x0u ðt; r Þ are differentiable with respect to t for all 0  r  1 and
    DaCgH xðt; r Þ ¼ min DaC xl ðt; r Þ; DaC xu ðt; r Þ ; max DaC xl ðt; r Þ; DaC xu ðt; r Þ where 1 ¼ Cð1  aÞ

DaC xl ðs; r Þ DaC xu ðt; r Þ

1 ¼ Cð1  aÞ

Zs t0

x0l ðt; r Þ dt; ðs  tÞa

Zs t0

x0u ðt; r Þ dt ðs  t Þa

The proof is straight, DaCgH xðs; r Þ

1  ¼ C ð 1  aÞ

Zs t0

x0gH ðt; r Þ dt ðs  t Þa

We have
    x0gH ðs; r Þ ¼ min x0l ðs; r Þ; x0u ðs; r Þ ; max x0l ðs; r Þ; x0u ðs; r Þ Then

" 0

xgH ðs; r Þ ¼

( min (

max

1 Cð1aÞ 1 Cð1aÞ

Rs t0

Rs t0

x0l ðt;rÞ ðstÞa

x0l ðt;r Þ ðstÞa

1 dt; Cð1a Þ

1 dt; Cð1a Þ

Rs x0u ðt;rÞ t0

ðstÞa

Rs x0u ðt;rÞ t0

ðstÞa

) dt ; )#

dt

This is exactly,
    DaCgH xðt; r Þ ¼ min DaC xl ðt; r Þ; DaC xu ðt; r Þ ; max DaC xl ðt; r Þ; DaC xu ðt; r Þ Also, in case 0\a\1, we have two cases of differentiability, • i  gH differentiable x0igH ðtÞ ¼ lim

h!0

xðt þ hÞH xðtÞ h

102

3 Fuzzy Fractional Operators

Its level-wise form,
 DaCigH xðs; r Þ ¼ DaC xl ðs; r Þ; DaC xu ðs; r Þ

• ii  gH differentiable x0iigH ðtÞ ¼ lim

h!0

xðtÞH xðt þ hÞ h

Its level-wise form,
 DaCiigH xðs; r Þ ¼ DaC xu ðs; r Þ; DaC xl ðs; r Þ Example Consider the same fuzzy xðtÞ ¼ ect ; t 2 ðt0 ; T ; c 2 FR ; cðr Þ ¼ ðcl ðr Þ; cu ðr ÞÞ: DaCgH ecs

1  ¼ Cð1  aÞ

Zs t0

exponential

function

c  ect dt ðs  t Þa

We know that xl ðt; r Þ ¼ ecl ðrÞt ; xu ðt; r Þ ¼ ecu ðrÞt x0l ðt; r Þ ¼ cl ðr Þecl ðrÞt ; x0u ðt; r Þ ¼ cu ðr Þecu ðrÞt Because t [ 0 and the exponential function is increasing. Thus, h i x0gH ðt; r Þ ¼ cl ðr Þecl ðrÞt ; cu ðr Þecu ðrÞt Finally, the i—differential is found as,
 DaCigH xðs; r Þ ¼ DaC xl ðs; r Þ; DaC xu ðs; r Þ Where, DaC xl ðs; r Þ

1 ¼ C ð 1  aÞ

Zs t0

cl ðr Þecl ðrÞt cl ð r Þ a dt ¼ Cð1  aÞ ðs  t Þ

Zs t0

ecl ðrÞt dt ðs  t Þa

let v ¼ s  t, and then dt ¼R dv and ect ¼ ecðsvÞ ; s  v 0. By these condis tions, the integral equation, t0 ðs  tÞa ect dt is always i  gH differentiable, and

3.4 Fuzzy Caputo Fractional Derivative

DaC xl ðs; r Þ

cl ð r Þ ¼ C ð 1  aÞ

Zst0 0

103

ecl ðrÞðsvÞ cl ðr Þecl ðrÞs dv ¼ a v Cð1  aÞ

Zst0

ecl ðrÞv dv va

0

The integral equation, 1 Cð1  aÞ where

R st0 0

Zst0 0

ecl ðrÞv 1 dv ¼ a Cð1  aÞ v

Zst0

va ecl ðrÞv dv

0

va ecl ðrÞv dv is an incomplete cðv; tÞ function thus, DaC xl ðs; r Þ ¼

cl ðr Þecl ðrÞs cðv; tÞ Cð1  aÞ

Using the same process, DaC xu ðs; r Þ ¼

cl ðr Þecl ðrÞs cðv; tÞ; 0\a\1 Cð1  aÞ

Now if for any r 2 ½0; 1, the conditions of the interval are satisfied then the Caputo derivative for this exponential function exists. This is exactly depending on the sign of cðv; tÞ and it can be variational. Then it is not easy to claim that, DaC xl ðs; r Þ  DaC xu ðs; r Þ; 0  r  1

3.5

Fuzzy Riemann-Liouville Generalized Fractional Derivative

In this section, the concept is going to be covered for order of 0\a\1 on absolutely continuous fuzzy number valued functions. First, the fuzzy RiemannLiouville generalized fractional integral is defined.

3.5.1

Fuzzy Riemann-Liouville Generalized Fractional Integral

To define, assume that x 2 FR and integrable on ½t0 ; T ; p [ 0.

104

3 Fuzzy Fractional Operators

I a;p RL xðtÞ

p1a  ¼ C ð aÞ

Zt

sp1 ðtp  sp Þa1 xðsÞds

t0

In level-wise form,
 a;p   a;p  a;p a;p I a;p RL xðt; r Þ ¼ min I RL xl ðt; r Þ; I RL xu ðt; r Þ ; max I RL xl ðt; r Þ; I RL xu ðt; r Þ where

min



 a;p I a;p RL xl ðt; r Þ; I RL xu ðt; r Þ

¼

I a;p RL xl ðt; r Þ

p1a ¼ C ð aÞ t 1a Z

  a;p p a;p max I a;p RL xl ðt; r Þ; I RL xu ðt; r Þ I RL xu ðt; r Þ ¼ CðaÞ

Zt

sp1 ðtp  sp Þa1 xl ðs; r Þds

t0

sp1 ðtp  sp Þa1 xu ðs; r Þds

t0

Properties For two fuzzy number valued functions x; y and two fractional orders a; b 2 ð0; 1Þ; 

a;p a;p I a;p RL ðx  yÞðt Þ ¼ I RL xðt Þ  I RL yðtÞ



 a;p length I a;p RL xðt; r Þ ¼ I RL lengthðxðt; r ÞÞ

ða þ bÞ;p

b;p I a;p RL I RL xðt Þ ¼ I RL



The proofs of two first items are clear and the third can be proved as, b;p I a;p RL I RL xðtÞ

p  ð a þ bÞ þ 2  ¼ CðaÞCðbÞ ¼

p  ð a þ bÞ þ 2  CðaÞCðbÞ

Zt p1

s

p a1

ðt  s Þ p

Zs 

t0

u

Zt

Zs up1  xðuÞ 

t0

up1 ðtp  up Þb1 xðuÞduds

u

sp1 ðup  tp Þb1 ðsp  tp Þa1 dsdu

3.5 Fuzzy Riemann–Liouville Generalized Fractional Derivative

By changing some variables like, suppose, Zs s

p1

p b1

ðu  t Þ p

p a1

ðs  t Þ p

up tp sp tp

105

¼ v; where s; u are fixed. Then

ðsp  up Þa þ b1 ds ¼ p

u

Z1

ð1  vÞa1 vb1 dv

0

¼

p a þ b1

ðs  u Þ p p

CðaÞCðbÞ Cða þ bÞ

By substituting, b;p I a;p RL I RL xðt Þ

pða þ bÞ þ 1  ¼ Cða þ bÞ ¼

Zt

up1 ðsp  up Þa þ b1 xðuÞdu

t0

ða þ bÞ;p I RL xð t Þ

Now using the Fuzzy Riemann-Liouville generalized fractional integral I a;p RL we are going to define its fractional derivative based on gH differentiability.

3.5.2

Riemann Liouville–Katugampola gH—Fractional Derivative

With the same assumptions in the integral operator, the Da;p RL is defined as the following form,  0 ð1aÞ;p 1p I RL x ðtÞ; t 2 ½t0 ; T ; 0\a\1 Da;p RLgH xðtÞ ¼ t gH

where ð1aÞ;p I RL xðtÞ



ð1aÞ;p

I RL

x

pa  ¼ Cð1  aÞ

0 gH

ðtÞ ¼ lim

h!0

Zt

sp1 ðtp  sp Þa xðsÞds

t0

ð1aÞ;p ð1aÞ;p I RL xðt þ hÞgH I RL xðtÞ

h

It is supposed that the gH difference exists. Moreover, based on the nature of gH difference we have two cases,

106

3 Fuzzy Fractional Operators

• i  gH differentiability 

ð1aÞ;p

I RL

0

I

ð1aÞ;p

ðtÞ ¼ lim RL h!0 gH  0 ð1aÞ;p a;p 1p DRLigH xðtÞ ¼ t I RL x ðtÞ; x

ð1aÞ;p

xðt þ hÞH I RL h

gH

xðtÞ

t 2 ½t0 ; T ; 0\a\1

Where Zt pa  sp1 ðtp  sp Þa xðsÞds ¼ Cð1  aÞ t0 h i a;p a;p x ð t; r Þ DRLigH xðt; r Þ ¼ DRLgH xl ðt; r Þ; Da;p u RLgH

ð1aÞ;p I RL xðtÞ

• ii  gH differentiability 

ð1aÞ;p

I RL



0

I

ð1aÞ;p

xðtÞ I

ð1aÞ;p

xðt þ hÞ

H RL ðtÞ ¼ lim RL h h!0   ð 1a Þ;p a;p DRLiigH xðtÞ ¼ t1p I RL x iigH 0 ðtÞ; t 2 ½t0 ; T ; 0\a\1

x

iigH

where Rt p1 p pa xðtÞ ¼ Cð1a ðt  sp Þa xðsÞds Þ s h t0 i a;p Da;p x ð t; r Þ ¼ Da;p RLiigH RLgH xu ðt; r Þ; DRLgH xl ðt; r Þ

ð1aÞ;p

I RL

Note If the length function of xðt; r Þ is increasing (decreasing) for all r, then we have i  gH differentiability (ii  gH differentiability). Example Let consider p ¼ 1; a ¼ 12, and x : ð0; 1 ! FR and is given by xð t Þ ¼

pffi pffi t  1; 0; 1  t ; t 2 ð0; 1

The derivative of length of xðtÞ is, d r1 ðlengthðxðt; r ÞÞÞ ¼ pffi  0; r 2 ½0; 1 dt t

3.5 Fuzzy Riemann–Liouville Generalized Fractional Derivative

107

There for the length of xðtÞ is decreasing. See the following Fig. 3.2. And for t 2 ð0; 1, I aRL xðtÞ

1 ¼ 1  C 2

Zt 0

p pffi pffi p  1 1 ðt  sÞ2 xðsÞds ¼ pffiffiffi  t  2 t; 0; 2 t  t ; 2 2 p

And     d a 2ð r  1Þ p 1 0; t 2
0; p42 ; increasing length I RL xðt; r Þ ¼ pffiffiffi  pffi :¼ \0; t 2 p42 ; 1 ; decreasing length dt 2 p t It is shown in the following Fig. 3.3. Remark For xðtÞ 2 FR and 0\a\bh1; pi0 we have,   a;p Da;p RLgH I RL x ðtÞ ¼ xðtÞ   ðbaÞ;p b;p xð t Þ Da;p RLgH I RL x ðtÞ ¼ I RL To show the first item,  0 ð1aÞ;p a;p 1p Da;p I RL I a;p ðt Þ RLgH I RL xðt Þ ¼ t RL x gH

¼

pa t1p Cð1aÞ

Fig. 3.2 Decreasing length of xðtÞ



Rt t0

s

p1

p a

ðt  s Þ p

!0 I a;p RL xðsÞds gH

108

Fig. 3.3

3 Fuzzy Fractional Operators

d a dt I RL xðt; r Þ

length

where I a;p RL xðsÞ

p1a  ¼ C ð aÞ

Zs

up1 ðtp  up Þa1 xðuÞdu

t0

By substituting and by Dirichlet formula, the known formula for the Beta p sp function, and setting, v ¼ utp s p, a;p Da;p RLgH I RL xðt Þ

0 t 0 1 10 s Z p1 1a Z p1 pa s p u 1p t ¼ @ @ xðuÞduAdsA Cð1  aÞ ðt p  sp Þa CðaÞ ðSp  up Þ1a t0

t0

0 t 10 Z Zt p p a1 p ðu  s Þ 1p p1 p1 t ¼  @ xðsÞs  u dudsA CðaÞCð1  aÞ ðtp  up Þa 0 ¼

t0

s

10

C BZ t Z1 B p ð1  vÞa va1 C C B t1p  B xðsÞsp1 ds  du C C B CðaÞCð1  aÞ p A @ t0 0 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

0 t 10 Z Bð1  a; aÞ 1p @ p1 A t ¼  xðsÞs ds ¼ x ðt Þ CðaÞCð1  aÞ t0

The second property,

gH

1

gH

gH

gH

3.5 Fuzzy Riemann–Liouville Generalized Fractional Derivative

b;p Da;p RLgH I RL xðtÞ

109

0 t 10 Z Z1 a a1 p1 pab s ð 1  v Þ v 1p t duA ¼ @  xðsÞds  CðbÞCð1  aÞ p ðtp  sp Þab t0

¼

ab

p  Cðb  a þ 1Þ

Zt t0

s ðt p

0

p1

 sp Þab

ðbaÞ;p

 xðsÞds ¼ I RL

gH

xðtÞ

Remark If in the interval ðt0 ; T  the type of differentiability does not change, it means, it is either i  gH differentiable or ii  gH differentiable. Then the following relation as a relation of fractional integral and derivative operators is stablished. a;p I a;p RL DRLgH xðt Þ ¼ xðtÞgH

ð1aÞ;p

Where I RL

p a1 1a t  t0p p ð1aÞ;p  I RL xðt0 Þ; t 2 ðt0 ; T  CðaÞ

xðt0 Þ exists and ð1aÞ;p

lim I RL

t!t0 þ

ð1aÞ;p

xðtÞ ¼ I RL

xð t 0 Þ

To show the assertion, we consider two cases, • i  gH differentiability h i a;p a;p a;p a;p a;p I a;p RL DRLigH xðt; r Þ ¼ I RL DRLgH xl ðt; r Þ; I RL DRLgH xu ðt; r Þ where a;p I a;p RL DRLgH xl ðt; r Þ ¼ xl ðt; r Þ 

ðtp t0p Þ

a;p I a;p RL DRLgH xu ðt; r Þ ¼ xu ðt; r Þ 

ðt

a1 1a

p

CðaÞ

p

a1 1a t0p p

Þ

CðaÞ

ð1aÞ;p

I RL

xl ðt0 ; r Þ

ð1aÞ;p

I RL

xu ð t 0 ; r Þ

a;p Now, since xl ðt; r Þ is increasing then I a;p RL DRLgH xl ðt; r Þ is also increasing and is decreasing because xu ðt; r Þ is decreasing.

a;p I a;p RL DRLgH xu ðt; r Þ

3.6

Caputo–Katugampola gH–Fractional Derivative

If the RL derivative Da;p RLgH xðtÞ exists in ½to ; T  and 0\a\1, a;p  Da;p CK gH xðt Þ ¼ DRLgH xðtÞgH xðt0 Þ

110

3 Fuzzy Fractional Operators

In the sequel, some relations between fuzzy type Riemann–Liouville– Katugampola generalized fractional derivative and fuzzy type Caputo-Katugampola fractional derivative are shown. Remark Assume xðtÞ is absolutely continuous fuzzy number valued function which does have increasing or decreasing length i.e. it is i  gH differentiable or ii  gH differentiable, then Da;p CK gH xðt Þ pa ¼ Cð1  aÞ

Zt

ðtp  sp Þa x0gH ðsÞds; t 2 ½t0 ; T ; 0\a\1

t0 ð1aÞ;p

To show the relation, we know the fuzzy function I RL continuous because in its relation, ð1aÞ;p I RL xðtÞ

pa  ¼ Cð1  aÞ

Zt

xðtÞ is absolutely

sp1 ðtp  sp Þa xðsÞds

t0 a

p The coefficients sp1 ðtp  sp Þa [ 0; Cð1a Þ [ 0 and the fuzzy function xðsÞ is  0 ð1aÞ;p ðtÞ exists and finally, Da;p absolutely continuous thus I RL x RLgH xðtÞ exists for gH

t 2 ðt0 ; T ,

 0 ð1aÞ;p 1p Da;p x ð t Þ ¼ t I x ðtÞ: RLgH RL gH

Now, let us consider a constant fuzzy function like y 2 FR which is yðtÞ:¼xðt0 Þ. Then ð1aÞ;p

I RL

ð1aÞ;p

yðtÞ ¼ I RL

xð t 0 Þ ¼

p 1a a1 t  t0p p  xðt 0 Þ Cð2  aÞ

And if a :! 1 þ a; Da;p RLgH yðtÞ

¼

I a;p RL yðtÞ

p a t  t0p pa  xð t 0 Þ ¼ C ð 1  aÞ ð1aÞ;p

This is the reason for gH differentiability of I RL it follows,

x (in two cases) on ðt0 ; T  and

3.6 Caputo–Katugampola gH—Fractional Derivative

111

a;p  a;p  Da;p CK gH xðtÞ ¼ DRLgH xðt ÞgH xðt0 Þ ¼ DRLgH xðt ÞgH yðt Þ p a t  t0p pa a;p a;p a;p  xð t 0 Þ ¼ DRLgH xðtÞgH DRLgH yðtÞ ¼ DRLgH xðtÞgH C ð 1  aÞ So, Da;p CK gH xðt Þ

¼

Da;p RLgH xðtÞgH

p a t  t0p pa  xðt0 Þ Cð1  aÞ

Two sides in the level-form and based on the first case of gH difference, we have, Case i  gH difference, Da;p RLgH xðt; r Þ

¼

Da;p CK gH xðt; r Þ

p a t  t0p pa xð t 0 ; r Þ  Cð1  aÞ

In the level-wise form,



a;p  a;p a;p Da;p RL xl ðt; r Þ; DRL xu ðt; r Þ ¼ DCK xl ðt; r Þ; DCK xu ðt; r Þ p a t  t0p pa ½xl ðt0 ; r Þ; xu ðt0 ; r Þ þ C ð 1  aÞ

There for any r 2 ½0; 1 and t 2 ðt0 ; T , a

ðtp t0p Þ pa a;p Da;p RLgH xðtÞ ¼ DCK gH xðtÞ  Cð1aÞ  xðt0 Þ a   ðtp t0p Þ pa ð1aÞ;p d ¼ I RL s1p ds x ðtÞ  Cð1a Þ  xðt0 Þ Substituting in the following, Da;p CK gH xðt Þ

¼

Da;p RLgH xðtÞgH

We get, Da;p CK gH xðtÞ ¼

p a t  t0p pa  xðt0 Þ Cð1  aÞ

 d ð1aÞ;p I RL s1p x ðtÞ ds

Based on the definition of ð1aÞ;p

I RL

xð t Þ ¼

pa  Cð1  aÞ

Zt t0

sp1 ðtp  sp Þa xðsÞds

112

3 Fuzzy Fractional Operators

It is concluded,  Zt pa ð1aÞ;p 1p d x ðt Þ ¼ I RL s  ðtp  sp Þa x0 ðsÞds ds Cð1  aÞ t0

Thus, the proof is completed in case i  gH difference. Case ii  gH difference, Da;p CK gH xðt; r Þ

¼

Da;p RLgH xðt; r Þ

p a t  t0p pa  xð t 0 ; r Þ  ð1Þ C ð 1  aÞ

Because in level-wise form, since the function xðtÞ is ii  gH differentiable, so
a;p  a;p Da;p RLgH xðt; r Þ ¼
DRL xu ðt; r Þ; DRL xl ðt; r Þ ; a;p a;p Da;p CK gH xðt; r Þ ¼ DCK xu ðt; r Þ; DCK xl ðt; r Þ h i h i a;p a;p a;p Da;p x ð t; r Þ; D x ð t; r Þ ¼ D x ð t; r Þ; D x ð t; r Þ u l u l RL RL CK gH CK gH gH gH p p a a t  t0 p þ ð1Þ ½xl ðt0 ; r Þ; xu ðt0 ; r Þ C ð 1  aÞ There for for any r 2 ½0; 1 and t 2 ðt0 ; T ,



ð1aÞ;p 1p

I RL

s

 
 d d ð1aÞ;p a;p xl ðt; r Þ; I RL s1p xu ðt; r Þ ¼ Da;p RL xu ðt; r Þ; DRL xl ðt; r Þ ds ds p a t  t0p pa ½xl ðt0 ; r Þ; xu ðt0 ; r Þ þ ð1Þ Cð1  aÞ

So, we have 

ð1aÞ;p I RL s1p

p a t  t0p pa d a;p x ðtÞ ¼ DRLgH xðtÞ  ð1Þ  xðt0 Þ ds Cð1  aÞ

Substituting in the following, Da;p CK gH xðt Þ

¼

Da;p RLgH xðtÞgH

We get, Da;p CK gH xðtÞ

p a t  t0p pa  xðt0 Þ Cð1  aÞ

 ð1aÞ;p 1p d x ðt Þ ¼ I RL s ds

3.6 Caputo–Katugampola gH—Fractional Derivative

113

Based on the definition of ð1aÞ;p I RL xðtÞ

pa  ¼ Cð1  aÞ

Zt

sp1 ðtp  sp Þa xðsÞds

t0

It is concluded,  Zt pa ð1aÞ;p 1p d x ðt Þ ¼  ðtp  sp Þa x0 ðsÞds I RL s ds Cð1  aÞ t0

Thus, the proof is also completed in case ii  gH difference. Remark The RL integral operator is bounded. I a;p RL xðtÞ ¼

p1a  C ð aÞ

Zt

sp1 ðtp  sp Þa1 xðsÞds

t0

It can be shown by Hausdorff distance.

sup t2½t0 ;T 



DH I a;p RL xðtÞ; 0  DH ðxðsÞ; 0Þ

 a pa DH ðxðsÞ; 0Þ T p  t0p ¼ Cða þ 1Þ

p1a CðaÞ

Zt

sp1 ðtp  sp Þa1 ds

t0

As it is mentioned before, the existence of the Caputo- Katugampola depends on the continuity of xðtÞ and it has been supposed that is absolutely continuous. Then we claim the following remark. Remark Da;p CK gH xðtÞ ¼ 0 at t ¼ 0: ð1aÞ;p

Since I RL

is bounded then the Caputo-Katugampola derivative is continuous. Da;p CK gH xðtÞ ¼

 d ð1aÞ;p I RL s1p x ðtÞ ds

To show the assertion, it is enough to show that the upper bound of the derivative goes to zero at the point t ¼ 0 Then

114

3 Fuzzy Fractional Operators

     ð1aÞ;p 1p 0 ð t Þ; 0 DH Da;p x ð t Þ; 0 ¼ D I s x H RL CK gH 1a   pa1 DH s1p x0 ; 0 tp  t0p Cð2  aÞ   1a pa1 sup DH s1p x0 ; 0 tp  t0p  Cð2  aÞ s2½t0 ;T 



The supremum sups2½t0 ;T  DH ðs1p x0 ; 0Þ is a real number like k then   DH Da;p CK gH xðt Þ; 0 

 1a pa1 k tp  t0p Cð2  aÞ

It is clear that, at the point t ¼ 0, the distance goes zero and it completes the proof. Remark If the function xðtÞ is i  gH or ii  gH differentiable on ðt0 ; T ; for 0\a\1 we have, 

a;p I a;p RL DCK gH xðtÞ ¼ xðtÞgH xðt0 Þ



a;p Da;p CK gH I RL xðtÞ ¼ xðtÞ

The first item,   ð1aÞ;p 1p d a;p a;p I a;p D x ð t Þ ¼ I I s x ðt Þ RL RL CK gH RL ds   Rt 0 1p d ¼ I 1;p x ðsÞds ¼ xðtÞgH xðt0 Þ RL s ds x ðt Þ ¼ t0

Because in the definition, I a;p RL xðtÞ

p1a  ¼ C ð aÞ

Zt

sp1 ðtp  sp Þa1 xðsÞds

t0

If a ¼ 1; p ¼ 1 then, Zt I 1;1 RL xðtÞ

¼

xðsÞds t0

3.6 Caputo–Katugampola gH—Fractional Derivative

115

To prove the second item, we have the following relations, For xðt0 Þ as a constant fuzzy, we have ð1aÞ;p I RL xðt0 Þ

pa  ¼ Cð1  aÞ

Zt s t0

p1

p 1a a1 t  t0p p ðt  s Þ xðt0 Þds ¼  xðt0 Þ Cð2  aÞ p a

p

And if a :! 1 þ a; a

ðtp t0p Þ pa a;p Da;p RLgH xðt0 Þ ¼ I RL yðtÞ ¼ Cð1aÞ  xðt0 Þ a ðtp t0p Þ pa a;p  a;p Da;p CK gH xðtÞ ¼ DRLgH xðtÞgH xðt0 Þ ¼ DRLgH xðtÞgH Cð1aÞ  xðt0 Þ Now by substituting I a;p RL xðtÞ ! xðt Þ we get, a;p Da;p CK gH I RL xðtÞ

¼

a;p Da;p RLgH I RL xðt ÞgH

p a t  t0p pa  a;p  I RL x ðt0 Þ C ð 1  aÞ

 If we show I a;p RL x ðt0 Þ ¼ 0 the proof is completed. To do this, we use the distance.  p1a DH I a;p RL x ðt0 Þ; 0  k CðaÞ

Zt

sp1 ðtp  sp Þa1 ds ¼ k

t0

p a t  t0p pa Cð1 þ aÞ

Where supt2½t0 ;T  DH ðxðtÞ; 0Þ ¼ k. Then DH

3.7



I a;p RL x ðt0 Þ; 0  k

p a t  t0p pa !0 C ð 1 þ aÞ

Riemann-Liouville gH-Fractional Derivative of a 2 ð1; 2Þ

To discuss the subject, we need some primary definitions like the second order of gH-differentiability of a fuzzy number valued function xðtÞ 2 FR that is absolutely continuously gH-differentiable. It means the lower and upper parametric functions xl ðt; r Þ; xu ðt; r Þ in the level-wise form are absolutely continuous. This is because this continuity is evaluated by Hausdorff distance. As we know, the definition of second 00 order of gH-differential on ½t0 ; T  is as follows, such that xgH ðtÞ 2 FR

116

3 Fuzzy Fractional Operators

x0 ðt þ hÞgH x0 ðtÞ h!0 h

x00gH ðtÞ ¼ lim

0

Remark With the same assumptions for xðtÞ and more over xðtÞ; xgH ðtÞ do have monotone length, we have xðtÞgH xðt0 ÞgH ðt 

t0 Þx0gH ðt0 Þ

Z t Zs ¼ t0

x00gH ðuÞduds; t0  u  s  t  T

t0

To show the relation, Z t Zs t0

x00gH ðuÞduds

t0

Zt ¼

 0 x ðsÞgH x0 ðt0 Þ ds

t0

Let us know about the length of x0 ðsÞgH x0 ðt0 Þ,   0    0 0 0 length x0 ðsÞgH x0 ðt0 Þ ¼ xu ðs; r Þ  xu ðt0 ; r Þ  xl ðs; r Þ  xl ðt0 ; r Þ  0   0  0 0 ¼ xu ðs; r Þ  xl ðs; r Þ  xu ðt0 ; r Þ  xl ðt0 ; r Þ ¼ lengthðx0 ðsÞÞ  lengthðx0 ðt0 ÞÞ Finally, we have,  length x0 ðsÞgH x0 ðt0 Þ ¼ lengthðx0 ðsÞÞ  lengthðx0 ðt0 ÞÞ  • If lengthðx0 ðsÞÞ lengthðx0 ðt0 ÞÞ then length x0 ðsÞgH x0 ðt0 Þ 0  • If lengthðx0 ðsÞÞ  lengthðx0 ðt0 ÞÞ then length x0 ðsÞgH x0 ðt0 Þ  0 These two cases mean, if the length of x0 ðsÞ is increasing (decreasing) then the length of x0 ðsÞgH x0 ðt0 Þ is increasing (decreasing) too. Then the sign of gH-difference x0 ðsÞgH x0 ðt0 Þ is constant, Rt Rs

00

xgH ðuÞduds ¼

t0 t0

It is completed.

Rt  t0

Rt Rt x0 ðsÞgH x0 ðt0 Þ ds ¼ x0 ðsÞdsgH x0 ðt0 Þds t0

¼ xðtÞgH xðt0 ÞgH ðt  t0 Þ  x0 ðt0 Þ

t0

3.7 Riemann-Liouville gH-Fractional Derivative of a 2 ð1; 2Þ

117

Definition—The fuzzy Riemann-Liouville gH-fractional derivative of order a 2 ð1; 2Þ  00 DaRL[gH1 xðtÞ ¼ I 2a RL xðt Þ ; t 2 ½t0 ; T  Subject to the second derivative in the sense of gH-differentiability exists. In the other words, these limits exist. 0 0 2a 00 ðI2a RL xÞ ðt þ hÞgH ðI RL xÞ ðt Þ I 2a x ð t Þ ¼ lim RL h h!0  2a 0 I2a x ð t þ h Þ I 2a gH RL xðt Þ I RL xðtÞ ¼ lim RL h



h!0

Where I 2a RL xðtÞ

1 ¼ Cð2  aÞ

Zt ðt  sÞ1a xðsÞds; t 2 ½t0 ; T  t0

 2a 0 • Suppose that the functions I 2a RL xðtÞ and I RL xðtÞ are i  gH differentiable on ½t0 ; T : The level-wise form of the derivative is as,  00 DaRL[gH1 xðt; r Þ ¼ I 2a RL xðt; r Þ ; r 2 ½0; 1 Then DaRL[gH1 xl ðt; r Þ

¼



I 2a RL xl ðt; r Þ

00

1 ¼ Cð2  aÞ

 00 DaRL[gH1 xu ðt; r Þ ¼ I 2a RL xu ðt; r Þ ¼

Zt ðt  sÞ1a xl ðs; r Þds t0

1 C ð 2  aÞ

Zt ðt  sÞ1a xu ðs; r Þds t0

 2a 0 • Suppose that the functions I 2a RL xðtÞ and I RL xðtÞ are ii  gH differentiable on ½t0 ; T : The level-wise form of the derivative is as, 

00
2a  2a I 2a RL xðt; r Þ ¼ I RL xu ðt; r Þ; I RL xl ðt; r Þ  00 DaRL[gH1 xðt; r Þ ¼ I 2a RL xðt; r Þ ; r 2 ½0; 1

In this case to define the derivative, the Caputo operative is also ii  gH differentiable, then

118

3 Fuzzy Fractional Operators

h i a[1 DaRL[gH1 xðt; r Þ ¼ DaRL[gH1 xu ðt; r Þ; DRL x ð t; r Þ l gH where  00 DaRL[gH1 xl ðt; r Þ ¼ I 2a RL xl ðt; r Þ ¼ DaRL[gH1 xu ðt; r Þ

¼



I 2a RL xu ðt; r Þ

00

1 Cð2  aÞ

Zt ðt  sÞ1a xl ðs; r Þds t0

1 ¼ C ð 2  aÞ

Zt ðt  sÞ1a xu ðs; r Þds t0

In fact, the type of gH-differentiability of Caputo depends on the type of second order differentiability of RL integral operator. • Assume the fuzzy functions x0 ðtÞ is i  gH differentiable and xðtÞ is ii  RLgH 1 differentiable, DaRL[iigH , in this case, h 00 i 00 x00 ðt; r Þ ¼ xl ðt; r Þ; xu ðt; r Þ Then


2a 00  00 00 I 2a xl ðt; r Þ; I 2a RL x ðt; r Þ ¼ I RL RL xu ðt; r Þ
 1 a[1 xðt; r Þ ¼ DRL xu ðt; r Þ; DaRL[ 1 xl ðt; r Þ DaRL[iigH

where 00

00

a[1 2a DaRL[ 1 xl ðt; r Þ ¼ I 2a RL xu ðt; r Þ; DRLgH xu ðt; r Þ ¼ I RL xl ðt; r Þ

Basically, this case forms a system of fractional differential equations and it is occasionally is easy to solve. Another case is similar to this case. The following properties can be proved for any t 2 ½t0 ; T : Properties 

DaRL[gH1 I aRL xðtÞ ¼ xðtÞ



DaRL[gH1 I bRL xðtÞ ¼ I ba RL xðtÞ; b [ a

3.7 Riemann-Liouville gH-Fractional Derivative of a 2 ð1; 2Þ

119

For the first one, by using the definition,   00  2 00 a DaRL[gH1 I aRL xðtÞ ¼ DaRL[gH1 I aRL xðtÞ ¼ I 2a RL I RL xðtÞ ¼ I RL xðt Þ And 

00

0

I 2RL xðtÞ ¼ @

Z t Zs t0

Finally,

0 DaRL[gH1 I aRL xðtÞ ¼ @

xðuÞdudsA

t0

Z t Zs t0

100

100 xðuÞdudsA ¼ xðtÞ

t0

The second property, 

00 2a b DaRL[gH1 I bRL xðtÞ ¼ I RL I RL xðtÞ ¼

0 t 100 Z 1  @ ðt  sÞ1a I bRL xðsÞdsA Cð2  aÞ t0

0 t 100 Zs Z 1 1a b1 ¼ ðs  uÞ xðuÞdudsA  @ ðt  sÞ Cð2  aÞCðbÞ t0

t0

If we suppose v ¼ us ts then ðt  sÞ1a ¼ ðu  sÞ1a va1 ; ðs  uÞb1 ¼ vb1 ðs  tÞb1 DaRL[gH1 I bRL xðtÞ ¼

1 Cð2  aÞCðbÞ 0 t 100 Z Z1  @ ðt  sÞ1 þ ba xðsÞds ð1  vÞ2a1 vb1 dvA t0

0

On the other hand, Z1 0

ð1  vÞ2a1 vb1 dv ¼

Cð2  aÞCðbÞ Cðb  a þ 2Þ

120

3 Fuzzy Fractional Operators

Then DaRL[gH1 I bRL xðtÞ

0 t 100 Z 1 1 þ ba  @ ðt  sÞ ¼ xðsÞdsA Cðb  a þ 2Þ t0

Now we are going to find the

Rt

!00 ðt  sÞ

1 þ ba

xðsÞds

,

t0

We have, 0 @

10

Zt ðt  sÞ

1 þ ba

xðsÞdsA ¼

t0

Zt

ð1 þ b  aÞðt  sÞba xðsÞds

t0

Also, 0 @

10

Zt ð1 þ b  aÞðt  sÞ

ba

xðsÞdsA ¼ ð1 þ b  aÞðb  aÞ

t0

Zt

ðt  sÞba1 xðsÞds

t0

By substituting in, 0 DaRL[gH1 I bRL xðtÞ ¼

1 @ C ð b  a þ 2Þ

Zt

100 ðt  sÞ1 þ ba xðsÞdsA

t0

ð1 þ b  aÞðb  aÞ  ¼ Cðb  a þ 2Þ

Zt

ðt  sÞba1 xðsÞds

t0

Since, ð1 þ b  aÞðb  aÞ 1 ¼ Cðb  a þ 2Þ Cðb  aÞ DaRL[gH1 I bRL xðtÞ

1  ¼ C ð b  aÞ

Zt

ðt  sÞba1 xðsÞds ¼ I ba RL xðt Þ

t0

Example Consider the fuzzy function xðtÞ ¼ c  et ; c 2 FR ; t 2 ½0; 1; a ¼ 32

3.7 Riemann-Liouville gH-Fractional Derivative of a 2 ð1; 2Þ

121

 1 00 3 D2RLgH xðtÞ ¼ I 2RL c  et ; where c I RL c  e ¼ 1  C 2 1 2

3.7.1

Z1

t

ðt  sÞ2 es ds; t 2 ½t0 ; T  1

0

Definition—Fuzzy Caputo Fractional Derivative of Order a 2 ð1; 2Þ

Considering the same assumptions on the fuzzy number valued function xðtÞ,  1 DCa [ xðtÞ ¼ DaRL[gH1 xðtÞgH xðt0 ÞgH ðt  t0 Þ  x0 ðt0 Þ ; t 2 ½t0 ; T  gH We obtained the following relation, Z t Zs t0

00

xgH ðuÞduds ¼ xðtÞgH xðt0 ÞgH ðt  t0 Þ  x0 ðt0 Þ

t0

By substituting,  a[1

0



DRLgH xðtÞgH xðt0 ÞgH ðt  t0 Þ  x0 ðt0 Þ ¼ DaRL[gH1 @

Z t Zs t0

1 xgH ðuÞdudsA 00

t0

Indeed, 0 1 DaC[ xðtÞ ¼ DaRL[gH1 @ gH

Z t Zs t0

1

 00 xgH ðuÞdudsA ¼ DaRL[gH1 I 2RL x00 ðtÞ ¼ I 2a RL x ðt Þ 00

t0

Finally, 1 DCa [ xð t Þ gH

In level-wise form,

¼

00 I 2a RL x ðtÞ

1 ¼ Cð2  aÞ

Zt t0

ðt  sÞ1a x00 ðsÞds

122

3 Fuzzy Fractional Operators

• Assume the fuzzy functions xðtÞ; x0 ðtÞ are i  gH differentiable, h 00 i 00 x00 ðt; r Þ ¼ xl ðt; r Þ; xu ðt; r Þ Then h i 00 2a 00 2a 00 I 2a RL x ðt; r Þ ¼ I RL xl ðt; r Þ; I RL xu ðt; r Þ where 00 I 2a RL xl ðt; r Þ

00

1 ¼ Cð2  aÞ

I 2a RL xu ðt; r Þ ¼

1 Cð2  aÞ

Zt

00

ðt  sÞ1a xl ðsÞds t0

Zt

00

ðt  sÞ1a xu ðsÞds t0

h i 1 a[1 a[1 DaC[ x ð t; r Þ ¼ D x ð t; r Þ; D x ð t; r Þ l u CgH CgH gH where 00

00

1 a[1 2a DaC[ xl ðt; r Þ ¼ I 2a RL xl ðt; r Þ; DCgH xu ðt; r Þ ¼ I RL xu ðt; r Þ gH

• Assume the fuzzy functions xðtÞ; x0 ðtÞ are ii  gH differentiable, h 00 i 00 x00 ðt; r Þ ¼ xu ðt; r Þ; xl ðt; r Þ Then
2a 00  2a 00 00 I 2a RL x ðt; r Þ ¼ hI RL xu ðt; r Þ; I RL xl ðt; r Þ i 1 a[1 a[1 x ð t; r Þ ¼ D x ð t; r Þ; D x ð t; r Þ DaC[ u l CgH CgH gH where 00

00

1 a[1 2a DaC[ xl ðt; r Þ ¼ I 2a RL xl ðt; r Þ; DCgH xu ðt; r Þ ¼ I RL xu ðt; r Þ gH

3.7 Riemann-Liouville gH-Fractional Derivative of a 2 ð1; 2Þ

123

• Assume the fuzzy functions x0 ðtÞ is ii  gH differentiable and xðtÞ is i  CgH 1 differentiable, DaC[ , in this case, igH h 00 i 00 x00 ðt; r Þ ¼ xu ðt; r Þ; xl ðt; r Þ Then


2a 00  2a 00 00 I 2a RL x ðt; r Þ ¼
I RL xu ðt; r Þ; I RL xl ðt; r Þ  1 DaC[ xðt; r Þ ¼ DaC[ 1 xl ðt; r Þ; DaC[ 1 xu ðt; r Þ igH

where 00

00

a[1 2a DaC[ 1 xl ðt; r Þ ¼ I 2a RL xu ðt; r Þ; DCgH xu ðt; r Þ ¼ I RL xl ðt; r Þ

Basically, this case forms a system of fractional differential equations and it is occasionally is easy to solve. Another case is similar to this case. Example Consider the fuzzy function xðtÞ ¼ c  t2 ; ; c 2 FR ; t 2 ½0; 1; a ¼ 32 It is clear the function xðtÞ; x0 ðtÞ are i  gH differentiable and 1 DaC[ gH



ct

2



¼

I 2a RL ð2t

pffi 4t t  cÞ ¼  1  c C 2

1 ðc  t2 Þ is i  CgH differentiable. It can be seen that DaC[ gH

Remark 1 xðtÞ ¼ xðtÞgH xðt0 ÞgH ðt  t0 Þ  x0gH ðt0 Þ I aRL[ 1 DaC[ gH

To show it, we have,   1 a[1 0 DaC[ x ð t Þ ¼ D x ð t Þ x ð t Þ ð t  t Þ  x ð t Þ gH 0 gH 0 0 RLgH gH gH using the RL fractional integration in both sides, we have 1 xðtÞ ¼ xðtÞgH xðt0 ÞgH ðt  t0 Þ  x0gH ðt0 Þ I aRL[ 1 DaC[ gH

This relation is also can be considered in the level-wise form, 0

• If xðtÞgH xðt0 Þ exists in case of i  gH, then xigH ðt0 Þ exists and if xðtÞ is 1 i  CgH differentiable, DaC[ , in this case, igH 0

1 xðtÞ ¼ xðtÞigH xðt0 ÞigH ðt  t0 Þ  xigH ðt0 Þ I aRL[ 1 DaC[ igH

124

3 Fuzzy Fractional Operators

In level-wise form, 0

1 I aRL[ 1 DaC[ xl ðt; r Þ ¼ xl ðt; r Þ  xl ðt0 ; r Þ  ðt  t0 Þxl ðt0 ; r Þ gH 0 a[1 a[1 I RL DCgH xu ðt; r Þ ¼ xu ðt; r Þ  xu ðt0 ; r Þ  ðt  t0 Þxu ðt0 ; r Þ 0

• If xðtÞgH xðt0 Þ exists in case of ii  gH, then xiigH ðt0 Þ exists and if xðtÞ is 1 ii  CgH differentiable, DaC[ , again in this case, iigH 0

1 xðtÞ ¼ xðtÞiigH xðt0 ÞigH ðt  t0 Þ  xiigH ðt0 Þ I aRL[ 1 DCa [ iigH

In level-wise form, 0

1 I aRL[ 1 DaC[ xl ðt; r Þ ¼ xl ðt; r Þ  xl ðt0 ; r Þ  ðt  t0 Þxu ðt0 ; r Þ gH 0 a[1 a[1 I RL DCgH xu ðt; r Þ ¼ xu ðt; r Þ  xu ðt0 ; r Þ  ðt  t0 Þxl ðt0 ; r Þ 0

• If xðtÞgH xðt0 Þ exists in case of i  gH, then xigH ðt0 Þ exists and if xðtÞ is 1 ii  CgH differentiable, DaC[ , again in this case, iigH 0

1 xðtÞ ¼ xðtÞigH xðt0 ÞigH ðt  t0 Þ  xigH ðt0 Þ I aRL[ 1 DaC[ iigH

In level-wise form, 0

1 I aRL[ 1 DaC[ xu ðt; r Þ ¼ xl ðt; r Þ  xl ðt0 ; r Þ  ðt  t0 Þxl ðt0 ; r Þ gH 0 a[1 a[1 I RL DCgH xl ðt; r Þ ¼ xu ðt; r Þ  xu ðt0 ; r Þ  ðt  t0 Þxu ðt0 ; r Þ

This case will be very difficult to solve, because it is a system of fractional differential equations.

3.8

Generalized Fuzzy ABC Fractional Derivative

The generalized ABCgH derivative of a fuzzy number valued function xðtÞ on c interval ½t0 ; T  starting at t0 with kernel Ea;l ðk; tÞ where 0\ah1; ReðlÞi0; c 2 a R; k ¼ 1a is defined in the following form, Da;l:c ABCgH xðt Þ

BðaÞ  ¼ 1a

Zt t0

0

c xgH ðsÞ  Ea;l ðk; t  sÞds

3.8 Generalized Fuzzy ABC Fractional Derivative

125

where BðaÞ [ 0 is a normalizing function and is defined as, BðaÞ ¼ 1  a þ

a CðaÞ

With properties Bð0Þ ¼ Bð1Þ ¼ 1. The level-wise form of the derivative is defined in the following form with considering the type of gH-differential. Case 1. If x is i  gH differentiable, Da;l:c ABCgH xl ðt; r Þ Da;l:c ABCgH xu ðt; r Þ

BðaÞ  ¼ 1a

Zt

0

c xl ðs; r Þ  Ea;l ðk; t  sÞds

t0

BðaÞ  ¼ 1a

Zt

0

c xu ðs; r Þ  Ea;l ðk; t  sÞds

t0

Case 2. If x is ii  gH differentiable, Da;l:c ABCgH xl ðt; r Þ Da;l:c ABCgH xu ðt; r Þ

BðaÞ  ¼ 1a B ð aÞ  ¼ 1a

Zt

0

c xu ðs; r Þ  Ea;l ðk; t  sÞds

t0

Zt

0

c xl ðs; r Þ  Ea;l ðk; t  sÞds

t0

Note In case 2, the fuzzy ABC differential cannot be defined because the left side and right side do not have the same function. We only claim that how can we have an interval with this derivative in the level-wise form. Case 1. xðtÞ is i  gH differentiable,
a;l:c  a;l:c Da;l:c ABCigH xðt; r Þ ¼ DABC xl ðt; r Þ; DABC xu ðt; r Þ Case 2. xðtÞ is ii  gH differentiable,
a;l:c  a;l:c Da;l:c ABCiigH xðt; r Þ ¼ DABC xu ðt; r Þ; DABC xl ðt; r Þ where t  a BðaÞ R 0 a Da;l:c ABC xl ðt; r Þ ¼ 1a xl ðs; r ÞEa  1a ðt  sÞ ds

Da;l:c ABC xu ðt; r Þ

¼

t0 t BðaÞ R 1a t0

 a 0 xu ðs; r ÞEa  1a ðt  sÞa ds

126

3 Fuzzy Fractional Operators

Its corresponding AB fractional integral operator is defined as, I a;l:c AB xðtÞ

¼

c  X c i¼0

i

ai BðaÞð1  aÞi1

I a;l;c xðtÞ

Subject to, a;l:c I a;l:c AB DABCgH xðtÞ ¼ xðtÞgH xðt0 Þ

Based on the definition of the gH-difference, Case 1. xðtÞ is i  gH difference, a;l:c I a;l:c AB DABCgH xðtÞ ¼ xðt ÞH xðt0 Þ a;l:c xðtÞ ¼ I a;l:c AB DABCgH xðt Þ  xðt0 Þ a;l:c xl ðt; r Þ ¼ xl ðt0 ; r Þ þ I a;l:c AB DABCgH xl ðt; r Þ a;l:c xu ðt; r Þ ¼ xu ðt0 ; r Þ þ I a;l:c AB DABCgH xu ðt; r Þ

Case 2. xðtÞ is ii  gH difference, a;l:c I a;l:c AB DABCgH xðtÞ ¼ H ð1ÞxðtÞ  ð1Þxðt0 Þ a;l:c xðt0 Þ ¼ xðtÞ  ð1ÞI a;l:c AB DABCgH xðtÞ a;l:c xu ðt; r Þ ¼ xu ðt0 ; r Þ þ I a;l:c AB DABCgH xl ðt; r Þ a;l:c xl ðt; r Þ ¼ xl ðt0 ; r Þ þ I a;l:c AB DABCgH xu ðt; r Þ

It is seen that in case 2 we have a system of fractional differential equations and analytical solving of such system is so difficult and we should find the numerical solutions. The numerical method for this problem is explained in Chap. 5.

Chapter 4

Fuzzy Fractional Differential Equations

4.1

Introduction

Different materials and processes in many applied sciences like electrical circuits, biology, biomechanics, electrochemistry, electromagnetic processes and, others are widely recognized to be well predicted by using fractional differential operators in accordance with their memory and hereditary properties. For the complex phenomena, the modeling and their results in diverse widespread fields of science and engineering, are also so complicated and for achieving the accurate method the only powerful tool is fractional calculus. Indeed, the fractional calculus is not only a very important and productive topic, it also represents a new point of view that how to construct and apply a certain type of non-local operators to real-world problems. In general, the majority of the fuzzy fractional differential equations as same as fuzzy differential equations do not have exact solutions. This is why, approximate and numerical procedures are important to be developed. On the other hand, the complicity of many parameters in mathematical modeling of natural phenomena appears as an uncertain fractional model and they play an important role in various disciplines. Hence, it motivates the researchers to investigate effective numerical methods with error analysis to approximate the fuzzy differential equations. The objectives of this chapter are to consider fractional operators such as differentials and integral operators on fuzzy number valued functions. Then the fuzzy fractional differential equations are introduced and solved and analyzed by some theoretical methods.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. Allahviranloo, Fuzzy Fractional Differential Operators and Equations, Studies in Fuzziness and Soft Computing 397, https://doi.org/10.1007/978-3-030-51272-9_4

127

128

4.2

4 Fuzzy Fractional Differential Equations

Fuzzy Fractional Differential Equations— Caputo-Katugampola Derivative

In this section, we are going to cover the topic with different types of fuzzy fractional operators. The fuzzy fractional differential equations can be defined by each operator which are defined before. In each type of differentiability, the existence and uniqueness of the solutions are discussed, and some analytical methods are introduced to obtain the solutions. One of the popular methods that is going to be explained is Laplace transforms. Thus, these transforms are introduced and discussed with details in this section as well. Definition—Fuzzy fractional differential equations Consider the following fuzzy fractional differential equation with fuzzy initial value. Da;p CK gH xðtÞ ¼ f ðt; xðtÞÞ;

xðt0 Þ ¼ x0 ;

t 2 ½t0 ; T ;

0\a\1:

where f : ½t0 ; T   FR ! FR is a fuzzy number valued function and xðtÞ is a continuous fuzzy set valued solution and Da;p CK gH is the fractional Caputo- Katugampola fractional operator. The first discussion is about the relation between the solution of fractional differential equations and solution of its corresponding fuzzy fractional differential equation. It means the fuzzy solution is the common solution of two fractional operators, differential and integral. In the following remark, we suppose that, the fuzzy solution is i  gH differentiable or ii  gH differentiable on ðt0 ; T : Remark Considering the mentioned above assumptions, the continuous fuzzy function xðtÞ is the solution of Da;p CK gH xðtÞ ¼ f ðt; xðtÞÞ;

xðt0 Þ ¼ x0 ;

t 2 ½t0 ; T ;

0\a\1

If and only if, xðtÞ satisfies the following integral equation, p1a xðtÞgH xðt0 Þ ¼ CðaÞ

Zt

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds;

t 2 ½t0 ; T 

t0

Before to show the assertion, it is mentioned that, the function xðtÞ can be i  gH differentiable (increasing length) or ii  gH differentiable (decreasing length) on ðt0 ; T  but the fractional integral operator I a;p RL always does have increasing length. To prove the remark, first suppose that xðtÞ is the solution of differential equation, and moreover, suppose xðtÞgH xðt0 Þ := yðtÞ: The length of yðtÞ is increasing and it is i  gH differentiable, because if xðtÞ is i  gH differentiable, then

4.2 Fuzzy Fractional Differential Equations—Caputo-Katugampola Derivative

129

t0 \t ) xu ðt0 ; r Þ  xl ðt0 ; r Þ\xu ðt; r Þ  xl ðt; r Þ Then yl ðt; r Þ ¼ xl ðt; r Þ  xl ðt0 ; r Þ\xu ðt; r Þ  xu ðt0 ; r Þ ¼ yu ðt; r Þ It means yðt; r Þ is an interval for any r 2 ½0; 1 and concludes it satisfies the type (1) of gH-difference. As it is shown before, a;p I a;p RL DCK gH xðtÞ ¼ xðtÞgH xðt0 Þ

And Da;p CK gH xðtÞ ¼ f ðt; xðt ÞÞ then the following relation is concluded, I a;p RL f ðt; xðt ÞÞ ¼ xðt ÞgH xðt0 Þ On the other hand, based on the definition of fractional integral I a;p RL xðtÞ

p1a  ¼ C ð aÞ

Then I a;p RL f ðt; xðtÞÞ

p1a  ¼ CðaÞ

Zt

sp1 ðtp  sp Þa1 xðsÞds

t0

Zt

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds

t0

So, the necessary condition is now proved. Next, the sufficient condition is going to be proved. Suppose we have, p1a xðtÞgH xðt0 Þ ¼ CðaÞ

Zt

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds

t0

Effecting the fractional derivative Da;p RLgH to both sides,   Da;p RLgH xðtÞgH xðt0 Þ

¼

Da;p RLgH

p1a CðaÞ

Zt t0

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds

130

4 Fuzzy Fractional Differential Equations

thus   a;p a;p Da;p RLgH xðt ÞgH xðt0 Þ ¼ DRLgH I RL f ðt; xðtÞÞ ¼ f ðt; xðtÞÞ Therefor,   Da;p RLgH xðtÞgH xðt0 Þ ¼ f ðt; xðtÞÞ Since we had,  a;p  Da;p CK gH xðt Þ ¼ DRLgH xðtÞgH xðt0 Þ Thus, Da;p CK gH xðtÞ ¼ f ðt; xðtÞÞ And the proof is completed. Note Based on the definition of the gH-difference in the integral equation, p1a xðtÞgH xðt0 Þ ¼ CðaÞ

Zt

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds;

t 2 ½t0 ; T 

t0

• In case the gH-difference is i  gH difference, p1a xðtÞ ¼ xðt0 Þ  C ð aÞ

Zt

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds;

t 2 ½t0 ; T 

t0

• In case the gH-difference is ii  gH difference, p1a xðtÞ ¼ xðt0 ÞH ð1Þ CðaÞ

Zt

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds;

t 2 ½t0 ; T 

t0

Example Consider the following fuzzy initial value problem,   pffi 1 2 DCK gH xðtÞ ¼ t; pffi ; pffi ¼ f ðtÞ; t t 1 2;1

xð0Þ ¼ ð2; 0; 1Þ;

t 2 ð0; 1;

4.2 Fuzzy Fractional Differential Equations—Caputo-Katugampola Derivative

131

Now using the fractional derivative operator in both sides, 1

;p

1

;p

2 2 DCK xðtÞ ¼ xðtÞgH xðt0 Þ I RL gH

with p1a xðtÞgH xðt0 Þ ¼ I RL f ðtÞ ¼ CðaÞ 1 2;p

Zt

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds;

t 2 ð0; 1

t0

And p ¼ 1 we have, 1 I RL f ðtÞ ¼ 1  C 2 1 2;1

Zt

pffiffiffi    pffiffi 1 2 p pffiffiffi pffiffiffi t; p; 2 p ðt  sÞ  s; pffiffi ; pffiffi ds ¼ 2 s s 12

0

then xðtÞgH xðt0 Þ ¼

pffiffiffi  p pffiffiffi pffiffiffi t; p; 2 p 2

By substituting the initial value, pffiffiffi  p pffiffiffi pffiffiffi xðtÞgH ð2; 0; 1Þ ¼ t; p; 2 p 2 The length of

pffiffi pffiffiffi pffiffiffi  pffiffiffi p t 2 t; p; 2 p is ð1  r Þ p 2  2 ; t 2 ð0; 1; r 2 ½0; 1: The

length function is decreasing because, pffiffiffi dh t i ð1  r Þ p 2  \0; dt 2

r 2 ½0; 1

In case i  gH difference, (Fig. 4.1) pffiffiffi    pffiffiffi pffiffiffi p pffiffiffi pffiffiffi p pffiffiffi t; p; 2 p ¼ 2 þ t; p; 1 þ 2 p xðtÞ ¼ ð2; 0; 1Þ  2 2 The level-wise form of the solution is, xl ðt; r Þ ¼

 pffiffiffi  pffiffiffi pffiffiffi p t ðr  1Þ; pþ pþ2  2

xu ðt; r Þ ¼

pffiffiffi  pffiffiffi p þ 1 þ p ð1  r Þ

132

4 Fuzzy Fractional Differential Equations

Fig. 4.1 The length of xðtÞgH xðt0 Þ is decreasing

Fig. 4.2 Length of xðtÞ is decreasing

And lengthðxðtÞÞ ¼

 pffiffiffi  pffiffiffi p t ð1  r Þ; 3þ2 p  2

pffiffiffi

pffiffiffi p d t \0; r 2 ½0; 1 3þ2 p  2 dt

It is seen from the figures both xðtÞ and xðtÞgH x0 do have decreasing length and the solution is as, (Fig. 4.2)

4.2 Fuzzy Fractional Differential Equations—Caputo-Katugampola Derivative

133

Fig. 4.3 The solution xðtÞ at t ¼ 1

  pffiffiffi pffiffiffi p pffiffiffi t; p; 1 þ 2 p xðtÞ ¼ x0  I RL f ðtÞ ¼ 2 þ 2 1 2;1

The figure shows that the solution is fuzzy number at each point like t ¼ 1: To check the function is the solution of the problem, (Fig. 4.3) Da;p CK gH xðt Þ

  pffiffiffi pffiffiffi 0 1 p pffiffiffi s; p; 1 þ 2 p ds ðt  sÞ2  2 þ 2 0 pffi pffiffiffi 3  pffiffiffi pffi !  2 2 p ffi 2 ð 1 þ 2 pÞ t pffi 1 2 3 pt  4 t pffiffiffi pffiffiffi ; 2 t; 6¼ t; pffi ; pffi p p t t

1 ¼ 1 C 2 ¼

Zt

Despite the fuzzy function xðtÞ is a fuzzy number valued function but it is not the solution of fuzzy fractional differential equation. The reason is the length of xðtÞgH xðt0 Þ is not increasing. Example Consider the following fuzzy initial value problem, 1

;1

2 DCK xð t Þ ¼ gH

 pffiffiffi 2 p t ; 0; 2t  t2 ¼ f ðtÞ;

xð0Þ ¼ ð2; 0; 2Þ;

t 2 ð0; 1;

134

4 Fuzzy Fractional Differential Equations

Now using the fractional derivative operator in both sides, 1

;p

1

;p

2 2 DCK xðtÞ ¼ xðtÞgH xðt0 Þ I RL gH

with p1a xðtÞgH xðt0 Þ ¼ I RL f ðtÞ ¼ CðaÞ 1 2;p

Zt

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds;

t0

And p ¼ 1 we have, Zt  pffiffiffi 1 1 I RL f ðtÞ ¼ 1  ðt  sÞ2  p s2 ; 0; 2s  s2 ds C 2 0   16 5 8 3 16 5 ¼  t2 ; 0; t2  t2 15 3 15 1 2;1

then xðtÞgH xðt0 Þ ¼

  16 5 8 3 16 5  t2 ; 0; t2  t2 15 3 15

Now, two types of the solution we should have, (Fig. 4.4)

Fig. 4.4 Length of xðtÞgH xðt0 Þ is increasing

t 2 ð0; 1

4.2 Fuzzy Fractional Differential Equations—Caputo-Katugampola Derivative

Fig. 4.5 Length of xigH ðtÞ is increasing

• i  gH solution 

16 5 8 3 16 5 xigH ðtÞ ¼ ð2; 0; 2Þ   t2 ; 0; t2  t2 15 3 15   16 5 8 3 16 5 ¼ 2  t2 ; 0; t2  t2 þ 2 15 3 15



• ii  gH solution (Fig. 4.5)   16 5 8 3 16 5 2 2 2 xiigH ðtÞ ¼ ð2; 0; 2ÞH ð1Þ  t ; 0; t  t 15 3 15   8 3 16 5 16 5 t2  t2  2; 0; 2  t2 ¼ 3 15 15 To check the solutions, (Fig. 4.6)

135

136

4 Fuzzy Fractional Differential Equations

Fig. 4.6 Length of xiigH ðtÞ is decreasing

  16 5 8 3 16 5 2 2 2 DCK gH xigH ðtÞ ¼ DCK gH 2  t ; 0; t  t þ 2 15 3 15      1 1 16 5 8 3 16 5 2;1 2;1 2 2 2 ¼ DCK 2  t ; 0; DCK t  t þ 2 15 3 15 0 1 t t Z Z 1 3 3 2 4s2  83 s2 1 8s 1 pffiffiffiffiffiffiffiffiffiffi dsA pffiffiffiffiffiffiffiffiffiffi ds; 0; 1 ¼ @ 1 C 2 C 2 ts 3 ts 0 0  pffiffiffi 2 ¼ p t ; 0; 2t  t2 1 2;1

1 2;1

Then the i  gH solution satisfies the fractional problem. The same process can be done for checking ii  gH solution.   1 1 8 3 16 5 16 5 2;1 2;1 2  2  2; 0; 2  2 t t t DCK x ð t Þ ¼ D CK gH gH iigH 3 15 15      1 1 16 5 16 5 ;1 8 3 2;1 ¼ DCK 2  t2 ; 0; D2CK t2  t2 þ 2 15 3 15  pffiffiffi 2 2 ¼ p t ; 0; 2t  t Both are i  gH and ii  gH solutions of the problem because the length of xðtÞgH xðt0 Þ is increasing.

4.2 Fuzzy Fractional Differential Equations—Caputo-Katugampola Derivative

4.2.1

137

Existence and Uniqueness of the Solution

In this section the existence and uniqueness results of solution to fuzzy fractional differential equations by using an idea of successive approximations under generalized Lipschitz condition of the right-hand side are investigated. Furthermore, the formula of solution to the linear fuzzy Caputo—Katugampola fractional differential equation is given. Since the real intervals in the level-wise form are used for any arbitrary level, so to reach the aims, we should consider the following theorem in the real case of Caputo fractional derivative. Theorem—Existence and uniqueness in real fractional differential equation Consider the initial value problem as follows,

is a ball Let g [ 0 be a given constant and with radius g. Also assume a real valued function g : ½t0 ; T   ½0; g ! around R þ satisfies the following conditions, 1. 2.

for . is nondecreasing function with respect to x for any t 2 ½t0 ; T .

all

Then the mentioned fractional problem has at least one solution on ½t0 ; T  and . Proof The solution of the mentioned above fractional differential equation is the solution of following fractional integral equation.

The following successive method for approximation of the solution of fractional differential equation is defined,

Such that

138

4 Fuzzy Fractional Differential Equations

For n ¼ 0 and t 2 ½t0 ; T   we have,

So, we found,

For n ¼ 1 and t 2 ½t0 ; T  ;

Since g is a nondecreasing function in and . Now we have thus

, then

Proceeding recursively, we will have

It follows that the sequence other hand,

is uniformly bounded for n 0. On the

Then in the interval ½t0 ; T  , we can use the mean value theorem for t1 ; t2 2 ½t0 ; T  ; n 0,

Therefor the sequence

is equi-continuous and then

Hence, by the Arzela–Ascoli Theorem and the monotonicity of the sequence we conclude the convergency of the sequence and .

4.2 Fuzzy Fractional Differential Equations—Caputo-Katugampola Derivative

139

Theorem—Existence and uniqueness in fuzzy fractional differential equation Consider the fuzzy initial value problem as follows, Da;p CK gH xðtÞ ¼ f ðt; xðtÞÞ;

xðt0 Þ ¼ x0 ;

t 2 ½t0 ; T ;

0\a\1

Let g [ 0 be a given constant and Bðx0 ; gÞ ¼ fx 2 R; jx  x0 j gg is a ball around x0 with radius g. Also assume a fuzzy number valued function f : ½t0 ; T   ½0; g ! FR satisfies the following conditions, i. f 2 C ð½t0 ; T   ½0; g; FR Þ; 9Mg 0; DH ðgðt; xÞ; 0Þ Mg ; ðt; xÞ 2 ½t0 ; T   Bðx0 ; gÞ. ii. For any C ð½t0 ; T   ½0; g; R þ Þ in the problem Da;p CK xðtÞ ¼ gðt; xðtÞÞ;

xðt0 Þ ¼ x0 ¼ 0;

for

all

where g 2

t 2 ½t0 ; T 

has only solution xðtÞ  0 on ½t0 ; T  (the previous theorem of existence and uniqueness in real case). Then the following successive approximations given by x0 ðtÞ ¼ x0 p1a xn ðtÞgH x0 ¼ CðaÞ

Zt

sp1 ðtp  sp Þa1  f ðs; xn1 ðsÞÞds; n ¼ 1; 2; . . .;

t0

Converges to a unique solution of Da;p CK gH xðtÞ ¼ f ðt; xðtÞÞ;

xð t 0 Þ ¼ x0 ;

0\a\1

on t 2 ½t0 ; T  ; T  2 ðt0 ; T , provided that xn ðtÞgH x0 does have increasing length. Proof Let us consider the point t , " #1p 1 gCða þ 1Þ a a p p þa ; t0 \t

M 

Now consider the sequence fxn ðtÞgn ; x0 ðtÞ ¼ x0 ; t 2 ½t0 ; T  . xn ðtÞgH x0 ¼

p1a CðaÞ

Zt t0

 M ¼ max Mf ; Mg ;

of

fuzzy

T  ¼ minft ; T g

continuous

functions

sp1 ðtp  sp Þa1  f ðs; xn1 ðsÞÞds; n ¼ 1; 2; . . .;

140

4 Fuzzy Fractional Differential Equations

First, we prove the xn ðtÞ 2 Cð½t0 ; T  ; Bðx0 ; gÞÞ. To this end, assume t1 ; t2 2 ½t0 ; T   and t1 \t2 .   DH xn ðt1 ÞgH x0 ; xn ðt2 ÞgH x0 Zt1 h a1  p a1 i p1a

sp1 t1p  sp  t2  sp DH ðf ðs; xn ðsÞÞ; 0Þds CðaÞ t0

þ

t2 1a Z

p CðaÞ

h a1 i sp1 t2p  sp DH ðf ðs; xn ðsÞÞ; 0Þds

t1

And p1a C ð aÞ

Zt2 t1

t1 1a Z

p C ð aÞ ¼

h a1 i sp1 t2p  sp ds ¼

a pa  p t2  t1p Cða þ 1Þ

h a1  p a1 i sp1 t1p  sp  t2  sp ds

t0

a  a  pa  p t1  t0p  t2p  t1p Cða þ 1Þ

And   DH ðf ðs; xn ðsÞÞ; 0Þ Mf ; DH xn ðt1 ÞgH x0 ; xn ðt2 ÞgH x0 ¼ DH ðxn ðt1 Þ; xn ðt2 ÞÞ Hence, DH ðxn ðt1 Þ; xn ðt2 ÞÞ

a  a  a  pa Mf  p t2  t1p þ t1p  t0p þ t2p  t1p C ð a þ 1Þ

Finally, DH ðxn ðt1 Þ; xn ðt2 ÞÞ

a 2pa Mf  p t2  t1p g C ð a þ 1Þ

This means, if t2 ! t1 then DH ðxn ðt1 Þ; xn ðt2 ÞÞ ! 0 and follows the function xn ðtÞ is continuous on ½t0 ; T  : In addition, it follows for n 1; t 2 ½t0 ; T  ; xn ðtÞ 2 Bðx0 ; gÞ; xn ðtÞ gH x0 2 Bðx0 ; gÞ

4.2 Fuzzy Fractional Differential Equations—Caputo-Katugampola Derivative

141

Now, if xn1 ðtÞ 2 Bðx0 ; gÞ;   DH xn ðtÞgH x0 ; 0

t p1a R p1 p ðt  sp Þa1 DH ðf ðs; xn1 ðsÞÞ; 0Þds CðaÞ s t0 a pa M 

Cða þ 1f Þ tp  t0p g

In conclusion, the fuzzy function xn ðtÞ 2 Bðx0 ; gÞ for all n 1 and all t 2 ½t0 ; T  : Now our next step is proving the convergence, for xn ðtÞ; xðtÞ 2 C ð½t0 ; T  ; Bðx0 ; gÞÞ; lim xn ðtÞ ¼ xðtÞ

n!0

To this purpose, we need some relations,

Since Da;p CK gH xðtÞ ¼ xðt ÞgH x0 Thus, based on the properties of the distance,

Finally,

Continuing,

142

4 Fuzzy Fractional Differential Equations

Then

Hence,

And

In general, assume for m n, DH ðxm ðtÞ; xn ðtÞÞ ! 0 Then using the definition of the Cauchy sequence the sequence fxn ðtÞgn ! xðtÞ. Uniqueness. To show it, let’s suppose that is another solution of fuzzy fractional differential equation, and assume,

Finally, we have,

The only solution (because of , the maximal solution) is

.

4.2 Fuzzy Fractional Differential Equations—Caputo-Katugampola Derivative

143

Remark In conclusion, if the fuzzy function f : ½t0 ; T   FR ! FR in the following fuzzy fractional initial value problem, Da;p CK gH xðtÞ ¼ f ðt; xðt ÞÞ;

xð t 0 Þ ¼ x0 ;

t 2 ½t0 ; T 0\a\1

Satisfies in the Lipchitz condition, DH ðf ðt; xðtÞÞ; f ðt; yðtÞÞÞ LDH ðxðtÞ; yðtÞÞ; DH ðf ðt; xðtÞÞ; 0Þ Mf Then the following successive approximations converge uniformly to the unique solution of the problem on ½t0 ; T , subject to xn ðtÞgH x0 does have increasing length. p1a xn ðtÞgH x0 ¼ CðaÞ

Zt

sp1 ðtp  sp Þa1  f ðs; xn1 ðsÞÞds;

n ¼ 1; 2; . . .;

t0

Example Consider Da;p CK gH xðt Þ ¼ k  xðtÞ  hðtÞ;

xð t 0 Þ ¼ x0 ;

k 2 R;

t 2 ðt 0 ; T 

a;p Such that xðtÞ; hðtÞ 2 Cððt0 ; T ; FR Þ: Since I a;p RL DCK gH xðtÞ ¼ xðtÞgH xðt0 Þ then a;p xðtÞgH xðt0 Þ ¼ kI a;p RL xðtÞ  I RL hðt Þ a;p If consider the k 0 then kI a;p RL xðtÞ  I RL hðtÞ has increasing length because xðtÞ and hðtÞ are two fuzzy number valued functions. Now we can use the successive approximation method, a;p xn ðtÞgH xðt0 Þ ¼ kI a;p RL xn1 ðtÞ  I RL hðt Þ;

n ¼ 1; 2; . . .;

If n ¼ 1, Let us consider k 0 and x is i  gH differentiable (increasing length)  a k tp  t0p x1 ðtÞgH xðt0 Þ ¼ x0  a  I a;p RL hðtÞ; p C ð a þ 1Þ Let us consider k\0 and x is ii  gH differentiable (decreasing length)  a k tp  t0p  I a;p ð1Þ xðt0 ÞgH x1 ðtÞ ¼ x0  a RL hðtÞ; p Cða þ 1Þ 

If n ¼ 2,



144

4 Fuzzy Fractional Differential Equations

Let us consider k 0 and x is i  gH differentiable (increasing length) "  a  2a # k tp  t0p k tp  t0p þ x2 ðtÞgH xðt0 Þ ¼ x0  a p Cða þ 1Þ p2a Cð2a þ 1Þ 2a;p  I a;p RL hðtÞ  I RL hðtÞ;

Let us consider k\0 and x is ii  gH differentiable (decreasing length)   ð1Þ xðt0 ÞgH x2 ðtÞ

"  a  2a # k tp  t0p k tp  t0p þ ¼ x0  a p Cða þ 1Þ p2a Cð2a þ 1Þ 2a;p  I a;p RL hðt Þ  I RL hðt Þ;

If it is proceeding to more n ! 1;  ia Z t 1 1 i1 p X X ki tp  t0p k ðt  sp Þia1 p1   hðsÞds xn ðtÞgH xðtÞ ¼ x0  s pia Cðia þ 1Þ pia1 CðiaÞ i¼1 i¼1 t0

 ia Z t 1 1 X X ki tp  t0p ki ðtp  sp Þia þ ða1Þ  sp1 ¼ x0   hðsÞds ia p Cðia þ 1Þ pia þ ða1Þ Cðia þ aÞ i¼1 i¼0 t0

 ia Z t 1 1 p p a1 X X k tp  t0p ki ðtp  sp Þia p1 ðt  s Þ   hðsÞds s ¼ x0  pia Cðia þ 1Þ pa1 pia Cðia þ aÞ i¼1 i¼0 i

t0

Let us consider k 0 and x is i  gH differentiable (increasing length),  p a  t  t0p xðtÞ ¼ x0 Ea;1 k p  p   Zt 1 t  sp a  a1 sp1 ðtp  sp Þa1 Ea;a k  hðsÞds p p t0

Let us consider k\0 and x is ii  gH differentiable (decreasing length),  p a  t  t0p xðtÞ ¼ x0 Ea;1 k p  p   Zt 1 t  sp a p1 p p a1 H ð1Þ a1 s ðt  s Þ Ea;a k  hðsÞds p p t0

4.2 Fuzzy Fractional Differential Equations—Caputo-Katugampola Derivative

145

Such that, the following function is called Mittag-Leffler function, Ea;b ðtÞ ¼

1 X i¼0

4.2.2

ti ; Cðia þ bÞ

a [ 0;

b[0

Some Properties of Mittag-Leffler Function

The basic Mittag-Leffler function is defined as, Ea ðtÞ ¼

1 X i¼0

If a ¼ 1; E 1 ðt Þ ¼

1 X i¼0

ti ; Cðia þ 1Þ

a[0

1 i X ti t ¼ ¼ et Cði þ 1Þ i! i¼1

Hence Ea ðtÞ is a generalization of exponential series. One generalization of Ea ðtÞ as a two-parameter generalization of is Ea ðtÞ is, Ea;b ðtÞ ¼

1 X i¼0

ti ; Cðia þ bÞ

a [ 0;

b[0

c A three-parameter generalization of Ea ðtÞ is denoted by Ea;b ðtÞ and is defined as, c Ea;b ðtÞ ¼

1 X i¼0

ð cÞ i ; i!Cðia þ bÞ

a [ 0;

b[0

where ðcÞi is the Pochhammer symbol standing for ðcÞi ¼ cðc þ 1Þðc þ 2Þ. . .ðc þ i  1Þ;

ðcÞ0 ¼ 1;

c 6¼ 0

Here, no other condition on ðcÞi , Here c could be a negative integer also. In that case the series is going to terminate into a polynomial. But, if ðcÞi is to be written in terms of a gamma function as ð cÞ i ¼

Cðc þ iÞ ; CðcÞ

c[0

146

4 Fuzzy Fractional Differential Equations

If more parameters are to be incorporated, then we can consider c1 ;c2 ;...;cp ð xÞ Ea;b;d 1 ;...;dq

  1 ðc Þ c X xi 1 i  p i ; ¼ Cðia þ bÞ i¼0 i!ðd1 Þi dq i

a [ 0;

b[0

where dj 6¼ 0; 1; 2; . . .;

j ¼ 1; 2; . . .; q:

No other restrictions on c1 ; . . .; cp and d1 ; . . .; dq are there other than the conditions for the convergence of the series. A dj can be a negative integer provided there is a cr , a negative integer such that ðcr Þk ¼ 0 first before ðdr Þk ¼ 0, such as c2 ¼ 3 and d1 ¼ 5 so that ðc2 Þ4 ¼ 0 and ðd1 Þ4 6¼ 0. Example Consider the following fuzzy fractional initial value problem with p ¼ 1; 1

k ¼ 1;

;1

2 DCK xðtÞ ¼ xðtÞ; gH

1 a¼ ; 2

hðtÞ ¼ 0;

t 2 ð0; 1

xðt0 ; r Þ ¼ x0 ðr Þ ¼ ð2; 2  r Þ 1

;1

1

;1

a1

2 2 2 Such that xðtÞ 2 Cððt0 ; T ; FR Þ: Since I RL DCK xðtÞ ¼ xðtÞgH x0 ¼ I RL xðtÞ and gH it has increasing length then we can use the successive approximation method,

a;1

xn ðtÞgH x0 ¼ I RL2 xn1 ðtÞ;

n ¼ 1; 2; . . .;

The i  gH differentiable solution (with increasing length) is as, 0 1 Zt  1 2 2 2 xðtÞ ¼ x0  E12;1 t2 ¼ x0  @1 þ pffiffiffi es dsAet p 0

Example Consider the following fuzzy fractional initial value problem with p ¼ 1; 1

k 2 f1; 1g;

1 a¼ ; 2

;1

2 DCK xðtÞ ¼ k  xðtÞ  hðtÞ; gH

hð t Þ ¼ c  t 2 ;

t 2 ð0; 1

c; x0 2 FR

Let us consider k ¼ 1 and x is i  gH differentiable,  1 Z t   1 1 xigH ðtÞ ¼ x0  Ea;1 t2  ðt  sÞ2 E12;12 ðt  sÞ2  c  s2 ds 0

4.2 Fuzzy Fractional Differential Equations—Caputo-Katugampola Derivative

147

where i 1  1 X t2 i ; E12;1 t2 ¼ i¼0 C 2 þ 1

0 1 i Zt 1   X 2 pffi 1 ð t  s Þ 2 1 2 i  ¼ t@1 þ pffiffiffi es dsA þ 1 E12;12 ðt  sÞ2 ¼ 1 p C 2þ2 C 2 i¼0 0

The solution is obtained as, 0 2 xigH ðtÞ ¼ x0  @1 þ pffiffiffi p

1

Zt e

s

2

2 dsAet 

0

Zt

  1 1 ðt  sÞ2 E12;12 ðt  sÞ2  c  s2 ds

0

Now let us consider k ¼ 1 and x is ii  gH differentiable, 0 2 xiigH ðtÞ ¼ x0  @1 þ pffiffiffi p  s2 ds

4.3

1

Zt e

s

2

2 dsAet H ð1Þ

0

Zt

  1 1 ðt  sÞ2 E12;12 ðt  sÞ2  c

0

Fuzzy Fractional Differential Equations—Laplace Transforms

In this section we suppose that the Laplace operator acts on a fuzzy number valued function and this is the reason we call it fuzzy Laplace transform. As like as before, let us consider the function x is a fuzzy number valued function and s is a positive real parameter. The fuzzy Laplace transform is defined as follows: Z1 X ðsÞ ¼ LðxðtÞÞ ¼

est  xðtÞdt;

s[0

0

Or Zs X ðsÞ ¼ LðxðtÞÞ ¼ lim

s!1 0

est  xðtÞdt;

s[0

148

4 Fuzzy Fractional Differential Equations

If we consider the Laplace operator in the level-wise form of xðtÞ as Lðxðt; r ÞÞ ¼ X ðs; r Þ ¼ ½Lðxl ðt; r ÞÞ; Lðxu ðt; r ÞÞ ¼ ½Xl ðs; r Þ; Xu ðs; r Þ So, the level-wise form of the Laplace operator is as follows, Zs X ðs; r Þ ¼ lim

s!1

est  xðt; r Þdt

0

And

2 ½Xl ðs; r Þ; Xu ðs; r Þ ¼ 4 lim

Zs

s!1

est xl ðt; r Þdt; lim

Zs

s!1

0

3 est xu ðt; r Þdt5

0

Then Zs Xl ðs; r Þ ¼ lim

s!1

est xl ðt; r Þdt

0

Zs Xu ðs; r Þ ¼ lim

s!1

est xu ðt; r Þdt

0

To define this operator the important condition is, the integral must converge to a real number. It means it should be bounded. However, there are some integrals that are not convergence. Example Suppose the fuzzy number valued function xðtÞ ¼ c  et 2 ; c 2 FR . Then Zs Xl ðs; r Þ ¼ lim

s!1

est cl ðr Þet 2 dt ! 0

0

Zs Xu ðs; r Þ ¼ lim

s!1

est cu ðr Þet 2 dt ! 0

0

The integral grows without bound for any s as s ! 1. If you remember we defined the absolute value of fuzzy number. Now in the same way we can define the absolute value of a fuzzy number valued function. The absolute value of the same function in level-wise form is defined as, jxðt; r Þj ¼ ½minfjxl ðt; r Þj; jxu ðt; r Þjg; maxfjxl ðt; r Þj; jxu ðt; r Þjg

4.3 Fuzzy Fractional Differential Equations—Laplace Transforms

149

It can be defined in two cases, Type I. Type 1 absolute value fuzzy number function jxðt; r Þj ¼ ½jxl ðt; r Þj; jxu ðt; r Þj In another word, if xl ðt; r Þ 0 for all r then x is type 1 absolute value fuzzy number function. Type II. Type 2 absolute value function jxðt; r Þj ¼ ½jxu ðt; r Þj; jxl ðt; r Þj In another word, if xu ðt; r Þ\0 for all r then x is type 2 absolute value fuzzy number function. Moreover, the other conditions of a fuzzy number in level-wise form should be satisfied. Note The absolute value of a fuzzy number function is always a positive fuzzy number valued function. Example Consider the fuzzy number function xðt; r Þ ¼ c½r et in the level-wise form where c½r  ¼ ½2 þ r; 4  r . As we know jxl ðt; r Þj ¼ jð2 þ r Þet j ¼ ð2 þ r Þet ;

jxu ðt; r Þj ¼ jð4  r Þet j ¼ ð4  r Þet

And jxðt; r Þj ¼ ½ð2 þ r Þet ; ð4  r Þet  Then then x is type 1 absolute value fuzzy number function. Example Consider the fuzzy number function xðt; r Þ ¼ c½r et in the level-wise form where c½r  ¼ ½4 þ r; 2  r . As we know jxl ðt; r Þj ¼ jð4 þ r Þet j ¼ ð4  r Þet ; jfu ðt; r Þj ¼ jð2  r Þet j ¼ ðr þ 2Þet And jxðt; r Þj ¼ ½ðr þ 2Þet ; ð4  r Þet  Then then x is type 2 absolute value fuzzy number function.

150

4.3.1

4 Fuzzy Fractional Differential Equations

Definition—Absolutely Convergence

The integral operator in the Laplace transformation, Zs X ðsÞ ¼ lim

s!1

est  xðtÞdt

0

Is said to be absolutely convergent if, Zs lim

s!1

jest  xðtÞjdt

0

Exists. Considering the level-wise form, both of the following integrals exist. Zs lim

s!1

est jxl ðt; r Þjdt; lim

Zs

s!1

0

est jxu ðt; r Þjdt

0

Theoretically, in order to apply the fuzzy Laplace transform to physical problems, it is necessary to involve the inverse transform. If X ðsÞ ¼ LðxðtÞÞ is the Laplace transform the L1 is called as inverse Laplace transform and we have L1 ðX ðsÞÞ ¼ xðtÞ;

t 0

where 1 L ðX ðsÞÞ ¼ 2pi 1

cZ þ i1

est  X ðsÞds;

c2R

ci1

As same as the Laplace transform the inverse transform is also a linear transform operator.

4.3.2

Definition—Exponential Order

A fuzzy function xðtÞ is said to be of exponential order a [ 0 on 0 t\1 if there exist positive constants K and T such that for all t [ T; DðxðtÞ; 0Þ Keat . Remark Consider the function xðtÞ is a fuzzy continuous or peace-wise continuous function on any finite interval, ðt0 ; T Þ, and of exponential order eat ,

4.3 Fuzzy Fractional Differential Equations—Laplace Transforms

0 DðX ðsÞ; 0Þ ¼ D@

1

Z1 e

st

 xðtÞdt; 0A

0

Z1

Z1

151

est  DðxðtÞ; 0Þdt:

0

est  Keat dt ¼ K

0

Z1

eðasÞt dt ¼

K sa

0

then the fuzzy Laplace transform exists for all s [ a:

4.3.3

Some Properties of Laplace

1. For two fuzzy functions x; y subject to Z1 Lða  xðtÞ  b  yðtÞÞ ¼

est  ða  xðtÞ  b  yðtÞÞdt

0

Z1 ¼a

e

st

Z1  xðtÞdt  b 

0

est  yðtÞdt

0

¼ a  LðxðtÞÞ  b  LðyðtÞÞ In level-wise form, • If a; b 0 Consider a  xðtÞ  b  yðtÞ ¼ hðtÞ hðt; r Þ ¼ ½a Lðxl ðt; r ÞÞ þ b Lðyl ðt; r ÞÞ; a Lðxu ðt; r ÞÞ þ b Lðyu ðt; r ÞÞ • a; b 0 hðt; r Þ ¼ ½a Lðxu ðt; r ÞÞ þ b Lðyu ðt; r ÞÞ; a Lðxl ðt; r ÞÞ þ b Lðyl ðt; r ÞÞ • a 0; b 0 hðt; r Þ ¼ ½a Lðxl ðt; r ÞÞ þ b Lðyu ðt; r ÞÞ; a Lðxu ðt; r ÞÞ þ b Lðyl ðt; r ÞÞ

152

4 Fuzzy Fractional Differential Equations

And also, LðxðtÞÞ ¼ X ðsÞ;

LðyðtÞÞ ¼ Y ðsÞ

Then L1 ða  X ðsÞ  b  Y ðsÞÞ ¼ a  L1 ðX ðsÞÞ  b  L1 ðY ðsÞÞ ¼ a  xð t Þ  b  yð t Þ 2. For any k 2 R Z1 Lðk  xðtÞÞ ¼ k 

est  xðtÞdt ¼ k  LðxðtÞÞ

0

In level-wise form, • k 0 Suppose, k  xðtÞ ¼ hðtÞ, Lðhðt; r ÞÞ ¼ ½kLðxl ðt; r ÞÞ; kLðxu ðt; r ÞÞ • k\0 hðt; r Þ ¼ ½kLðxu ðt; r ÞÞ; kLðxl ðt; r ÞÞ A generalized version of this property can be explained for non-negative function instead of k: Suppose, yðtÞ  xðtÞ ¼ hðtÞ, Z1 LðyðtÞ  xðtÞÞ ¼ yðtÞ 

est  xðtÞdt ¼ yðtÞ  LðxðtÞÞ

0

• yð t Þ 0 Lðhðt; r ÞÞ ¼ ½kyðtÞLðxl ðt; r ÞÞ; kyðtÞLðxu ðt; r ÞÞ

4.3 Fuzzy Fractional Differential Equations—Laplace Transforms

153

• yðtÞ\0 Lðhðt; r ÞÞ ¼ ½yðtÞLðxu ðt; r ÞÞ; yðtÞLðxl ðt; r ÞÞ Derivative property of Laplace Assume that xðtÞ is a fuzzy continuous function for t 0 and of exponential eat and x0gH ðtÞ is a peace-wise continuous in every finite closed interval. If the type of gH-differentiability does not change then for any s [ a we have, • i  gH differentiability   L x0igH ðtÞ ¼ s  X ðsÞH xð0Þ • ii  gH differentiability   L x0iigH ðtÞ ¼ ð1Þxð0ÞH ð1Þs  X ðsÞ To show the first item, according to the definition of Laplace, Zs   0 L xigH ðtÞ ¼ lim est  x0igH ðtÞdt s!1

0

Now we are going to investigate the result of integral. Since, the function est [ 0 and its derivative is negative, sest \0, then 0

00

ðest  xðtÞÞigH ¼ est  x0igH ðtÞH ð1Þðest Þ xðtÞ ¼ est  x0igH ðtÞH sest  xðtÞ Because, the level-wise form of the right-side is,

est x0l ðt; r Þ  sest  xl ðt; r Þ; est x0u ðt; r Þ  sest  xu ðt; r Þ

0 0 ¼ ðest xl ðt; r ÞÞ ; ðest xu ðt; r ÞÞ



 And ðest xl ðt; r ÞÞ0 ; ðest xu ðt; r ÞÞ0 is the level-wise form of ðest  xðtÞÞ0igH .

154

4 Fuzzy Fractional Differential Equations

Now, Zs ðe

st



0 xðtÞÞigH dt

Zs   ¼ est  x0igH ðtÞH sest  xðtÞ dt

0

0

Zs ¼

e

st



x0igH ðtÞdtH s

Zs 

0

est  xðtÞdt

0

On the other hand, Zs

0

ðest  xðtÞÞigH dt ¼ ess  xðsÞH xð0Þ

0

By substituting we have,

e

ss

Zs  xðsÞH xð0Þ ¼

e

st



x0igH ðtÞdtH s

Zs 

0

est  xðtÞdt

0

Based on the definition of H-difference, Zs e

st



x0igH ðtÞdt

¼e

ss

Zs  xðsÞH xð0Þ  s 

0

est  xðtÞdt

0

Zs   0 L xigH ðtÞ ¼ lim est  x0igH ðtÞdt s!1

0

0 ¼ lim @e s!1

ss

1

Zs  xðsÞH xð0Þ  s 

e

st

 xðtÞdtA

0

¼ lim ðess  xðsÞH xð0ÞÞ  s  lim s!1

Zs

s!1 0

¼ s  X ðsÞH xð0Þ Finally, the proof is completed.   L x0igH ðtÞ ¼ s  X ðsÞH xð0Þ

est  xðtÞdt

4.3 Fuzzy Fractional Differential Equations—Laplace Transforms

155

For proving the second item, again according to the definition of Laplace, Zs   0 L xiigH ðtÞ ¼ lim est  x0iigH ðtÞdt s!1

0

It is enough to find integral. Since, the function est [ 0 and its derivative is negative, sest \0, then 0

0

ðest  xðtÞÞiigH ¼ est  x0iigH ðtÞ  ðest Þ xðtÞ ¼ est  x0iigH ðtÞ  ð1Þsest  xðtÞ Because, the level-wise form of the right-side is,

est x0u ðt; r Þ  sest  xu ðt; r Þ; est x0l ðt; r Þ  sest  xl ðt; r Þ

0 0 ¼ ðest xu ðt; r ÞÞ ; ðest xl ðt; r ÞÞ



 And ðest xu ðt; r ÞÞ0 ; ðest xl ðt; r ÞÞ0 is the level-wise form of ðest  xðtÞÞ0iigH . Now, Zs ðe

st



0 xðtÞÞiigH dt

Zs   ¼ est  x0iigH ðtÞ  ð1Þsest  xðtÞ dt

0

0

Zs ¼

e

st



x0iigH ðtÞdt

Zs  ð1Þs 

0

est  xðtÞdt

0

On the other hand, Zs

0

ðest  xðtÞÞiigH dt ¼ ð1Þxð0ÞH ð1Þess  xðsÞ

0

By substituting we have, ð1Þxð0ÞH ð1Þess  xðsÞ Zs Zs st 0 ¼ e  xiigH ðtÞdt  ð1Þs  est  xðtÞdt 0

0

156

4 Fuzzy Fractional Differential Equations

Based on the definition of H-difference in type 2, Zs

est  x0iigH ðtÞdt

0

¼ ð1Þxð0ÞH ð1Þe

ss

Zs  xðsÞð1Þs 

est  xðtÞdt

0

Zs   0 L xiigH ðtÞ ¼ lim est  x0iigH ðtÞdt s!1

0

0 ¼ lim @ð1Þxð0ÞH ð1Þe s!1

ss

1

Zs  xðsÞð1Þs 

e

st

 xðtÞdtA

0

¼ ð1Þxð0Þð1Þs  X ðsÞ Finally, the proof is completed.   L x0iigH ðtÞ ¼ ð1Þxð0Þð1Þs  X ðsÞ

4.3.4

Convolution Theorem

If xðtÞ is a fuzzy peace-wise continuous function on ½0; 1 and of exponential order a, then Lððx yÞðtÞÞ ¼ LðxðtÞÞ  LðyðtÞÞ where yðtÞ is a peace-wise continuous real function on ½0; 1Þ. To prove, 0 LðxðtÞÞ  LðyðtÞÞ ¼ @

Z1

1

0

ess  xðsÞdsA  @

0

Z1 0

1 esr  yðrÞdrA

0 1 Z1 Z1 ¼ @ esðs þ rÞ  xðsÞdsA  yðrÞdr 0

0

4.3 Fuzzy Fractional Differential Equations—Laplace Transforms

157

Let us to hold s fixed in the interior integral, substituting t ¼ s þ r and dr ¼ dt, we obtain 0 1 Z1 Z1 LðxðtÞÞ  LðyðtÞÞ ¼ @ est  xðsÞ  yðt  sÞdtAds r

0

Z1 Z1 ¼ 0

Z1 ¼

r

est  xðsÞ  yðt  sÞdtds 0

est  @

0

Zt

1 xðt  rÞ  yðrÞdsAdr

0

¼ Lððx yÞðtÞÞ One of the important functions occurring in some electrical systems is the delay and it can be displayed as a unit step function like,  ua ðtÞ := uðt  aÞ ¼

1; 0;

t a t\a

For instance, in an electric circuit for a voltage at a particular time t ¼ a we write such a situation using unit step functions as Fig. 4.7. V ð t Þ ¼ uð t Þ  uð t  aÞ It is a shifted unit step. It is clear that uðtÞ ¼ uðt  aÞ ¼ 1 and V ðtÞ ¼ 0 for t a 0 and uðtÞ ¼ 1; uðt  aÞ ¼ 0 and V ðtÞ ¼ 1 for a [ t 0 (Fig. 4.8).

Fig. 4.7 Unit step function ua ðtÞ

158

4 Fuzzy Fractional Differential Equations

Fig. 4.8 Shifted unit step function V ðtÞ

4.3.5

First Translation Theorem

If X ðsÞ ¼ LðxðtÞÞ for s [ a then X ðs  aÞ ¼ Lðeat  xðtÞÞ such that a is a real number. The proof is clear from the definition of Laplace transform, Z1 X ð s  aÞ ¼

e

ðsaÞt

Z1  xðtÞdt ¼

0

4.3.6

est eat  xðtÞdt ¼ Lðeat  xðtÞÞ

0

Second Translation Theorem

If X ðsÞ ¼ LðxðtÞÞ for s [ a 0 then eas  X ðsÞ ¼ Lðua ðtÞ  xðt  aÞÞ According to the definition, Z1 Lðua ðtÞ  xðt  aÞÞ ¼

est ua ðtÞ  xðt  aÞdt

0

Since ua ðtÞ ¼ 0 for 0\t\a and ua ðtÞ ¼ 1 for t a then Z1 L ð ua ð t Þ  x ð t  aÞ Þ ¼ a

est  xðt  aÞdt

4.3 Fuzzy Fractional Differential Equations—Laplace Transforms

159

Let us suppose that t  a ¼ s Z1 e

st

 xðt  aÞdt ¼ e

a

sa

Z1 

ess  xðsÞds ¼ esa  X ðsÞ

a

Finally Lðua ðtÞ  xðt  aÞÞ ¼ esa  X ðsÞ

4.3.7

Remark—Laplace Forms of Fractional Derivatives

If the fuzzy function xðtÞ is continuous and integrable on ½t0 ; T  then its RL-gH and C-gH derivatives are explained as follows, RL-gH derivative,     L DaRLgH xðtÞ ¼ sa  LðxðtÞÞgH L Da1 x ð t Þ 0 RLgH where DaRLgH xðtÞ

1 d  ¼ Cð1  aÞ dt

Zt t0

xðsÞ ds; ð t  sÞ a

0\a\1

  Z1 a L DRLgH xðtÞ ¼ est  DaRLgH xðtÞdt 0

And       L DaRLgH xðt; r Þ ¼ L DaRL xl ðt; r Þ ; L DaRL xu ðt; r Þ Z1 ¼

 est DaRL xl ðt; r Þ; DaRL xu ðt; r Þ dt

0

2 ¼4

Z1 0

est DaRL xl ðt; r Þdt;

Z1 0

3 est DaRL xu ðt; r Þdt5

160

4 Fuzzy Fractional Differential Equations

Finally,

L



DaRL xl ðt; r Þ



Z1 ¼

est DaRL xl ðt; r Þdt

0

  L DaRL xu ðt; r Þ ¼

Z1

est DaRL xu ðt; r Þdt

0

• If xðtÞ is i  RLgH differentiable,     x ð t Þ L DaRLigH xðtÞ ¼ sa  LðxðtÞÞH L Da1 0 RLigH • If xðtÞ is ii  RLgH differentiable,     a L DaRLiigH xðtÞ ¼ ð1ÞL Da1 RLiigH xðt0 Þ H ð1Þs  Lðxðt ÞÞ It can be investigated in level-wise form very easily. • If xðtÞ is i  RLgH differentiable, The level-wise form of right side is,   sa  Lðxðt; r ÞÞH L Da1 RLigH xðt0 ; r Þ

  a  a1  ¼ sa Lðxl ðt; r ÞÞ  L Da1 RL xl ðt0 ; r Þ ; s Lðxu ðt; r ÞÞ  L DRL xu ðt0 ; r Þ  

    ¼ L DaRL xl ðt; r Þ ; L DaRL xu ðt; r Þ ¼ L DaRLigH xðt; r Þ Because       L DaRLigH xðt; r Þ ¼ L DaRL xl ðt; r Þ ; L DaRL xu ðt; r Þ • If xðtÞ is ii  RLgH differentiable, The level-wise form of right side is,   ð1ÞL Da1 x ð t ; r Þ H ð1Þsa  Lðxðt; r ÞÞ 0 RLiigH

  a  a1  a  ¼ L Da1 RL xl ðt0 ; r Þ þ s Lðxl ðt; r ÞÞ; L DRL xu ðt0 ; r Þ þ s Lðxu ðt; r ÞÞ  

    ¼ L DaRL xl ðt; r Þ ; L DaRL xu ðt; r Þ ¼ L DaRLigH xðt; r Þ

4.3 Fuzzy Fractional Differential Equations—Laplace Transforms

161

Because,       L DaRLigH xðt; r Þ ¼ L DaRL xl ðt; r Þ ; L DaRL xu ðt; r Þ Caputo-gH derivative Also some similar relation about the Laplace transform of Caputo-gH derivative can be obtained as follow, DaCgH xðtÞ

1  ¼ Cð1  aÞ

  L DaCgH xðtÞ ¼

Z1

Zt t0

x0gH ðsÞ ds ð s  sÞ a

est  DaCgH xðtÞdt

0

And       L DaCgH xðt; r Þ ¼ L DaC xl ðt; r Þ ; L DaC xu ðt; r Þ Z1 ¼

 est DaC xl ðt; r Þ; DaC xu ðt; r Þ dt

0

2 ¼4

Z1

est DaC xl ðt; r Þdt;

0

Z1

3 est DaC xu ðt; r Þdt5

0

Finally,   L DaC xl ðt; r Þ ¼

Z1

est DaC xl ðt; r Þdt

0

  L DaC xu ðt; r Þ ¼

Z1

est DaC xu ðt; r Þdt

0

• i  gH differentiable

 DaCigH xðs; r Þ ¼ DaC xl ðs; r Þ; DaC xu ðs; r Þ   L DaCigH xðtÞ ¼ sa  LðxðtÞÞH sa1 xðt0 Þ

162

4 Fuzzy Fractional Differential Equations

• ii  gH differentiable

 DaCiigH xðs; r Þ ¼ DaC xu ðs; r Þ; DaC xl ðs; r Þ   L DaCigH xðtÞ ¼ ð1Þsa1 xðt0 ÞH ð1Þsa  LðxðtÞÞ The process to show these two cases is similar to the RL-derivative. It can be shown by level-wise form. Laplace—Fuzzy fractional differential equation—RL derivative Consider the following fractional differential equation with fuzzy initial value and RL-derivative, (

DaRLgH xðtÞ ¼ f ðt; xðtÞÞ Da1 RLgH xðt0 Þ ¼ x0

The approach is, using the Laplace transform. To this end, we take the Laplace operator form both sides of the equation. So we have,  8  < L DaRL xðt; r Þ ¼ Lðf ðt; xðtÞÞÞ gH   : L Da1 xðt0 Þ ¼ Lðx0 Þ RLgH On the other hand,     L DaRLgH xðtÞ ¼ sa  LðxðtÞÞgH L Da1 RLgH xðt0 Þ So,

  8 < sa  LðxðtÞÞgH L Da1 RLgH xðt0 Þ ¼ Lðf ðt; xðt ÞÞÞ   : L Da1 xðt0 Þ ¼ Lðx0 Þ RLgH

Finally, sa  LðxðtÞÞgH Lðx0 Þ ¼ Lðf ðt; xðtÞÞÞ In two types of gH-difference, • i  gH difference sa  LðxðtÞÞH Lðx0 Þ ¼ Lðf ðt; xðtÞÞÞ

4.3 Fuzzy Fractional Differential Equations—Laplace Transforms

163

In the level-wise form,   sa Lðxl ðt; r ÞÞ  L x0;l ðr Þ ¼ Lðfl ðt; xðt; r ÞÞÞ   sa Lðxu ðt; r ÞÞ  L x0;u ðr Þ ¼ Lðfu ðt; xðt; r ÞÞÞ   sa Lðxl ðt; r ÞÞ ¼ L x0;l ðr Þ þ Lðfl ðt; xðt; r ÞÞÞ   sa Lðxu ðt; r ÞÞ ¼ L x0;u ðr Þ þ Lðfu ðt; xðt; r ÞÞÞ

• ii  gH difference Lðx0 Þ ¼ sa  LðxðtÞÞ  ð1ÞLðf ðt; xðtÞÞÞ In the level-wise form,   L x0;l ðr Þ ¼ sa Lðxl ðt; r ÞÞ  Lðfu ðt; xl ðt; r Þ; xu ðt; r ÞÞÞ   L x0;u ðr Þ ¼ sa Lðxu ðt; r ÞÞ  Lðfl ðt; xl ðt; r Þ; xu ðt; r ÞÞÞ   sa Lðxl ðt; r ÞÞ ¼ L x0;l ðr Þ þ Lðfu ðt; xl ðt; r Þ; xu ðt; r ÞÞÞ   sa Lðxu ðt; r ÞÞ ¼ L x0;u ðr Þ þ Lðfl ðt; xl ðt; r Þ; xu ðt; r ÞÞÞ where fl ðt; xl ðt; r Þ; xu ðt; r ÞÞ ¼ minff ðt; uÞju 2 xðt; r Þg fu ðt; xl ðt; r Þ; xu ðt; r ÞÞ ¼ maxff ðt; uÞju 2 xðt; r Þg Example Consider the following fractional differential equation with fuzzy initial value, (

DaRLgH xðtÞ ¼ k  xðtÞ; Da1 RLgH xðt0 Þ ¼ x0 2 FR

t 2 ½0; 1;

0\a\1

In general, we found that sa  LðxðtÞÞgH Lðx0 Þ ¼ Lðf ðt; xðtÞÞÞ Lðk  xðtÞÞ ¼ sa  LðxðtÞÞgH Lðx0 Þ Z1 Lðk  xðtÞÞ ¼ k  est  xðtÞdt ¼ k  LðxðtÞÞ 0

164

4 Fuzzy Fractional Differential Equations

Then k  LðxðtÞÞ ¼ sa  LðxðtÞÞgH Lðx0 Þ Case 1. k 0   sa Lðxl ðt; r ÞÞ ¼ L x0;l ðr Þ þ kLðxl ðt; r ÞÞ   sa Lðxu ðt; r ÞÞ ¼ L x0;u ðr Þ þ kLðxu ðt; r ÞÞ This is exactly the case i  gH difference. The solution can be obtained as,     ðsa  kÞLðxl ðt; r ÞÞ ¼ L x0;l ðr Þ ; ðsa  kÞLðxu ðt; r ÞÞ ¼ L x0;u ðr Þ Taking inverse Laplace, xl ðt; r Þ ¼ L1



 1 x0;l ðr Þ; sa  k

xl ðt; r Þ ¼ ta1 Ea;a ðkta Þx0;l ðr Þ;

xu ðt; r Þ ¼ L1



 1 x0;u ðr Þ sa  k

xu ðt; r Þ ¼ ta1 Ea;a ðkta Þx0;u ðr Þ

Case 2. k\0   sa Lðxl ðt; r ÞÞ ¼ L x0;l ðr Þ þ Lðkxu ðt; r ÞÞ   sa Lðxu ðt; r ÞÞ ¼ L x0;u ðr Þ þ Lðkxl ðt; r ÞÞ Since k\0 then   sa Lðxl ðt; r ÞÞ ¼ L x0;l ðr Þ þ kLðxl ðt; r ÞÞ   sa Lðxu ðt; r ÞÞ ¼ L x0;u ðr Þ þ kLðxu ðt; r ÞÞ For 0 r 1 the same solution is found. Example Consider the following fractional differential equation with fuzzy initial value, (

DaRLgH xðtÞ ¼ ð1Þ  xðtÞ þ t þ 1; Da1 RLgH xðt0 Þ ¼ x0 2 FR

t 2 ½0; 1;

0\a\1

In general, we found that Lðð1Þ  xðtÞ  t  1Þ ¼ sa  LðxðtÞÞgH Lðx0 Þ

4.3 Fuzzy Fractional Differential Equations—Laplace Transforms

165

On the other hand, based on the linearity property of the Laplace transform, Lðð1Þ  xðtÞ  t  1Þ ¼ ð1Þ  LðxðtÞÞ  LðtÞ  Lð1Þ By substituting, ð1Þ  LðxðtÞÞ  LðtÞ  Lð1Þ ¼ sa  LðxðtÞÞgH Lðx0 Þ

• i  gH difference ð1Þ  LðxðtÞÞ  LðtÞ  Lð1Þ ¼ sa  LðxðtÞÞH Lðx0 Þ In the level-wise form,    Lðxl ðt; r ÞÞ þ LðtÞ þ Lð1Þ ¼ sa Lðxl ðt; r ÞÞ  L x0;l ðr Þ    Lðxu ðt; r ÞÞ þ LðtÞ þ Lð1Þ ¼ sa Lðxu ðt; r ÞÞ  L x0;u ðr Þ Then   ð1 þ sa ÞLðxl ðt; r ÞÞ ¼ LðtÞ þ Lð1Þ þ L x0;l ðr Þ   ð1 þ sa ÞLðxu ðt; r ÞÞ ¼ LðtÞ þ Lð1Þ þ L x0;u ðr Þ   LðtÞ Lð1Þ 1 þ þ L x0;l ðr Þ ð1 þ sa Þ ð1 þ sa Þ ð1 þ sa Þ   LðtÞ Lð1Þ 1 þ þ L x0;u ðr Þ Lðxu ðt; r ÞÞ ¼ ð1 þ sa Þ ð1 þ sa Þ ð1 þ sa Þ

Lðxl ðt; r ÞÞ ¼

By taking inverse Laplace,      1 1 1 1 x0;l ðr Þ þ L þ L s2 ð1 þ sa Þ sð1 þ sa Þ 1 þ sa       1 1 1 1 x0;u ðr Þ xu ðt; r Þ ¼ L1 2 þ L þ L s ð1 þ sa Þ sð1 þ sa Þ 1 þ sa

xl ðt; r Þ ¼ L1

xl ðt; r Þ ¼ t

a1



a

Zt

Ea;a ðkt Þx0;l ðr Þ þ

ðt  uÞa1 Ea;a ðkðt  uÞa Þðu þ 1Þdu

0

xu ðt; r Þ ¼ t

a1

a

Zt

Ea;a ðkt Þx0;u ðr Þ þ 0

For 0 r 1:

ðt  uÞa1 Ea;a ðkðt  uÞa Þðu þ 1Þdu

166

4 Fuzzy Fractional Differential Equations

4.3.8

Fuzzy Fourier Transform Operator

Consider the function x : R ! FR is the fuzzy valued function. The fuzzy Fourier transform of xðtÞ denoted by ðF fxðtÞg : R ! FC Þ is given by the following integral, Z1 1 xðtÞ  eiwt dt ¼ F ðwÞ F fxðtÞg ¼ pffiffiffiffiffiffi 2p 1

Here FC is the set of all fuzzy nu90mbers on complex numbers. In the level-wise form, 2 1 F fxðt; r Þg ¼ 4pffiffiffiffiffiffi 2p 1 F fxl ðt; r Þg ¼ pffiffiffiffiffiffi 2p

Z1

1 xl ðt; r Þeiwt dt; pffiffiffiffiffiffi 2p

1

Z1 1

Z1

3 xu ðt; r Þeiwt dt5 ¼ F ðw; r Þ

1

1 xl ðt; r Þeiwt dt ; F fxu ðt; r Þg ¼ pffiffiffiffiffiffi 2p

Z1

xl ðt; r Þeiwt dt

1

where, F fxðt; r Þg ¼ ½F ðxl ðt; r ÞÞ; F ðxu ðt; r ÞÞ

4.3.9

Existence of Fourier Transform

Suppose that the fuzzy valued function xðtÞ is fuzzy absolutely integrable on ð1; 1Þ the fuzzy Fourier transform F fxðtÞg exists. Using the distance it will be proved and 0 1 DH ðF fxðtÞg; 0Þ ¼ DH @pffiffiffiffiffiffi 2p 1

pffiffiffiffiffiffi 2p

Z1

Z1

1 xðwÞ  eiwt dt; 0A

1



DH xðwÞ  e

iwt

1



1 ; 0 dt pffiffiffiffiffiffi 2p

Z1

 iwt  e DH ðxðtÞ; 0Þdt

1

Since jeiwt j ¼ 1, 1 DH ðF fxðtÞg; 0Þ pffiffiffiffiffiffi 2p The proof is completed.

Z1 DH ðxðtÞ; 0Þdt\1 1

4.3 Fuzzy Fractional Differential Equations—Laplace Transforms

167

Fuzzy inverse Fourier transform If F ðwÞ is the fuzzy Fourier transform of xðtÞ, then the fuzzy inverse Fourier transform of F ðwÞ is defined as, F

1

1 fF ðwÞg ¼ pffiffiffiffiffiffi 2p

Z1 xðwÞ  eiwt dw ¼ xðtÞ 1

In the same way we can show that if F ðwÞ fuzzy absolutely integrable then the fuzzy inverse Fourier transform F 1 fF ðwÞg exists. Similar to the fuzzy Laplace transforms, the fuzzy Fourier transformations are linear and this came from the linearity property of the fuzzy Riemann integral. Remark—Fuzzy Fourier transform of first derivative Let xðtÞ be fuzzy continuous, fuzzy absolutely integrable and converge to zero as jtj ! 0. Furthermore, let x0gH ðtÞ is fuzzy absolutely integrable on ð1; 1Þ. Then • i  gH differentiability n o F x0igH ðtÞ ¼ ðiwÞ  F fxðtÞg • ii  gH differentiability n o F x0iigH ðtÞ ¼ ð1ÞH ðiwÞ  F fxðtÞg Because, in case i  gH differentiability, F

n

o

x0igH ðtÞ

1 ¼ pffiffiffiffiffiffi 2p

Z1

x0igH ðtÞ  eiwt dt

1

0

Since eiwt [ 0 and ðeiwt Þ \0 then Zs



 x0igH ðtÞ  eiwt dt ¼ xðtÞ  eiwt s H xðtÞ  eiwt s

s

Zs  s



0 eiwt xðtÞdt

168

4 Fuzzy Fractional Differential Equations

So, 1 lim pffiffiffiffiffiffi s!1 2p

Zs

x0igH ðtÞ

e

iwt

s

1 dt ¼ ðiwÞ lim pffiffiffiffiffiffi s!1 2p

Zs

xðtÞ  eiwt dt

s

¼ ðiwÞ  F fxðtÞg Because, lim xðtÞ ¼ 0. s!1

In case ii  gH differentiability, Z1 n o 1 0 F xiigH ðtÞ ¼ pffiffiffiffiffiffi x0iigH ðtÞ  eiwt dt 2p 1

0

Since eiwt [ 0 and ðeiwt Þ \0 then Zs

x0igH ðtÞ  eiwt dt ¼

s



 ¼ xðtÞ  ð1Þeiwt s H xðtÞ  ð1Þeiwt s H

Zs

ðeiwt Þ0  xðtÞdt

s

So, 1 lim pffiffiffiffiffiffi s!1 2p

Zs

x0igH ðtÞ

e

iwt

s

1 dt ¼ ð1ÞH ðiwÞ lim pffiffiffiffiffiffi s!1 2p

Zs

xðtÞ  eiwt dt

s

¼ ð1ÞH ðiwÞ  F fxðtÞg Example If xðtÞ ¼ c  dðtÞ where d is a real Dirac function. Therefore 1 F fc  dðtÞg ¼ pffiffiffiffiffiffi 2p

Z1 c  dð t Þ  e 1

iwt

1 dt ¼ c  pffiffiffiffiffiffi 2p

Because 1 pffiffiffiffiffiffi 2p

Z1 1

dðtÞ  eiwt dt ¼ 1

Z1 1

dðtÞ  eiwt dt ¼ c

4.4 Fuzzy Solutions of Time-Fractional Problems

4.4

169

Fuzzy Solutions of Time-Fractional Problems

The main purpose of this section is to obtain an analytical solution for the time-fractional fuzzy equation. To do this, the time-fractional equation is transformed into an algebraic equation using the fuzzy Laplace and Fourier transforms. In this study, the fractional derivatives are described in the Caputo gH-differentiability. This section examines the explicit and fundamental solutions of the following fuzzy time fractional problem, DaCgH uðt; xÞ ¼ k 

@uðt; xÞ ; @x

t [ 0;

1\x\1

where 0 6¼ k 2 R; 0\a\1 and DaCgH uðt; xÞ

1  ¼ Cð1  aÞ

Zt

ð t  sÞ a 

t0

@uðs; xÞ ds @s

where pgH differentiability of the solution is defined as, @uðs; xÞ uðs þ h; xÞgH uðs; xÞ ¼ lim h!0 @s h Partial gH-differentiability The fuzzy number valued function of two variables uðt; xÞ 2 FR is called gH-partial differentiable p  gH at the point ðt; xÞ 2 D with respect to t and x and denoted by @tgH uðt; xÞ and @xgH uðt; xÞ if f ðt þ h; xÞgH f ðt; xÞ h f ðt; x þ k ÞgH f ðt; xÞ @xgH uðt; xÞ ¼ lim k!0 k

@tgH uðt; xÞ ¼ lim

h!0

Provided to both derivatives @tgH uðt; xÞ and @xgH uðt; xÞ are fuzzy number valued functions not fuzzy sets. Level-wise form of Partial gH-differentiability Suppose that the fuzzy number valued function uðt; xÞ 2 FR is p  gH differentiable at the point ðt; xÞ with respect to t and ul ðt; x; r Þ; uu ðt; x; r Þ are real valued functions and partial differentiable with respect to t. We say,

170

4 Fuzzy Fractional Differential Equations

• f ðt; xÞ is ði  p  gH Þ differentiable with respect to t at ðt; xÞ if @t;igH uðt; x; r Þ ¼ ½@t ul ðt; x; r Þ; @t uu ðt; x; r Þ

• f ðt; xÞ is ðii  p  gH Þ differentiable with respect to t at ðt; xÞ if @t;iigH uðt; x; r Þ ¼ ½@t uu ðt; x; r Þ; @t ul ðt; x; r Þ Please note that in each cases the conditions of the definition in level-wise form should be satisfied.

4.4.1

Fuzzy Explicit Solution of the Time-Fractional Problem

Now, we investigate the fuzzy explicit solution of the fuzzy linear partial fractional differential Equation with the following fuzzy boundary conditions, lim uðt; xÞ ¼ 0;

x! 1

uð 0 þ ; x Þ ¼ gð x Þ

Suppose that uðt; xÞ is the fuzzy explicit solution of fuzzy time fractional problem provided that the types of pgH differentiability with respect to t and x are the same. Consider the fuzzy Laplace transform of the solution with respect to t, for fixed x; Z1 Lt ðuðt; xÞÞ ¼

est  uðt; xÞdt ¼ Ut ðs; xÞ

0

The Laplace inverse of Ut ðs; xÞ with respect to first component is defined as, L1 s ðUt ðs; xÞÞ

1 ¼ 2pi

cZ þ i1

est  Ut ðs; xÞds; ci1

Such that L1 s Lt ðuðt; xÞÞ ¼ uðt; xÞ

c2R

4.4 Fuzzy Solutions of Time-Fractional Problems

171

Also consider fuzzy Fourier transform with respect to x for fixed t [ 0, 1 F x fuðt; xÞg ¼ pffiffiffiffiffiffi 2p

Z1

uðt; xÞ  eiwx dx ¼ U x ðt; wÞ

1

It inverse with respect to the second component is also as, F 1 w fU w ðt; wÞg

1 ¼ pffiffiffiffiffiffi 2p

Z1 U w ðt; wÞ  eiwt dw 1

Such that F 1 w F x ðuðt; xÞÞ ¼ uðt; xÞ Let LF denotes the space of all fuzzy number valued functions uðt; xÞ such that the fuzzy Laplace transform and the fuzzy Fourier transform exist with the following notation, Z1 1 F x Lt ðuðt; xÞÞ ¼ F x ðLt ðuðt; xÞÞÞ ¼ pffiffiffiffiffiffi Lt ðuðt; xÞÞ  eiwx dx 2p 1 0 1 Z1 Z1 1 @ est  uðt; xÞdtA  eiwx dx ¼ pffiffiffiffiffiffi 2p 1 ¼ pffiffiffiffiffiffi 2p

1 Z1

Z1

1

0

0

eðs þ wÞt  uðt; xÞdtdx := vðs; wÞ

Note Let gð xÞ be a fuzzy number valued function such that, F ðgð xÞÞ ¼ GðwÞ. If the fuzzy solution uðt; xÞ 2 LF of DaCgH uðt; xÞ ¼ k 

@uðt; xÞ @x

is i  gH Caputo differentiability with respect to t and i  gH partial differentiability with respect to x, then it satisfies the following relation,   1 a 1 1 a1 L1 s GðwÞ t F x ðs Þ  uðx; t ÞH ð1ÞkðiwÞ  uðx; tÞ ¼ Lt F x

172

4 Fuzzy Fractional Differential Equations

Subject to the integrals and difference exist. In this case if the fuzzy Laplace transform is applied for both sides,   Lt DaCigH uðt; xÞ ¼ sa  Lt ðuðt; xÞÞH sa1 gð xÞ Because uð0 þ ; xÞ ¼ gð xÞ   Lt k@xigH uðt; xÞ ¼ k@xigH Lt ðuðt; xÞÞ Since Z1 Lt ðuðt; xÞÞ ¼

est  uðt; xÞdt ¼ Ut ðs; xÞ

0

Then

  Lt k@xigH uðt; xÞ ¼ k@xigH Ut ðs; xÞ

Finally, sa  Lt ðuðt; xÞÞH sa1 gð xÞ ¼ k@xigH Ut ðs; xÞ Applying fuzzy Fourier to both sides,   sa  F x Lt ðuðt; xÞÞH sa1 F x ðgð xÞÞ ¼ kF x @xigH Ut ðs; xÞ Since,  F x @xigH Ut ðs; xÞ ¼ ðiwÞ  F x fUt ðs; xÞg By substituting, sa  F x Lt ðuðt; xÞÞH sa1 GðwÞ ¼ kðiwÞ  F fUt ðs; xÞg sa  F x Lt ðuðt; xÞÞH sa1 GðwÞ ¼ kðiwÞ  F x fLt ðuðt; xÞÞg Based on the definition of H-difference, sa  F x Lt ðuðt; xÞÞH ð1ÞkðiwÞ  F x Lt ðuðt; xÞÞ ¼ sa1 GðwÞ Now first, the inverse Fourier the inverse Laplace are applied, then   1 a 1 1 a1 L1 s GðwÞ t F x ðs Þ  uðx; t ÞH ð1ÞkðiwÞ  uðx; tÞ ¼ Lt F x

4.4 Fuzzy Solutions of Time-Fractional Problems

173

Now, assume the solution is ii  gH Caputo differentiability with respect to t and ii  gH partial differentiability with respect to x, then it satisfies the following relation, 1 a sa1  L1 t F x ðGðwÞÞ ¼ s  uðt; xÞH ðiwÞ  uðt; xÞ

Subject to the integrals and difference exist. In this case if the fuzzy Laplace transform is applied for both sides,   Lt DaCiigH uðt; xÞ ¼ ð1Þsa1  gð xÞH ð1Þsa  Lt ðuðt; xÞÞ Because uð0 þ ; xÞ ¼ gð xÞ   Lt k@xiigH uðt; xÞ ¼ k@xiigH Ut ðs; xÞ Finally, ð1Þsa1  gð xÞH ð1Þsa  Lt ðuðt; xÞÞ ¼ k  @xiigH Ut ðs; xÞ Applying fuzzy Fourier to both sides,   ð1Þsa1  F ðgð xÞÞH ð1Þsa  F Lt ðuðt; xÞÞ ¼ k  F @xiigH Ut ðs; xÞ Since,  F @xiigH Ut ðs; xÞ ¼ H ð1ÞðiwÞ  F fUt ðs; xÞg By substituting, ð1Þsa1  GðwÞH ð1Þsa  F x Lt ðuðt; xÞÞ ¼ H ð1ÞðiwÞ  F x fUt ðs; xÞg sa1  GðwÞH sa  F x Lt ðuðt; xÞÞ ¼ H ðiwÞ  F x Lt ðuðt; xÞÞ sa1  GðwÞH sa  vðs; wÞ ¼ H ðiwÞ  vðs; wÞ Based on the definition of H-difference, sa1  GðwÞ ¼ sa  F x Lt ðuðt; xÞÞH ðiwÞ  F x Lt ðuðt; xÞÞ Now first, the inverse Fourier the inverse Laplace are applied, 1 a sa1  L1 t F x ðGðwÞÞ ¼ s  uðt; xÞH ðiwÞ  uðt; xÞ

174

4.5

4 Fuzzy Fractional Differential Equations

Fuzzy Impulsive Fractional Differential Equations

In this section, the concept of fuzzy fractional impulsive differential equations is considered, and its solution is going to be determined under some conditions. In the next chapter several numerical methods will be introduced. To this end, consider the following fuzzy fractional differential equations with not instantaneous impulsive, or impulsive fractional differential equations with fuzzy initial value. 8 a;p < DCK gH xðtÞ ¼ f ðt; xðtÞÞ; t 2 ðtk ; sk ; k ¼ 0; 1; 2; . . .; m xðtÞ ¼ Ik ðt; xðtÞÞ; t 2 ðsk1 ; tk ; k ¼ 1; 2; . . .; m : xðt0 Þ ¼ x0 where p [ 0; 0\a\1 and the functions f : ½t0 ; T   FR ! FR and Ik : ðtk ; sk   FR ! FR are jointly continuous at the points t0 \s0 \t1 \s1 \tm \sm ¼ T. The fuzzy solution of the fuzzy impulsive problem xðtÞ is a peace-wise continuous function on ðtk ; sk  and ½t0 ; T  and Caputo-Katugampola fractional    generalized  differentiable. By this assumption, the values x tk and x tkþ exist and they are         equal, x tk ¼ x tkþ . Indeed, in general, x tkþ H x tk ¼ Ik ðt; xðtÞÞ. In the first discussion of Sect. 4.2, we discussed about the following remark about the equivalency of the solution of two differential and integral equations. In the other word, the continuous fuzzy function xðtÞ is the solution of Da;p CK gH xðtÞ ¼ f ðt; xðtÞÞ;

xðt0 Þ ¼ x0 ;

t 2 ½t0 ; T ;

0\a\1

If and only if, xðtÞ satisfies the following integral equation, p1a xðtÞgH xðt0 Þ ¼ CðaÞ

Zt

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds;

t 2 ½t0 ; T 

t0

Note Based on the definition of the gH-difference in the integral equation, xðtÞgH xðt0 Þ ¼

p1a CðaÞ

Zt

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds;

t 2 ½t0 ; T 

t0

• In case the gH-difference is i  gH difference, p1a xðtÞ ¼ xðt0 Þ  CðaÞ

Zt t0

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds;

t 2 ½t0 ; T 

4.5 Fuzzy Impulsive Fractional Differential Equations

175

• In case the gH-difference is ii  gH difference, p1a xðtÞ ¼ xðt0 ÞH ð1Þ C ð aÞ

Zt

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds;

t 2 ½t0 ; T 

t0

In a similar way and assuming the jointly continuity of xðtÞ, we can show the same corresponding relation for the following fractional equations. Note The jointly continuous and gH-differentiable fuzzy function xðtÞ is the solution of the following CK-fractional differential equations with fuzzy initial values 8 a;p < DCK gH xðtÞ ¼ f ðt; xðtÞÞ; t 2 ½t0 ; T  x ð t Þ ¼ x 1 2 FR ; t 1 [ t 0 : 1 x ð t 0 Þ ¼ x 0 2 FR If and only if, the same function xðtÞ is the solution of the following fuzzy fractional integral equation, p1a xðtÞgH x ðt1 Þ ¼ CðaÞ

Zt

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds;

t 2 ðt 0 ; T 

 a1 sp1 t1p  sp f ðs; xðsÞÞds;

t 2 ðt 0 ; T 

t0

where xðtÞgH x ðt1 Þ exists and p1a x ðt1 ÞgH x1 ¼ CðaÞ

Zt1 t0

To show the assertion, it is enough to take the RL-derivative both sides of Da;p CK gH xðtÞ ¼ f ðt; xðtÞÞ Since we have this relation in the previous Chap. 3, a;p I a;p RL DCK gH xðtÞ ¼ xðtÞgH xðt0 Þ

then xðtÞgH xðt0 Þ ¼ I a;p RL f ðt; xðt ÞÞ

176

4 Fuzzy Fractional Differential Equations

Also, we have, I a;p RL f ðt; xðtÞÞ

Zt

p1a  ¼ CðaÞ

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds

t0

Therefor, Zt

p1a  xðtÞgH xðt0 Þ ¼ CðaÞ

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds

t0

The proof is completed. Remark Based on the definition of gH-difference the following items can be concluded, • Case 1, xðtÞigH x ðt1 Þ and x ðt1 ÞigH x1 . p1a xðtÞ ¼ x ðt1 Þ  CðaÞ x ð t 1 Þ ¼ x1 

Zt t0

t1 1a Z

p CðaÞ

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds

 a1 sp1 t1p  sp f ðs; xðsÞÞds

t0

Then p1a xð t Þ ¼ x1  C ð aÞ 

t 1a Z

p CðaÞ

Zt1

 a1 sp1 t1p  sp f ðs; xðsÞÞds

t0

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds

t0

• Case 2, xðtÞigH x ðt1 Þ and x ðt1 ÞiigH x1 p1a xðtÞ ¼ x ðt1 Þ  C ð aÞ x ðt1 Þ ¼ x1 H ð1Þ

Zt

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds

t0

p1a CðaÞ

Zt1 t0

 a1 sp1 t1p  sp f ðs; xðsÞÞds

4.5 Fuzzy Impulsive Fractional Differential Equations

Then p1a xðtÞ ¼ x1 H ð1Þ C ð aÞ 

t 1a Z

p C ð aÞ

Zt1

 a1 sp1 t1p  sp f ðs; xðsÞÞds

t0

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds

t0

• Case 3, xðtÞiigH x ðt1 Þ and x ðt1 ÞigH x1 p1a xðtÞ ¼ x ðt1 ÞH ð1Þ C ð aÞ x ð t 1 Þ ¼ x1 

p1a CðaÞ

Zt1

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds

t0

 a1 sp1 t1p  sp f ðs; xðsÞÞds

t0

Then p1a xð t Þ ¼ x1  CðaÞ H ð1Þ

Zt

Zt1

 a1 sp1 t1p  sp f ðs; xðsÞÞds

t0

p1a CðaÞ

Zt

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds

t0

• Case 4, xðtÞiigH x ðt1 Þ and x ðt1 ÞiigH x1 p1a xðtÞ ¼ x ðt1 ÞH ð1Þ C ð aÞ x ðt1 Þ ¼ x1 H ð1Þ

Zt t0

t1 1a Z

p CðaÞ

t0

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds

 a1 sp1 t1p  sp f ðs; xðsÞÞds

177

178

4 Fuzzy Fractional Differential Equations

Then p1a xðtÞ ¼ x1 H ð1Þ C ð aÞ H ð1Þ

t 1a Z

p C ð aÞ

Zt1

 a1 sp1 t1p  sp f ðs; xðsÞÞds

t0

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds

t0

Now we are going to discuss the solution of extended conditions on our fuzzy impulsive fractional differential equation. Where p [ 0; 0\a\1 and the functions f : ½t0 ; T   FR ! FR and Ik : ðtk ; sk   FR ! FR are jointly continuous at the points t0 \s0 \t1 \s1 \tm \sm ¼ T. Remark With the same conditions on the jointly continuous fuzzy functions and for [ 0; 0\a\1; f : ½t0 ; T   FR ! FR , and Ik : ðtk ; sk   FR ! FR ; the peace-wise function xðtÞ is the solution of the fuzzy fractional impulsive equation, 8 a;p < DCK gH xðtÞ ¼ f ðt; xðtÞÞ; t 2 ðtk ; sk ; k ¼ 0; 1; 2; . . .; m xðtÞ ¼ Ik ðt; xðtÞÞ; t 2 ðsk1 ; tk ; k ¼ 1; 2; . . .; m : xðt0 Þ ¼ x0 If and only if, xðtÞ satisfies the following equations and conditions. 8 t 1a R > > > xðtÞgH x0 ¼ Cp ðaÞ sp1 ðtp  sp Þa1 f ðs; xðsÞÞds; t 2 ðt0 ; s0  > > < t0 xðtÞ ¼ Ik ðt; xðtÞÞ; t 2 ðsk1 ; tk  > t > R 1a > p p1 p p a1 > > : xðtÞgH x ðtk Þ ¼ CðaÞ s ðt  s Þ f ðs; xðsÞÞds; t 2 ðtk ; sk  t0

    where x tkþ H x tk ¼ Ik ðt; xðtÞÞ and p1a x ðtk Þ ¼ Ik ðtk ; xðtk ÞÞH CðaÞ

Ztk

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds

t0

  If length xðtÞgH x ðtk Þ is increasing function or xðtÞigH x ðtk Þ exists on ðtk ; sk . Also, Ztk p1a x ðtk Þ ¼ Ik ðtk ; xðtk ÞÞ  ð1Þ sp1 ðtp  sp Þa1 f ðs; xðsÞÞds CðaÞ t0

4.5 Fuzzy Impulsive Fractional Differential Equations

179

  If length xðtÞgH x ðtk Þ is decreasing function or xðtÞiigH x ðtk Þ exists on ðtk ; sk . Proof We know, the continuous fuzzy function xðtÞ is the solution of Da;p CK gH xðtÞ ¼ f ðt; xðtÞÞ;

xðt0 Þ ¼ x0 ;

t 2 ½t0 ; T ;

0\a\1

If and only if, xðtÞ satisfies the following integral equation, p1a xðtÞgH xðt0 Þ ¼ CðaÞ

Zt

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds;

t 2 ½t0 ; T 

t0

And on the interval ðs0 ; t1  we define the function xðt1 Þ ¼ I1 ðt1 ; xðt1 ÞÞ, so for t 2 ðt1 ; s1  we have the following impulsive equation, 

Da;p CK gH xðtÞ ¼ f ðt; xðtÞÞ; xðt1 Þ ¼ I1 ðt1 ; xðt1 ÞÞ

t 2 ðt 1 ; s1 

And it is equivalent to p1a xðtÞgH x ðt1 Þ ¼ CðaÞ

Zt

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds;

t 2 ðt1 ; s1 

t0

Where p1a x ðt1 Þ ¼ I1 ðt1 ; xðt1 ÞÞH CðaÞ 

Zt1

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds

t0



If length xðtÞgH x0 is increasing function or xðtÞigH x0 exists on ðt1 ; s1 . Also, Zt1 p1a x ðt1 Þ ¼ I1 ðt1 ; xðt1 ÞÞ  ð1Þ sp1 ðtp  sp Þa1 f ðs; xðsÞÞds CðaÞ t0

  If length xðtÞgH x0 is decreasing function or xðtÞiigH x0 exists on ðt1 ; s1 . Also, on the interval ðs1 ; t2  we define the function xðt2 Þ ¼ I2 ðt2 ; xðt2 ÞÞ, so for t 2 ðt2 ; s2  we have the following impulsive equation, 

Da;p CK gH xðtÞ ¼ f ðt; xðtÞÞ; xðt2 Þ ¼ I2 ðt2 ; xðt2 ÞÞ

t 2 ðt 2 ; s2 

180

4 Fuzzy Fractional Differential Equations

And it is equivalent to p1a xðtÞgH x ðt2 Þ ¼ CðaÞ

Zt2

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds;

t 2 ðt2 ; s2 

t0

Where p1a x ðt2 Þ ¼ I2 ðt2 ; xðt2 ÞÞH CðaÞ 

Zt2

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds

t0



If length xðtÞgH x0 is increasing function or xðtÞigH x0 exists on ðt2 ; s2 . Also, Zt2 p1a sp1 ðtp  sp Þa1 f ðs; xðsÞÞds x ðt2 Þ ¼ I2 ðt2 ; xðt2 ÞÞ  ð1Þ CðaÞ t0

  If length xðtÞgH x0 is decreasing function or xðtÞiigH x0 exists on ðt2 ; s2 . By proceeding the process for t 2 ðsk1 ; tk ; k ¼ 3; 4; . . .; m, xðtk Þ ¼ Ik ðtk ; xðtk ÞÞ, so for t 2 ðtk ; sk  we have the following impulsive equation, 

Da;p CK gH xðtÞ ¼ f ðt; xðtÞÞ; xðtk Þ ¼ Ik ðtk ; xðtk ÞÞ

t 2 ðt k ; sk 

And it is equivalent to p1a xðtÞgH x ðtk Þ ¼ CðaÞ

Ztk

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds;

t 2 ðtk ; sk 

t0

where p1a x ðtk Þ ¼ Ik ðtk ; xðtk ÞÞH CðaÞ

Ztk

sp1 ðtp  sp Þa1 f ðs; xðsÞÞds

t0

  If length xðtÞgH x0 is increasing function or xðtÞigH x0 exists on ðtk ; sk . Also, Ztk p1a x ðtk Þ ¼ Ik ðtk ; xðtk ÞÞ  ð1Þ sp1 ðtp  sp Þa1 f ðs; xðsÞÞds CðaÞ t0





If length xðtÞgH x0 is decreasing function or xðtÞiigH x0 exists on ðtk ; sk . Thus, the proof is completed.

4.6 Concrete Solution of Fractional Differential Equations

4.6

181

Concrete Solution of Fractional Differential Equations

In this section, the fuzzy linear fractional differential equations under Riemann– Liouville gH-differentiability as the following fuzzy initial value problems are studied. (

DaRLgH xðtÞ ¼ k  xðtÞ  yðtÞ; Da1 RLgH xðt0 Þ ¼ x0 2 FR

t 2 ½to ; T ;

0\a\1

where xðtÞ; is continuous fuzzy number valued function that belongs to the space of all Lebesque integrable fuzzy number valued functions on ½t0 ; T : Also yðtÞ 2 FR . To this end, some of the previous results on solutions of these equations are concreted. The new solutions by using the fractional hyperbolic functions and their properties are obtained, in details.

4.6.1

Fractional Hyperbolic Functions

Here, the fractional hyperbolic functions and their properties that will be used in the next sections are pointed out. As it is mentioned in the chapter three, the Mittag-Leffler function frequently used in the solutions of fractional order systems and it is defined as, Ea;b ðtÞ ¼

1 X

tk ; Cðak þ bÞ k¼0

a [ 0;

b[0

The fractional hyperbolic functions are defined as, cos ha;b ðtÞ ¼ sin ha;b ðtÞ ¼

1 X

  t2k ¼ E2a;b t2 ; C ð 2ak þ b Þ k¼0

1 X

a [ 0;

  t2k þ 1 ¼ tE2a;a þ b t2 ; C ð 2ak þ a þ b Þ k¼0

b[0 a [ 0;

b[0

where cos ha;b ðtÞ 0 and even function for all t 2 R and sin ha;b ðtÞ is an odd function for all t 2 R and, 

sin ha;b ðtÞ 0; t 0 cos ha;b ðtÞ\0; t\0

Note Please note that in case, a ¼ b, cos ha;a ðtÞ := Cha ðtÞ and sin ha;a ðtÞ := Sha ðtÞ, and for all t 2 R and 0\a 1, the function Ea;a ðtÞ [ 0:

182

4 Fuzzy Fractional Differential Equations

Property It is easy to see that, Cha ðtÞ þ Sha ðtÞ ¼ Ea;a ðtÞ; Cha ðtÞ  Sha ðtÞ ¼ Ea;a ðtÞ     cos hðtÞ ¼ E2;1 t2 ¼ Ch1 ðtÞ; sin hðtÞ ¼ tE2;2 t2 ¼ Sh1 ðtÞ

4.6.2

Some Derivation Rules for the Fractional Hyperbolic Functions

Suppose that xðtÞ 2 ½t0 ; T ; k 2 R and c a we have,  ac1 t cos ha;ac ðkta Þ; a [ c • DcRLgH ðta1 Cha ðkta ÞÞ ¼ a1 kt Sh ðkta Þ; a¼c  ac1 a a t sin ha;ac ðkt Þ; a [ c • DcRLgH ðta1 Sha ðkta ÞÞ ¼ kta1 Cha ðkta Þ; a¼c The proofs are straightforward, based on the definition of RL-derivative, DaRLgH xðtÞ

1 d  ¼ Cð1  aÞ dt

Zt t0

xð sÞ ds ð t  sÞ a

Now, DaRLgH

 a1  t Cha ðkta Þ ¼

1 d  Cð1  cÞ dt 1 X

Zt t0

sa1 Cha ðksa Þ ds ðt  sÞc !

2k 2ka þ ac

d k t dt k¼0 Cð2ka þ a  c þ 1Þ 8P 1 2k 2ka þ ac1 k t > >  ac1 < Cð2ka þ acÞ ; a [ c cos ha;ac ðkta Þ; a [ c t k¼0 ¼ ¼ 1 a1 P k2k þ 2 t2ka þ 2a1 > kt Sha ðkta Þ; a¼c > : Cð2ka þ 2aÞ ; a ¼ c ¼

k¼0

The proof of the second case is similar to the first case. Note As an immediate results, DaRLgH

 a1  t Cha ðkta Þ ¼



tac1 Ea;ac ðkta Þ; kta1 Ea;a ðkta Þ;

a[c a¼c

4.6 Concrete Solution of Fractional Differential Equations

183

Now, we are going to find the concrete solution of our mentioned fractional differential equation. Consider, (

DaRLgH xðtÞ ¼ k  xðtÞ  yðtÞ; Da1 RLgH xðt0 Þ ¼ x0 2 FR

t 2 ½to ; T ;

0\a\1

Regarding to the definition of gH-differentiability, we have two types of differentiability and also two types of solutions, i  RLgH solution and ii  RLgH solution. Here, our discussion and strategy is using length function. First, we are going to cover the concept of types of differentiability in accordance with the definition of length function. Remark

  • length DaRLgH xðtÞ ¼ DaRLgH lengthðxðtÞÞ iff xðtÞ is i  RLgH differentiable   • length DaRLgH xðtÞ ¼ DaRLgH lengthðxðtÞÞ iff xðtÞ is ii  RLgH differentiable

To show the propositions, first we suppose that xðtÞ is i  RLgH differentiable, then DaRL lengthðxðtÞÞ ¼ DaRL ðxu ðt; r Þ  xl ðt; r ÞÞ ¼ DaRL xu ðt; r Þ  DaRL xl ðt; r Þ   ¼ DaRLu xðt; r Þ  DaRLl xðt; r Þ ¼ length DaRLgH xðtÞ And if xðtÞ is ii  RLgH differentiable, then DaRL lengthðxðtÞÞ ¼ DaRL ðxu ðt; r Þ  xl ðt; r ÞÞ ¼ DaRL xu ðt; r Þ  DaRL xl ðt; r Þ   ¼ DaRLl xðt; r Þ  DaRLu xðt; r Þ ¼ length DaRLgH xðtÞ Now converse,    length DaRL xðtÞ ¼ max DaRL xl ðt; r Þ; DaRL xu ðt; r Þ  min DaRL xl ðt; r Þ; DaRL xu ðt; r Þ 0 If you suppose that  max DaRL xl ðt; r Þ; DaRL xu ðt; r Þ ¼ DaRL xu ðt; r Þ  min DaRL xl ðt; r Þ; DaRL xu ðt; r Þ ¼ DaRL xl ðt; r Þ   Thus, DaRL xu ðt; r Þ  DaRL xl ðt; r Þ 0 and it means length DaRLgH xðtÞ 0 and it points out the differential is defined as i  RLgH differentiability.

184

4 Fuzzy Fractional Differential Equations

If you suppose that  max DaRL xl ðt; r Þ; DaRL xu ðt; r Þ ¼ DaRL xl ðt; r Þ  min DaRL xl ðt; r Þ; DaRL xu ðt; r Þ ¼ DaRL xu ðt; r Þ   Thus, DaRL xu ðt; r Þ  DaRL xl ðt; r Þ 0 and it means length DaRLgH xðtÞ 0 and it points out the differential is defined as ii  RLgH differentiability.   Concerning, length DaRLgH xðtÞ ¼ 0 at some point t, it can be concluded that the

derivative must be a scalar, DaRLgH xðtÞ ¼ fcg: Therefor,

  DaRL lengthðxðtÞÞ ¼ 0 ¼ length DaRLgH xðtÞ To discuss the solution of 

DaRL xðtÞ ¼ k  xðtÞ  yðtÞ; Da1 RLgH xðt0 Þ ¼ x0 2 FR

t 2 ½to ; T ;

0\a\1

The sign of k does have important role. If it is positive k [ 0 we have i  RLgH solution and in case k\0 we have ii  RLgH solution. • i  RLgH solution for k [ 0 xð t Þ ¼ t

a1

Zt

a

Ea;a ðkt Þ  x0 

ðt  sÞa1 Ea;a ðkðt  sÞa Þ  yðsÞds

t0

• ii  RLgH solution for k\0 xðtÞ ¼ ta1 Ea;a ðkta Þ  x0 H ð1Þ

Zt

ðt  sÞa1 Ea;a ðkðt  sÞa Þ  yðsÞds

t0

To show that xðtÞ is the solution, we use the length function in each case. Since the Mittag-Leffler function is positive for 0\a\1 then • i  RLgH solution for k [ 0 lengthðxðtÞÞ ¼ ta1 Ea;a ðkta Þlengthðx0 Þ Zt þ ðt  sÞa1 Ea;a ðkðt  sÞa ÞlengthðyðsÞÞds t0

4.6 Concrete Solution of Fractional Differential Equations

185

• ii  RLgH solution for k\0 lengthðxðtÞÞ ¼ ta1 Ea;a ðkta Þlengthðx0 Þ Zt H ð1Þ ðt  sÞa1 Ea;a ðkðt  sÞa ÞlengthðyðsÞÞds t0

Using the RL-derivative we have, • i  RLgH solution for k [ 0 DaRLgH lengthðxðtÞÞ Zt þk

ðt  sÞa1 Ea;a ðkðt  sÞa ÞlengthðyðsÞÞds

t0

þ lengthðyðtÞÞ ¼ klengthðyðsÞÞ þ lengthðyðtÞÞ   ¼ lengthðk  xðtÞ  yðtÞÞ ¼ length DaRLgH xðtÞ

• ii  RLgH solution for k\0

¼ kt

a1

a

Zt

Ea;a ðkt Þlengthðx0 Þ þ k

ðt  sÞa1 Ea;a ðkðt  sÞa ÞlengthðyðsÞÞds

t0

 lengthðyðtÞÞ ¼ klengthðyðsÞÞ  lengthðyðtÞÞ ¼ ðklengthðyðsÞÞ þ lengthðyðtÞÞÞ ¼ lengthðk  xðtÞ  yðtÞÞ   ¼ length DaRLgH xðtÞ The proof is completed. Note Please notice that, if the H-difference in ii  RLgH solution does not exist then, based on the above mentioned results, we cannot say anything about the existence of solution for the problem in case k\0. Furthermore, the behavior of the solution function should reflect the real behavior of a system. Therefore, if the solution of the equation is not unique, then we may sometimes choose a better solution between two solutions, for example, we can study the real system and choose the solution which has better reflects from behavior of the system. So it would be better to find the i  RLgH solution for k\0 and ii  RLgH solution for k [ 0. To this end, we first, search the solution functions as the following result.

186

4 Fuzzy Fractional Differential Equations

Remark If in the problem (

DaRLgH xðtÞ ¼ k  xðtÞ  yðtÞ; Da1 RLgH xðt0 Þ ¼ x0 2 FR

t 2 ½to ; T ;

0\a\1

The initial condition is a symmetric fuzzy number, x0;l ðr Þ ¼ x0;u ðr Þ and also suppose yðtÞ ¼ gð xÞ  x0 where gð xÞ; is a real continuous function on ½t0 ; T . Then • i  RLgH solution for k\0

x1 ð t Þ ¼ t

a1

a

Zt

ðt  sÞa1 Ea;a ðkðt  sÞa Þ  yðsÞds

Ea;a ðkt Þ  x0  t0

• ii  RLgH solution for k [ 0

x2 ð t Þ ¼ t

a1

Zt

a

Ea;a ðkt Þ  x0 H ð1Þ

ðt  sÞa1 Ea;a ðkðt  sÞa Þ  yðsÞds

t0

Provided that the H-difference exists. In fact, it seems that x1 ðtÞ is i  RLgH differentiable for k [ 0 or k\0 and x2 ðtÞ is ii  RLgH differentiable for k\0 or k [ 0. It is so easy to verify that xi ðt; r Þ ¼ ½xl ðt; r Þ; xu ðt; r Þ; i ¼ 1; 2 is symmetric because x0 is symmetric. The level-wise form of i  RLgH solution, x1;l ðt; r Þ ¼ t

a1

a

Zt

Ea;a ðkt Þ  x0;l ðr Þ 

ðt  sÞa1 Ea;a ðkðt  sÞa Þ  yl ðs; r Þds

t0

x1;u ðt; r Þ ¼ ta1 Ea;a ðkta Þ  x0;u ðr Þ 

Zt

ðt  sÞa1 Ea;a ðkðt  sÞa Þ  yu ðs; r Þds

t0

The corresponding differential equations on t 2 ½to ; T  are as, 

DaRL x1;l ðt; r Þ ¼ k  x1;l ðt; r Þ þ yl ðt; r Þ Da1 RL x1;l ðt0 ; r Þ ¼ x0;l ðr Þ  a DRL x1;u ðt; r Þ ¼ k  x1;u ðt; r Þ þ yu ðt; r Þ Da1 RL x1;u ðt0 ; r Þ ¼ x0;u ðr Þ

4.6 Concrete Solution of Fractional Differential Equations

187

Since xl ðt; r Þ ¼ xu ðt; r Þ then 8 a D x1;l ðt; r Þ ¼ k  x1;u ðt; r Þ þ yl ðt; r Þ > > < RL DaRL x1;u ðt; r Þ ¼ k  x1;l ðt; r Þ þ yu ðt; r Þ a1 x1;l ðt0 ; r Þ ¼ x0;l ðr Þ > > DRL : Da1 RL x1;u ðt0 ; r Þ ¼ x0;u ðr Þ We know that this a system of fractional differential equations and the solutions are the solution of the main problem when k\0. The same process can be done for the ii  RLgH solution, and we have 8 a D x2;l ðt; r Þ ¼ k  x2;u ðt; r Þ þ yl ðt; r Þ > > < RL DaRL x2;u ðt; r Þ ¼ k  x2;l ðt; r Þ þ yu ðt; r Þ x2;l ðt0 ; r Þ ¼ x0;l ðr Þ > Da1 > : RL Da1 RL x2;u ðt0 ; r Þ ¼ x0;u ðr Þ And the solution of this system is the solution of our problem in case k [ 0. The proof is completed. Note Here we mentioned that the initial condition is a symmetric fuzzy number, x0;l ðr Þ ¼ x0;u ðr Þ and also suppose yðtÞ ¼ gð xÞ  x0 with a real continuous function gð xÞ: Now assume that these two conditions are not met. In this situation, with considering three cases for k we should change the forms of x1 ðtÞ; x2 ðtÞ as follows, x1 ðtÞ ¼ ta1 ðCha ðkta Þ  x0 H Sha ðkta Þ  x0 Þ Zt  ðt  sÞa1 ðCha ðkðt  sÞa Þ  yðsÞH Sha ðkðt  sÞa Þ  yðsÞÞds t0

We connect the solution x1 ðtÞ to vector c ¼ ðc1 ; c2 ; c3 ; c4 Þ; jci j ¼ 1; i ¼ 1; 2; 3; 4: x1 ðt; cÞ ¼ ta1 ðc1 Cha ðkta Þ  x0 H c2 Sha ðkta Þ  x0 Þ Zt  ðt  sÞa1 ðc3 Cha ðkðt  sÞa Þ  yðsÞH c4 Sha ðkðt  sÞa Þ  yðsÞÞds t0

188

4 Fuzzy Fractional Differential Equations

This is true because, lengthðx1 ðtÞÞ ¼ lengthðx1 ðt; cÞÞ; x [ 0. It means that x1 ðt; cÞ is i  gH solution as well. Since k\0,   x1;l ðt; r; cÞ ¼ ta1 Cha ðkta Þðc1  x0 Þl ðr Þ  Sha ðkta Þðc2  x0 Þl ðr Þ Zt   þ ðt  sÞa1 Cha ðkðt  sÞa Þðc3  yðsÞÞl ðr Þ  Sha ðkðt  sÞa Þðc4  yðsÞÞl ðr Þ ds t0

Taking the derivative, DaRL x1;l ðt; r; cÞ    ¼ k ta1 Sha ðkta Þðc1  x0 Þl ðrÞ  Cha ðkta Þðc2  x0 Þl ðrÞ 9 Zt =   þ ðt  sÞa1 Sha ðkðt  sÞa Þðc3  yðsÞÞl ðrÞ  Cha ðkðt  sÞa Þðc4  yðsÞÞl ðr Þ ds ; t0

þ c3 yl ðt; r Þ

The same process can be done for x1;u ðtÞ. DaRL x1;u ðt; r; cÞ    ¼ k ta1 Sha ðkta Þðc1  x0 Þu ðrÞ  Cha ðkta Þðc2  x0 Þu ðr Þ 9 Zt =   þ ðt  sÞa1 Sha ðkðt  sÞa Þðc3  yðsÞÞu ðrÞ  Cha ðkðt  sÞa Þðc4  yðsÞÞu ðr Þ ds ; t0

þ c3 yu ðt; r Þ

Please note that, if c1 ¼ c3 ¼ 1; c2 ¼ c4 ¼ 1 then this is exactly the following system, 8 a D x1;l ðt; r Þ ¼ kx1;u ðt; r Þ þ yl ðt; r Þ > > < RL DaRL x1;u ðt; r Þ ¼ kx1;l ðt; r Þ þ yu ðt; r Þ x1;l ðt0 ; r Þ ¼ x0;l ðr Þ > Da1 > : RL Da1 RL x1;u ðt0 ; r Þ ¼ x0;u ðr Þ Indeed we proved the following theorem or here we call it remark. Remark If xðtÞ is i  RLgH solution for k\0 the it is as the following form, x1 ðtÞ ¼ ta1 ðCha ðkta Þ  x0 H Sha ðkta Þ  x0 Þ Zt  ðt  sÞa1 ðCha ðkðt  sÞa Þ  yðsÞ  Sha ðkðt  sÞa Þ  yðsÞÞds t0

4.6 Concrete Solution of Fractional Differential Equations

189

In case k [ 0 and ii  RLgH differentiability, similar to case 1 ðk\0Þ we have x2 ð t Þ ¼ t

a1

a

Zt

Ea;a ðkt Þ  x0 H ð1Þ

ðt  sÞa1 Ea;a ðkðt  sÞa Þ  yðsÞds

t0

And connecting the vector c ¼ ðc1 ; c2 ; c3 ; c4 Þ; jci j ¼ 1; i ¼ 1; 2; 3; 4 we have, x2 ðt; cÞ ¼ ta1 ðCha ðkta Þc1  x0 H Sha ðkta Þc2  x0 Þ Zt H ðt  sÞa1 ðCha ðkðt  sÞa Þc3  yðsÞH Sha ðkðt  sÞa Þc4  yðsÞÞds t0

Subject to the H-difference exists. Now by differentiating, DaRL x2;l ðt; r; cÞ    ¼ k ta1 Sha ðkta Þðc1  x0 Þl ðr Þ  Cha ðkta Þðc2  x0 Þl ðr Þ 9 Zt  = a1  a a Sha ðkðt  sÞ Þðc3  yðsÞÞl ðr Þ þ Cha ðkðt  sÞ Þðc4  yðsÞÞl ðr Þ ds þ ðt  sÞ ; t0

 c3 yl ðt; rÞ

And for the upper function, DaRL x2;u ðt; r; cÞ    ¼ k ta1 Sha ðkta Þðc1  x0 Þu ðr Þ  Cha ðkta Þðc2  x0 Þu ðrÞ 9 Zt =   þ ðt  sÞa1 Sha ðkðt  sÞa Þðc3  yðsÞÞu ðr Þ þ Cha ðkðt  sÞa Þðc4  yðsÞÞu ðrÞ ds ; t0

 c3 yu ðt; r Þ

Please note that, if c1 ¼ c4 ¼ 1; c2 ¼ c3 ¼ 1 then this is exactly the following system, 8 a D x2;l ðt; r Þ ¼ kx2;u ðt; r Þ þ yl ðt; r Þ > > < RL DaRL x2;u ðt; r Þ ¼ kx2;l ðt; r Þ þ yu ðt; r Þ x2;l ðt0 ; r Þ ¼ x0;l ðr Þ > Da1 > : RL Da1 RL x2;u ðt0 ; r Þ ¼ x0;u ðr Þ Indeed we proved the following theorem or here we call it remark.

190

4 Fuzzy Fractional Differential Equations

Remark If xðtÞ is ii  RLgH solution for k [ 0 the it is as the following form, x2 ðtÞ ¼ ta1 ðCha ðkta Þ  x0 H ð1ÞSha ðkta Þ  x0 Þ Zt H ð1Þ ðt  sÞa1 ðCha ðkðt  sÞa Þ  yðsÞH Sha ðkðt  sÞa Þ  yðsÞÞds t0

Note Please note that Cha ðkta Þ  x0 H ð1ÞSha ðkta Þ  x0 is always exists for k [ 0 because lengthðCha ðkta Þ  x0 Þ ¼ Cha ðkta Þ  lengthðx0 Þ Is greater than lengthðð1ÞSha ðkta Þ  x0 Þ ¼ Sha ðkta Þ  lengthðx0 Þ The reason is, Cha ðkta Þ  Sha ðkta Þ ¼ Ea;a ðkta Þ [ 0 The case k ¼ 0 is very easy and similar to the previous discussion in chapter three. The problem is as, (

DaRLgH xðtÞ ¼ yðtÞ; t 2 ½to ; T ; Da1 RLgH xðt0 Þ ¼ x0 2 FR

0\a\1

And the solutions, • i  RLgH solution x1 ð t Þ ¼ t

a1

Zt  x0 

ðt  sÞa1 yðsÞds

t0

• ii  RLgH solution x2 ð t Þ ¼ t

a1

Zt  x0 H ð1Þ t0

ðt  sÞa1 yðsÞds

4.6 Concrete Solution of Fractional Differential Equations

191

Example Consider the following fractional differential equation with fuzzy initial value, (

DaRLgH xðtÞ ¼ k  xðtÞ; Da1 RLgH xðt0 Þ ¼ x0 2 FR

t 2 ½0; 1;

0\a\1

The solution is as xðtÞ ¼ ta1 Ea;a ðkta Þ  x0 And DaRLgH xðtÞ ¼ kta1 Ea;a ðkta Þ  x0   length DaRLgH xðtÞ ¼ kta1 Ea;a ðkta Þ  lengthðx0 Þ ¼ klengthðxðtÞÞ So the solution i  LRgH differentiable if k 0 and ii  LRgH differentiable if k\0: Also to find i  LRgH for k\0 and ii  LRgH for k 0, Case 1. k\0 and i  RLgH differentiability x1 ðtÞ ¼ ta1 ðCha ðkta Þ  x0  Sha ðkta Þ  x0 Þ Case 2. k [ 0 and ii  RLgH differentiability x1 ðtÞ ¼ ta1 ðCha ðkta Þ  x0 H ð1ÞSha ðkta Þ  x0 Þ In special case, a ¼ 0:5; k ¼ 1 and D0:5 RLgH xð0Þ ¼ ð1 þ r; 3  r Þ then  pffi  pffi  1 1 ta1 Cha ðkta Þ ¼ pffiffiffiffiffi þ et erf  t  et erf t pt 2  pffi  pffi  1 ta1 Sha ðkta Þ ¼ et erf  t  et erf t 2 Example Consider the following fractional differential equation with fuzzy initial value, (

DaRLgH xðtÞ ¼ ð1Þ  xðtÞ þ t þ 1; Da1 RLgH xðt0 Þ ¼ x0 2 FR

t 2 ½0; 1;

0\a\1

192

4 Fuzzy Fractional Differential Equations

The i  RLgH solution is, x1 ðtÞ ¼ ta1 ðCha ðta Þ  x0 H Sha ðta Þ  x0 Þ Zt  ðt  sÞa1 ðCha ððt  sÞa Þ  ðs þ 1Þ  Sha ððt  sÞa Þ  ðs þ 1ÞÞds t0

Chapter 5

Numerical Solution of Fuzzy Fractional Differential Equations

5.1

Introduction

In this chapter, first, some numerical methods such as generalized fuzzy fractional Taylor’s expansion and its application entitled fuzzy fractional Euler’s method are presented for fuzzy-valued function in the sense of Caputo differentiability. Then the fuzzy impulsive fractional differential equations are going to be considered for numerically solving by some semi analytically methods. Also the fuzzy fractional differential equation is solved by applying some other derivatives such as ABC derivative by using numerical methods.

5.2

Preliminaries

To discuss the subject, we need some definitions and other concepts. Definition—Partial ordering For two fuzzy numbers A; B 2 FR , A4B , Al ðr Þ  Bl ðr Þ & Au ðr Þ  Bu ðr Þ And A  B , Al ðr Þ\Bl ðr Þ & Au ðr Þ\Bu ðr Þ For any 0  r  1. Properties—Partial ordering • A4B ) B4  A • A4B & B4A ) A ¼ B © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. Allahviranloo, Fuzzy Fractional Differential Operators and Equations, Studies in Fuzziness and Soft Computing 397, https://doi.org/10.1007/978-3-030-51272-9_5

193

194

5 Numerical Solution of Fuzzy Fractional Differential Equations

• If x; y : ½a; b ! FR & xðtÞ4yðtÞ then

Rb a

xðtÞdt4

Rb a

yðtÞdt

All the properties can be proved by using the level-wise form of fuzzy numbers and fuzzy number valued functions. To prove fuzzy mean value theorems for Riemann-Liouville integral, the fuzzy intermediate value theorem and fuzzy mean value theorem for integrals are needed, so we first prove these theorems. Remark Suppose that xðtÞ is a continuous fuzzy number valued function on ½a; b and there exists a fuzzy number c such that xðaÞ4c4xðbÞ then there exists at least c 2 ½a; b such that xðcÞ ¼ c. To prove, define S ¼ ftjt 2 ½a; b; xðtÞ4cg, this set is bounded above by b and non-empty. Then it does have a supremum and suppose c ¼ sup S. First, assume that xðcÞ  c then by definition, xl ðc; r Þ [ cl ðr Þ & xu ðc; r Þ [ cu ðr Þ Since x is continuous, 8 [ 0; 9d [ 0; 8xðjt  cj\d ) DH ðxðtÞ; xðcÞÞ\Þ Now, for a fixed r 2 ½0; 1 let  ¼ minfxl ðc; r Þ  cl ðr Þ; xu ðc; r Þ  cu ðr Þg then from the definition of Husdorff distance, jxl ðt; r Þ  xl ðc; r Þjh ) xl ðt; r Þixl ðc; r Þ    cl ðr Þ jxu ðt; r Þ  xu ðc; r Þjh ) xu ðt; r Þixu ðc; r Þ    cu ðr Þ Thus for any t  ðc  d; c þ dÞ it is deduced that xl ðt; r Þ [ cl ðr Þ and xu ðt; r Þ [ cu ðr Þ. On the other hand we know that there is no point in ðc  d; d such that xl ðt; r Þ [ cl ðr Þ; xu ðt; r Þ [ cu ðr Þ such that c ¼ sup S and this is contradiction. Therefore xðcÞ4c. Now consider xðcÞ  c, again by continuity and definition of distance we have, jxl ðt; r Þ  xl ðc; r Þj\ ) xl ðt; r Þ\xl ðc; r Þ þ   cl ðr Þ jxu ðt; r Þ  xu ðc; r Þj\ ) xu ðt; r Þ\xu ðc; r Þ þ   cu ðr Þ Thus for any t 2 ðc  d; c þ dÞ it is deduced that xl ðt; r Þ [ cl ðr Þ and xu ðt; r Þ [ cu ðr Þ. Thus c þ d2 2 S which is contradiction. Hence for any r 2 ½0; 1, xl ðt; r Þ [ cl ðr Þ;

xu ðt; r Þ [ cu ðr Þ ) xðcÞ k!  xgH ðaÞ I RL DCgH xðtÞ; a  x  n > > k¼1 > > > m1 > P ðtaÞk > ðk Þ a a > < xðaÞ H ð1Þ k!  xgH ðaÞ H ð1ÞI RL DCgH xðtÞ; n  x  r k¼1 ¼ > m1 > X > ðt  aÞk ðk Þ > 0 >  xgH ðaÞ H ð1ÞI aRL DaCgH xðtÞ; ð a Þ ð 1 Þ x ð a Þ

ð x  a Þ  x > H gH > k! > > k¼2 > > > : rxb

Case 5 If xðtÞ is i  gH differentiable before n 2 ½a; b and ii  gH differentiable after n, and if x0 ðtÞ is ii  gH differentiable before r 2 ½n; b and i  gH differenðk Þ tiable after r, moreover xgH ðtÞ; k ¼ 2; 3; . . .; m are ii  gH differentiable on ½a; b, then xðtÞ 8 m1 P ðtaÞk > ðk Þ a a > > xðaÞ H ð1Þ > k!  xgH ðaÞ H ð1ÞI RL DCgH xðtÞ; a  x  n > > k¼1 > > > m1 > P ðtaÞk > ðk Þ a a > < xðaÞ

k!  xgH ðaÞ I RL DCgH xðtÞ; n  x  r k¼1 ¼ > m1 > X > ðt  aÞk ðk Þ > 0 >  xgH ðaÞ H ð1ÞIaRL DaCgH xðtÞ; ð 1 Þ ð x  a Þ  x ð a Þ

x ð a Þ > H gH > k! > > k¼2 > > > : rxb ðmÞ

To prove all cases, we need to stablish the following relations, since xgH ðtÞ is continuous fuzzy number functions and x is gH-Caputo differentiable, then   a ma ðmÞ a ma ðmÞ m ðmÞ I aRL DaCgH xðtÞ ¼ I aRL I a RL xðt Þ ¼ I RL I RL xgH ðt Þ ¼ I RL I RL xgH ðt Þ ¼ I RL xgH ðtÞ

5.3 Fuzzy Fractional Taylor’s Expansion with Caputo gH-Derivative

203

ðk Þ

Case 1 Proof All derivatives xgH ; k ¼ 1; 2; . . .; m are i  gH differentiable. ðmÞ

ðm1Þ

m1 I aRL DaCgH xðtÞ ¼ I m RL xgH ðt Þ ¼ I RL xgH

ðm1Þ

ðtÞ H I m1 RL xgH

ð aÞ

Because ðm1Þ xgH ðtÞ

¼

ðm1Þ xgH ðaÞ

Zt

FR

ðmÞ

xgH ðsÞds a

In the operator form it is, Zs

ðmÞ

ðmÞ

ðm1Þ

xgH ðtÞdt ¼ I 1RL xgH ðtÞ ¼ xgH

FR

ðm1Þ

ðtÞ H xgH

ð aÞ

a

And ðm1Þ

Im1 RL xgH

ðm1Þ

ðtÞ H Im1 RL xgH

ðm2Þ ðtÞ Im2 RL xgH ðm3Þ

¼ I m3 RL xgH

ðm2Þ

ðaÞ ¼ I m2 RL xgH

ðm2Þ H Im2 ðaÞ RL xgH ðm3Þ

ðtÞ H I m3 RL xgH

ðm2Þ

ðtÞ H I m2 RL xgH

ðm1Þ

ðaÞ H I m1 RL xgH

ðm1Þ H I m1 ðaÞ RL xgH ðm2Þ

ðaÞ H I m2 RL xgH

ðm1Þ

ðaÞ H I m1 RL xgH

ðaÞ

By substituting then, ðm3Þ

I aRL DaCgH xðtÞ ¼ I m3 RL xgH

ðm3Þ

ðtÞ H I m3 RL xgH

ðm1Þ ð aÞ H I m1 RL xgH

ðm2Þ

ðaÞ H I m2 RL xgH

Proceeding straight forward, I aRL DaCgH xðtÞ ¼ xðtÞ H xðaÞ H I 1RL x0gH ðaÞ H I 2RL x00gH ðaÞ ðm1Þ

H    H I m1 RL xgH

ð aÞ

where Zt Zt Zt I kRL

:¼

Zt 

a

a

a

dsds    ds ¼ a

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ktimes

ð t  aÞ k k!

ð aÞ

ðaÞ

204

5 Numerical Solution of Fuzzy Fractional Differential Equations

So we found out, I aRL DaCgH xðtÞ ¼ xðtÞ H xðaÞ H ðt  aÞx0gH ðaÞ H

ðt  aÞ2 00 xgH ðaÞ 2!

ðt  aÞm1 ðm1Þ x ð aÞ ðm  1Þ! gH

H    H

And finally based on the definition of H-difference, x ð t Þ ¼ x ð aÞ

m1 X ð t  aÞ k

k!

k¼1

ðk Þ

 xgH ðaÞ I aRL DaCgH xðtÞ

ðk Þ

Case 2 Proof. All derivatives xgH ; k ¼ 1; 2; . . .; m are ii  gH differentiable. ðmÞ

ðm1Þ

m1 I aRL DaCgH xðtÞ ¼ I m RL xgH ðt Þ ¼ I RL xgH

ðm1Þ

ðtÞ H I m1 RL xgH

ð aÞ

Because in the operator form it is, Zs

ðmÞ

ðmÞ

ðm1Þ

ðtÞ H H ð1ÞxgH

ðm1Þ

ðtÞ ð1ÞxgH

xgH ðtÞdt ¼ I 1RL xgH ðtÞ ¼ H ð1ÞxgH

FR

ðm1Þ

ð aÞ

a

¼ H ð1ÞxgH

ðm1Þ

ð aÞ

Similar to the case (1), finally we get, ðm1Þ

I aRL DaCgH xðtÞ ¼ I 1RL x0gH ðtÞ H I 2RL x00gH ðaÞ H    H I m1 RL xgH

ð aÞ

Proceeding straight forward, I aRL DaCgH xðtÞ ¼ H ð1ÞxðtÞ ð1ÞxðaÞ H I 1RL x0gH ðaÞ H I 2RL x00gH ðaÞ ðm1Þ

H    H I m1 RL xgH

ð aÞ

Because, I 1RL x0gH ðtÞ

Zt ¼

x0gH ðsÞds ¼ ð H ð1ÞxðtÞÞ H ð H ð1ÞxðaÞÞ

a

¼ H ð1ÞxðtÞ ð1ÞxðaÞ

5.3 Fuzzy Fractional Taylor’s Expansion with Caputo gH-Derivative

205

where Zt Zt Zt I kRL

:¼

Zt 

a

a

a

dsds    ds ¼

ð t  aÞ k k!

a

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ktimes

Indeed we have, m1 X ðt  aÞk

I aRL DaCgH xðtÞ ¼ H ð1ÞxðtÞ ð1ÞxðaÞ H

k¼1

H ð1ÞxðtÞ ð1ÞxðaÞ ¼ I aRL DaCgH xðtÞ

ðk Þ

 xgH ðaÞ

k!

m1 X ð t  aÞ k

k!

k¼1

ðk Þ

 xgH ðaÞ

And finally based on the definition of gH-difference in type (2), xðtÞ ¼ xðaÞ H ð1Þ

m1 X ð t  aÞ k k¼1

k!

ðmÞ

 xgH ðaÞ H ð1ÞI aRL DaCgH xðtÞ

Case 3 If xðtÞ is i  gH differentiable and its higher-order derivatives are changing ð2kÞ in every other type periodically. In the other word, xgH ðtÞ; k ¼ 0; 1; . . .; m2 are ð2k1Þ

i  gH-differentiable and, xgH

ðtÞ; k ¼ 0; 1; . . .; m2 are ii  gH differentiable. ðmÞ

ðm1Þ

1 m1 I aRL DaCgH xðtÞ ¼ I m1 RL I RL xgH ðtÞ ¼ I RL xgH

ðm1Þ

ðtÞ H I m1 RL xgH

ðmÞ

By assumption xgH ðtÞ is in the i  gH sense and we have, Zt

ðmÞ

ðmÞ

ðm1Þ

xgH ðsÞds ¼ I 1RL xgH ðtÞ ¼ xgH

FR

ðm1Þ

ðtÞ H xgH

a

Thus h i ðm1Þ ðm1Þ I aRL DaCgH xðtÞ ¼ I m1 ðtÞ H xgH ðaÞ RL xgH ðm1Þ

¼ I m1 RL xgH

Since xðm1Þ ðtÞ is ii  gH differentiable,

ðm1Þ

ðtÞ H I m1 RL xgH

ðaÞ

ðaÞ

ð aÞ

206

5 Numerical Solution of Fuzzy Fractional Differential Equations ðm1Þ

I m1 RL xgH

ðm2Þ

ðtÞ ¼ H ð1ÞI m2 RL xgH

I aRL DaCgH xðtÞ

ðm1Þ H I m1 ð aÞ RL xgH

¼

ðm2Þ

ðtÞ ð1ÞI m2 RL xgH

ðm2Þ H ð1ÞI m2 ðt Þ RL xgH

ð aÞ ðm2Þ

ð1ÞI m2 RL xgH

ð aÞ

Since xðm2Þ ðtÞ is i  gH differentiable, ðm2Þ

I m2 RL xgH

ðm3Þ

ðtÞ ¼ I m3 RL xgH

ðm3Þ

ðtÞ H I m3 RL xgH

ð aÞ

Then ðm1Þ

I aRL DaCgH xðtÞ ¼ H I m1 RL xgH

ðm2Þ

ðaÞ ð1ÞI m2 RL xgH

ðm3Þ

ð1ÞI m3 RL xgH

ðm3Þ

ðaÞ H ð1ÞI m3 RL xgH

ðt Þ

ð aÞ

Since xðm3Þ ðtÞ is ii  gH differentiable, ðm3Þ

I m3 RL xgH

ðm4Þ

ðtÞ ¼ H ð1ÞI m4 RL xgH

ðm4Þ

ðtÞ ð1ÞI m4 RL xgH

ðaÞ

By substituting ðm1Þ

ðm2Þ

ðm3Þ

ðaÞ

ðm3Þ

ðaÞ

ðaÞ ð1ÞI m2 ðaÞ ð1ÞI m3 I aRL DaCgH xðtÞ ¼ H I m1 RL xgH RL xgH RL xgH h i ðm4Þ m4 ðm4Þ H ð1Þ H ð1ÞI m4 x ð t Þ

ð 1 ÞI x ð a Þ RL gH RL gH ðm1Þ

I aRL DaCgH xðtÞ ¼ H I m1 RL xgH

ðm4Þ

H I m4 RL xgH

ðm2Þ

ðaÞ ð1ÞI m2 RL xgH ðm4Þ

ðaÞ I m4 RL xgH

ðaÞ ð1ÞI m3 RL xgH

ðt Þ

Proceed straightforward and, ðm4Þ

I m4 RL xgH

ðm5Þ

ðtÞ ¼ I m5 RL xgH

I aRL DaCgH xðtÞ

¼

ðm5Þ

ðtÞ H I m5 RL xgH

ðm1Þ H I m1 ð aÞ RL xgH ðm4Þ

H I m4 RL xgH



ð aÞ

ðm2Þ ð1ÞI m2 ð aÞ RL xgH ðm5Þ

ðaÞ H I m5 RL xgH

ðm3Þ

ð1ÞI m3 RL xgH ðm5Þ

ðaÞ I m5 RL xgH

ð aÞ

ðt Þ

Finally, ðm1Þ

I aRL DaCgH xðtÞ ¼ H I m1 RL xgH I aRL DaCgH xðtÞ ¼ H

ðm2Þ

ðaÞ ð1ÞI m2 RL xgH

ðm3Þ

ðaÞ ð1ÞI m3 RL xgH

ð aÞ

ðt  aÞm1 ðt  aÞm2 ðm1Þ ðm2Þ  xgH ðaÞ ð1Þ  xgH ðaÞ ðm  1Þ! ðm  2Þ!

H    ð1Þ

ð t  aÞ 2 ð2Þ ð1Þ  xgH ðaÞ H ðt  aÞ  xgH ðaÞ xðtÞ H xðaÞ 2!

5.3 Fuzzy Fractional Taylor’s Expansion with Caputo gH-Derivative

207

Based on definition of H-difference, ð t  aÞ 2 ð2Þ  xgH ðaÞ 2! ðt  aÞm2 ðt  aÞm1 ðm2Þ ðm1Þ  xgH ðaÞ

 xgH ðaÞ

   H ð1Þ ðm  2Þ! ðm  1Þ! m1 m1 X X ð t  aÞ k ðt  aÞk ðk Þ ðk Þ xðtÞ ¼ xðaÞ H ð1Þ  xgH ðaÞ

 xgH ðaÞ k! k! k¼1;even k¼1;odd ð1Þ

xðtÞ H xðaÞ ¼ I aRL DaCgH xðtÞ ðt  aÞ  xgH ðaÞ H ð1Þ

I aRL DaCgH xðtÞ ðk Þ

Case 4 Now suppose that xgH ðtÞ; k ¼ 2; 3; . . .; m is i  gH differentiable, ðmÞ

ðm1Þ

m1 I aRL DaCgH xðtÞ ¼ I m RL xgH ðt Þ ¼ I RL xgH

ðm1Þ

ðtÞ H I m1 RL xgH

ð aÞ

By the same process in case 1, ð1Þ

ð1Þ

ð2Þ

ð2Þ

I aRL DaCgH xðtÞ ¼ I 1RL xgH ðtÞ H I 1RL xgH ðaÞ H I 2RL xgH ðtÞ H I 2RL xgH ðaÞ ðm1Þ

H    H I m1 RL xgH

ð aÞ

Since x0 ðtÞ is i  gH differentiable on ½a; r and ii  gH differentiable after ½r; b, ( ð2Þ I 2RL xgH ðtÞ

¼

ð1Þ

ð1Þ

I 1RL xgH ðtÞ H I 1RL xgH ðaÞ;

atr

ð1ÞI 1RL x1gH ðaÞ

rtb

ð1Þ H ð1ÞI 1RL xgH ðtÞ;

Also xðtÞ is ii  gH differentiable on ½a; n and i  gH differentiable after ½n; b, 8 < ð1ÞxðaÞ H ð1ÞxðtÞ; a  t  n ð1Þ I 1RL xgH ðtÞ ¼ xðtÞ H xðtÞ; ntr : xðtÞ H xðaÞ; rtb Substituting the recent equations in I aRL DaCgH xðtÞ the assertion of this case are obtained. In this section we are going to obtain a generalized Taylor’s expansion for fuzzy-valued functions by using the concept of Caputo generalized Hukuhara derivative. To this end, we need some results and definitions that are explained in the following. Remark Consider the fuzzy continuous function Dka CgH xðt Þ 2 FR in ða; bÞ for k ¼ 0; 1; . . .; n; such that 0\a  1. Then we have the following items,

208

5 Numerical Solution of Fuzzy Fractional Differential Equations ðn þ 1Þa

Case 1. If the derivatives Dna CgH xðtÞ; DCgH entiability for n  1, then ðn þ 1Þa

I aRL DCgH

xðtÞ are in the same types of differ-

na xðtÞ ¼ Dna CgH xðt Þ H DCgH xðaÞ ðn þ 1Þa

Case 2. If the derivatives Dna CgH xðt Þ; DCgH entiability for n  1, then ðn þ 1Þa

I aRL DCgH

xðtÞ are in different types of differ-

na xðtÞ ¼ ð1ÞDna CgH xðaÞ H ð1ÞDCgH xðtÞ

To show two cases, we have the following relation, ðn þ 1Þa

I aRL DCgH

    a na a a xðtÞ ¼ I aRL Dna D x ð t Þ ¼ D I D x ðtÞ ¼ Dna CgH CgH CgH RL CgH CgH xðtÞ     ðn1Þa a a ¼ DCgH DaCgH x ðtÞ ¼ Dna CgH I RL DCgH x ðtÞ

Since I aRL DaCgH xðtÞ ¼ xðtÞ gH xðaÞ     ðn þ 1Þa a a I aRL DCgH xðtÞ ¼ Dna I D x ð t Þ ¼ Dna CgH RL CgH CgH xðtÞ gH xðaÞ ðn þ 1Þa

I aRL DCgH

na xðtÞ ¼ Dna CgH xðtÞ gH DCgH xðaÞ

Two cases are obtained by the definition of gH-difference in two types. Remark Consider the fuzzy continuous function Dka CgH xðt Þ 2 FR in ða; bÞ for k ¼ 0; 1; . . .; n; such that 0\a  1. Then we have the following items, ðn þ 1Þa Case 1. If the derivatives Dna xðtÞ are in the same types of differCgH xðtÞ; DCgH entiability for n  1, then ðn þ 1Þa

na I na RL DCgH xðt Þ H I RL

ðn þ 1Þa

DCgH

xð t Þ ¼

ðn þ 1Þa

Case 2. If the derivatives Dna CgH xðt Þ; DCgH entiability for n  1, then ðn þ 1Þa

na I na RL DCgH xðt Þ ð1ÞI RL

ðn þ 1Þa

DCgH

Two cases can be proved as follows,

ðt  aÞna  Dna CgH xðaÞ Cðna þ 1Þ

xðtÞ are in different types of differ-

xð t Þ ¼

ðt  aÞna  Dna CgH xðaÞ Cðna þ 1Þ

5.3 Fuzzy Fractional Taylor’s Expansion with Caputo gH-Derivative ðn þ 1Þa

na I na RL DCgH xðt Þ gH I RL

ðn þ 1Þa

DCgH

209

  na na ðn þ 1Þa xðtÞ ¼ I na D x ð t Þ I D x ð t Þ gH RL CgH RL CgH

We proved in the previous remark, that is ðn þ 1Þa

I aRL DCgH

na xðtÞ ¼ Dna CgH xðtÞ gH DCgH xðaÞ

Substituting and ðn þ 1Þa

ðn þ 1Þa

na I na DCgH xðtÞ RL DCgH xðtÞ gH I RL    na na na na ¼ I na ¼ I na RL DCgH xðtÞ gH DCgH xðtÞ gH DCgH xðaÞ RL DCgH xðaÞ

¼

ðt  aÞna  Dna CgH xðaÞ Cðna þ 1Þ

So we proved, ðn þ 1Þa

na I na RL DCgH xðt Þ gH I RL

ðn þ 1Þa

DCgH

xð t Þ ¼

ðt  aÞna  Dna CgH xðaÞ Cðna þ 1Þ

In accordance with the definition of the gH-difference two cases are gotten.

5.3.2

Fuzzy Generalized Taylor’s Expansion

Let us consider the fuzzy continuous function Dka CgH xðtÞ 2 FR in ða; bÞ for k ¼ 0; 1; . . .; n; such that 0\a  1. Then we have the following items, Case 1 If xðtÞ is i  CgH differentiable of order ka then there exists n 2 ða; bÞ such that, xðtÞ ¼ xðaÞ

ðn þ 1Þa n X DCgH xðnÞ ðt  aÞka  ðt  aÞðn þ 1Þa  Dka x ð a Þ

CgH C ð ð n þ 1 Þa þ 1 Þ C ð ka þ 1 Þ k¼1

Case 2 If xðtÞ is ii  CgH differentiable of order ka then there exists n 2 ða; bÞ such that, xðtÞ ¼ xðaÞ H ð1Þ

ðn þ 1Þa n X DCgH xðnÞ ðt  aÞka  Dka x ð a Þ ð 1 Þ H CgH Cðka þ 1Þ Cððn þ 1Þa þ 1Þ k¼1

 ðt  aÞðn þ 1Þa

210

5 Numerical Solution of Fuzzy Fractional Differential Equations



Case 3 If xðtÞ is i  CgH differentiable of order 2ka; k ¼ 0; 1; . . .; n2 and also it is

ii  CgH differentiable of order ð2k  1Þa; k ¼ 0; 1; . . .; n2 then there exists n 2 ða; bÞ such that, xðtÞ ¼ xðaÞ H ð1Þ

n P k¼1;odd

ðn þ 1Þa

DC

ðtaÞka Cðka þ 1Þ

 Dka CgH xðaÞ

n P k¼1;even

ðtaÞka Cðka þ 1Þ

 Dka CgH xðaÞ

xðnÞ

ðn þ 1Þa H ð1Þ CððngH þ 1Þa þ 1Þ  ðt  aÞ

Case 4 If f 2 ½a; b; and xðtÞ is ii  CgH differentiable before f and i  CgH after it, and type of differentiability for Dka CgH xðt Þ; k ¼ 1; 2; . . .; n are i  CgH differentiable then there exists n 2 ða; bÞ such that,

xðtÞ ¼

8 > > > > > >
> > n > P > > : x ð aÞ

k¼1

DC

n P k¼2

xð n Þ

ðtaÞka Cðka þ 1Þ

ðn þ 1Þa ;

CððngH þ 1Þa þ 1Þ  ðt  aÞ ka

ðtaÞ Cðka þ 1Þ

ðn þ 1Þa

DC

 Dka CgH xðaÞ

atf

xðnÞ

ðn þ 1Þa gH  Dka ;ftb CgH xðaÞ Cððn þ 1Þa þ 1Þ  ðt  aÞ

Case 5 If f 2 ½a; b; and xðtÞ is i  CgH differentiable before f and ii  CgH after it, and type of differentiability for Dka CgH xðtÞ; k ¼ 1; 2; . . .; n are ii  CgH differentiable then there exists n 2 ða; bÞ such that, 8 n P Þa ðtaÞka ka >  DaCgH xðaÞ H ð1Þ xðaÞ Cððta > > a þ 1 Þ C ðka þ 1Þ  DCgH xðaÞ > > k¼2 > > ðn þ 1Þa > DC xðnÞ > ðn þ 1Þa < ; atf H ð1Þ CððngH þ 1Þa þ 1Þ  ðt  aÞ xðtÞ ¼ n ka P > ðtaÞ ka > xðaÞ H ð1Þ > Cðka þ 1Þ  DCgH xðaÞ H > > k¼1 > > ðn þ 1Þa > > DC xðnÞ : ðn þ 1Þa ð1Þ CððngH ; ftb þ 1Þa þ 1Þ  ðt  aÞ In general, in accordance with gH-difference, we have the following relations. Since

5.3 Fuzzy Fractional Taylor’s Expansion with Caputo gH-Derivative ðn þ 1Þa

ðn þ 1Þa

211

na

Þ na na I na DCgH xðtÞ ¼ Cððta RL DCgH xðtÞ gH I RL na þ 1Þ  DCgH xðaÞ   n P ka ka ðk þ 1Þa ðk þ 1Þa ðn þ 1Þa ðn þ 1Þa I RL DCgH xðtÞ gH I RL DCgH xðtÞ ¼ xðtÞ gH I RL DCgH xðtÞ k¼0   n n P ðtaÞka P ka ka ðk þ 1Þa ðk þ 1Þa ka I RL DCgH xðtÞ gH I RL DCgH xðtÞ ¼ Cðka þ 1Þ  DCgH xðaÞ k¼0

xð t Þ

ðn þ 1Þa ðn þ 1Þa gH I RL DCgH xðtÞ n P

xð t Þ ¼

k¼0

ðtaÞka Cðka þ 1Þ

xðtÞ ¼ xðaÞ

n P k¼1

¼

 Dka CgH xðaÞ ðtaÞka Cðka þ 1Þ

n P

k¼0

ðtaÞka Cðka þ 1Þ

 Dka CgH xðaÞ k¼0 ðn þ 1Þa ðn þ 1Þa

I RL DCgH xðtÞ ðn þ 1Þa

 Dka CgH xðaÞ I RL

ðn þ 1Þa

DCgH

xð t Þ

Using fractional mean value theorem, ðn þ 1Þa

I RL

ðn þ 1Þa

DCgH

xð t Þ ¼

¼

1  Cððn þ 1ÞaÞ

Zt

ðn þ 1Þa

ðt  sÞðn þ 1Þa1  DCgH

xðsÞds

a

ðn þ 1Þa DCgH xðnÞ

Cððn þ 1Þa þ 1Þ

 ðt  aÞðn þ 1Þa

Proof of Case 1 Since xðtÞ is i  CgH differentiable of order ka then all gH-differences are defined in the sense of i  gH difference and we have, xðtÞ ¼ xðaÞ

n X ðt  aÞka ðn þ 1Þa ðn þ 1Þa  Dka DCgH xðfÞ CgH xðaÞ I RL C ð ka þ 1 Þ k¼1

Proof of case 2 Since xðtÞ is ii  CgH differentiable of order ka then all gH-differences are defined in the sense of ii  gH difference and also fractional mean value theorem we have, ðn þ 1Þa

ðn þ 1Þa

na

Þ na na I na DCgH xðtÞ ¼ Cððta RL DCgH xðtÞ ð1ÞI RL na þ 1Þ  DCgH xðaÞ ðn þ 1Þa n DC xðnÞ P ðtaÞka ðn þ 1Þa gH ka  D x ð a Þ ð 1 Þ xðtÞ ¼ xðaÞ H ð1Þ H CgH Cððn þ 1Þa þ 1Þ  ðt  aÞ Cðka þ 1Þ k¼1



Proof of Case 3 If xðtÞ is i  CgH differentiable of order 2ka; k ¼ 0; 1; . . .; n2 and

also it is ii  CgH differentiable of order ð2k  1Þa; k ¼ 0; 1; . . .; n2 then there exists n 2 ða; bÞ.

212

5 Numerical Solution of Fuzzy Fractional Differential Equations

 n  P ðk þ 1Þa ðk þ 1Þa ka I ka DCgH xðtÞ ¼ xðtÞ ð1ÞI aRL DaCgH xðtÞ RL DCgH xðtÞ gH I RL

k¼0

2a H ð1ÞI aRL DaCgH xðtÞ H I 2a RL DCgH xðtÞ ðn þ 1Þa

na

   I na RL DCgH xðt Þ H ð1ÞI RL

ðn þ 1Þa

DCgH

xð t Þ

By similar reasoning in case 1, ðn þ 1Þa

xðtÞ ð1ÞI RL

ðn þ 1Þa

DCgH

xðtÞ ¼ xðaÞ H

n X

ka I ka RL DCgH xðaÞ

k¼1;odd

n X

ka Ika RL DCgH xðaÞ

k¼1;even

n n X X ðt  aÞka ðt  aÞka  Dka  Dka ¼ xðaÞ H CgH xðaÞ

CgH xðaÞ C ð ka þ 1 Þ C ð ka þ 1 Þ k¼1;odd k¼1;even

By using definition of H-difference, xðtÞ ¼ xðaÞ H ð1Þ

n P k¼1;odd

ðn þ 1Þa

DC

ðtaÞka Cðka þ 1Þ

 Dka CgH xðaÞ

n P k¼1;even

ðtaÞka Cðka þ 1Þ

 Dka CgH xðaÞ

xðnÞ

ðn þ 1Þa H ð1Þ CððngH þ 1Þa þ 1Þ  ðt  aÞ

The proof of the cases 4 and 5 are very similar to cases 1, 2 and 3. One of the applications of Fuzzy fractional Taylor, is Fuzzy Euler fractional method that is immediate consequence of the Taylor expansion. Now ware going to cover the Fuzzy fractional Euler method.

5.4

Fuzzy Fractional Euler with Caputo gH-Derivative

Consider the following fuzzy fractional differential equation, DaCgH xðtÞ ¼ f ðt; xðtÞÞ;

xðt0 Þ ¼ x0 ;

t 2 ½t0 ; T ;

0\a  1

where f : ½t0 ; T  FR ! FR is a fuzzy number valued function and xðtÞ is a continuous fuzzy set valued solution and DaCgH is the fractional Caputo fractional operator. In this method, Indeed, a sequence of approximations to the solution xðtÞ will be obtained at several points, called gride points. To derive Euler’s Method, the interval ½t0 ; T  is divided to into N equal subintervals, each of length h, by the grid 0 points ti ¼ t0 þ ih; i ¼ 0; 1; 2; . . .; N. The distance between points, h ¼ Tt N is called the grid size. To explain the method, we should discuss the type of differentiability of the solution and to this end we will have several cases like as Fuzzy Taylor expansion.

5.4 Fuzzy Fractional Euler with Caputo gH-Derivative

213

Case 1 Suppose that the fuzzy solution xðtÞ is i  CgH differentiable on ½t0 ; T . The Taylor expansion about tk on any subinterval ½tk ; tk þ 1  for k ¼ 0; 1; . . .; N  1 can be expressed as the following form, where h ¼ tk þ 1  tk ; 9fk 2 ½tk ; tk þ 1 . ðtk þ 1  tk Þa ðtk þ 1  tk Þ2a  DaCgH xðtk Þ

 D2a CgH xðfk Þ Cða þ 1Þ Cð2a þ 1Þ ha h2a  Dka  D2a x ð t Þ

xðtk þ 1 Þ ¼ xðtk Þ

k CgH CgH xðfk Þ C ð a þ 1Þ Cð2a þ 1Þ

xðtk þ 1 Þ ¼ xðtk Þ

Since DaCgH xðtk Þ ¼ f ðt; xðtk ÞÞ then by substituting, we have xðtk þ 1 Þ ¼ xðtk Þ

ha h2a  f ðtk ; xðtk ÞÞ

 D2a CgH xðfk Þ Cða þ 1Þ Cð2a þ 1Þ

2a

where the term Cð2ah þ 1Þ  D2a CgH xðfk Þ can be denoted as the error for the numerical method. As a result, the Fuzzy fractional Euler method on ½tk ; tk þ 1  can be introduced as, xð t k þ 1 Þ ¼ xð t k Þ

ha  f ðtk ; xðtk ÞÞ; h ¼ tk þ 1  tk ; k ¼ 0; 1; . . .; N  1 Cða þ 1Þ

Case 2 Suppose that the fuzzy solution xðtÞ is ii  CgH differentiable on ½t0 ; T . The Taylor expansion about tk on any subinterval ½tk ; tk þ 1  for k ¼ 0; 1; . . .; N  1 can be expressed as the following form, where h ¼ tk þ 1  tk ; 9fk 2 ½tk ; tk þ 1 . xðtk þ 1 Þ ¼ xðtk Þ H ð1Þ

ha ha  DaCgH xðtk Þ H ð1Þ  D2a CgH xðfk Þ C ð a þ 1Þ Cð2a þ 1Þ

Since DaCgH xðtk Þ ¼ f ðt; xðtk ÞÞ then by substituting, we have xðtk þ 1 Þ ¼ xðtk Þ H ð1Þ

ha h2a  f ðtk ; xðtk ÞÞ H ð1Þ  D2a CgH xðfk Þ Cða þ 1Þ Cð2a þ 1Þ

2a

where the term Cð2ah þ 1Þ  D2a CgH xðfk Þ can be denoted as the error for the numerical method. As a result, the Fuzzy fractional Euler method on ½tk ; tk þ 1  can be introduced as, ha  f ðtk ; xðtk ÞÞ; h ¼ tk þ 1  tk ; k Cða þ 1Þ ¼ 0; 1; . . .; N  1

xðtk þ 1 Þ ¼ xðtk Þ H ð1Þ

214

5 Numerical Solution of Fuzzy Fractional Differential Equations

Case 3 Suppose that the fuzzy solution xðtÞ has a switching point at n 2 ½t0 ; T  and it is i  CgH differentiable at the points t0 ; t1 ; ; . . .; tj and ii  CgH differentiable at the points tj þ 1 ; tj þ 2 ; . . .; tN on ½t0 ; T . The Taylor expansion about tk on any subinterval ½tk ; tk þ 1  for k ¼ 0; 1; . . .; N  1 can be expressed as the following form, (

a

k ¼ 1; 2; . . .; j xðtk þ 1 Þ ¼ xðtk Þ Cðahþ 1Þ  f ðtk ; xðtk ÞÞ; ha xðtk þ 1 Þ ¼ xðtk Þ H ð1Þ Cða þ 1Þ  f ðtk ; xðtk ÞÞ; k ¼ j þ 1; j þ 2; . . .; N

Case 4 Suppose that the fuzzy solution xðtÞ has a switching point at n 2 ½t0 ; T  and it is ii  CgH differentiable at the points t0 ; t1 ; ; . . .; tj and i  CgH differentiable at the points tj þ 1 ; tj þ 2 ; ; . . .; tN on ½t0 ; T . The Taylor expansion about tk on any subinterval ½tk ; tk þ 1  for k ¼ 0; 1; . . .; N  1 can be expressed as the following form, (

a

xðtk þ 1 Þ ¼ xðtk Þ H ð1Þ Cðahþ 1Þ  f ðtk ; xðtk ÞÞ; k ¼ 1; 2; . . .; j a k ¼ j þ 1; j þ 2; . . .; N xðtk þ 1 Þ ¼ xðtk Þ Cðahþ 1Þ  f ðtk ; xðtk ÞÞ;

Example Consider the following fuzzy fractional differential equation, DaCgH xðtÞ ¼ ð0; 1; 1:5Þ  Cða þ 1Þ;

xð0Þ ¼ 0;

t 2 ½0; 1;

0\a  1

The exact solution is, xðtÞ ¼ ð0; 1; 1:5Þ  ta :¼ c  ta This function is i  gH differentiable, because the length of xðtÞ is increasing, Note. The Caputo gH-derivative of xðtÞ ¼ ð0; 1; 1:5Þ  ta is, DaCgH ðð0; 1; 1:5Þ

1  t Þ¼ C ð 1  aÞ a

Zt 0

ð0; 1; 1:5Þa  ¼ Cð1  aÞ

ðð0; 1; 1:5Þ  sa Þ0 ds ð t  sÞ a

Zt 0

sa1 ds ð t  sÞ a

Since, Zt 0

pffiffiffi sa1 4a pCðaÞt2a   ds ¼ ð t  sÞ a C a þ 12

5.4 Fuzzy Fractional Euler with Caputo gH-Derivative

215

Then DaCgH ðð0; 1; 1:5Þ

pffiffiffi a4a pCðaÞt2a    t Þ ¼ ð0; 1; 1:5Þ  C a þ 12 Cð1  aÞ a

In the level-wise form, pffiffi a Þt2a DaC ðrta Þ ¼ r  Ca4a þ 1pCCðða1a Þ ð 2Þ pffiffi a Þt2a a a DC ðð1:5  1:5r Þt Þ ¼ ð1:5  1:5r Þ  Ca4a þ 1pCCðða1a Þ ð 2Þ

So, the Euler method is in the form of case 1, xð t k þ 1 Þ ¼ xð t k Þ

ha  ð0; 1; 1:5Þ  Cða þ 1Þ; Cða þ 1Þ

k ¼ 0; 1; . . .; N  1

For instance, consider the step size h ¼ 0:1 and a ¼ 0:5. The level-wise form of the Euler method is, xðtk þ 1 ; r Þ ¼ xðtk ; r Þ ð0:1Þ0:5 ðr; 1:5  1:5r Þ; k ¼ 0; 1; . . .; N  1   xðtk þ 1 ; r Þ ¼ xðtk ; r Þ ð0:1Þ0:5 r; ð0:1Þ0:5 ð1:5  1:5r Þ ; k ¼ 0; 1; . . .; N  1 In the level-wise form, xl ðtk þ 1 ; r Þ ¼ xl ðtk ; r Þ þ ð0:1Þ0:5 r; k ¼ 0; 1; . . .; N  1; 0  r  1 xu ðtk þ 1 ; r Þ ¼ xu ðtk ; r Þ þ ð0:1Þ0:5 ð1:5  1:5r Þ;

k ¼ 0; 1; . . .; N  1; 0  r  1

Using the recursive method and finding the values for xðtk Þ; k ¼ 0; 1; . . .; 9 finally we get, xl ðt10 ; r Þ ¼ 10ð0:1Þ0:5 r;

xu ðt10 ; r Þ ¼ 10ð0:1Þ0:5 ð1:5  1:5r Þ;

0r1

In Fig. 5.1 the exact solution and its Caputo gH-derivative with a ¼ 0:5 are presented at the grid points. In Table 5.1 the numerical solutions with order a ¼ 0:5 and h ¼ 0:1 are expressed. Also, in Fig. 5.2 the exact solution and its Caputo gH-derivative with a ¼ 0:75 are presented at the grid points. In Table 5.2 the numerical solutions with order a ¼ 0:75 and h ¼ 0:1 are expressed.

216

5 Numerical Solution of Fuzzy Fractional Differential Equations

Fig. 5.1 Exact solution (left) and its C-gH-derivative (right) at the grid points with a ¼ 0:5

Table 5.1 The numerical results for step size h ¼ 0:1

Fig. 5.2 Exact solution (left) and its C-gH-derivative (right) at the grid points with a ¼ 0:75

5.4 Fuzzy Fractional Euler with Caputo gH-Derivative

217

Table 5.2 The numerical results with order a ¼ 0:75 for step size h ¼ 0:1

Example Consider the following fuzzy fractional differential equation, DaCgH xðtÞ ¼ ð1Þ  xðtÞ;

xð0Þ ¼ ð0; 1; 2Þ;

t 2 ½0; 1;

0\a  1

The exact solution is, xðtÞ ¼ ð0; 1; 2Þ  Ea ðta Þ :¼ c  Ea ðta Þ where Ea ðta Þ ¼

1 X ðta Þi Cðia þ 1Þ i¼0

This function is ii  gH differentiable, because the length of xðtÞ is decreasing, So, the Euler method is in the form of case 2, xðtk þ 1 Þ ¼ xðtk Þ H ð1Þ

  ha  ð0; 1; 2Þ  Ea tka ; Cða þ 1Þ

k ¼ 0; 1; . . .; N  1

218

5 Numerical Solution of Fuzzy Fractional Differential Equations

For instance, consider the step size h ¼ 0:1 and a ¼ 0:5. The level-wise form of the Euler method is,   0:5a E0:5 tk0:5  ðr  1; 2  r Þ; xðtk þ 1 ; r Þ ¼ xðtk ; r Þ H ð1Þ Cð0:5 þ 1Þ For k ¼ 0; 1; . . .; N  1. 0:687498 0:775758 pffiffiffiffi ¼ pffiffiffiffi ð tk þ 1ÞCð0:5 þ 1Þ ð tk þ 1Þ Since the value

0:5a E0:5 ðt0:5 Þ Cð0:5 þ 1Þ

[ 0 then, in the level-wise form,

0:775758 ð2  r Þ; xl ðtk þ 1 ; r Þ ¼ xl ðtk ; r Þ þ pffiffiffiffi ð tk þ 1Þ

k ¼ 0; 1; . . .; N  1; 0  r  1

0:775758 ðr  1Þ; xu ðtk þ 1 ; r Þ ¼ xu ðtk ; r Þ þ pffiffiffiffi ð t k þ 1Þ

k ¼ 0; 1; . . .; N  1; 0  r  1

In Fig. 5.3 the exact solution and its Caputo gH-derivative with a ¼ 0:5 are presented at the grid points. In Table 5.3 the numerical solutions with order a ¼ 0:5 and h ¼ 0:1 are expressed. Also, in Fig. 5.4 the exact solution and its Caputo gH-derivative with a ¼ 0:75 are presented at the grid points. In Table 5.4 the numerical solutions with order a ¼ 0:75 and h ¼ 0:1 are expressed.

Fig. 5.3 Exact solution (left) and its C-gH-derivative (right) at the grid points with a ¼ 0:5

5.4 Fuzzy Fractional Euler with Caputo gH-Derivative

219

Table 5.3 The numerical results for step size h ¼ 0:1

Fig. 5.4 Exact solution (left) and its C-gH-derivative (right) at the grid points with a ¼ 0:75

Example Consider another fuzzy fractional differential equation, DaCgH xðtÞ ¼

  pt1a a 1  0; ; 1 Cð2 aÞ 2

 a 3 a 1 22 2 1 F1 1; 1  ;  ;  p t a ; 1  t  2 2 2 2 4

220

5 Numerical Solution of Fuzzy Fractional Differential Equations

Table 5.4 The numerical results with order a ¼ 0:75 for step size h ¼ 0:1

With fuzzy initial value xð1Þ ¼ ð0; 0:5; 1Þ  sin ap: where 1 F1 ða; b; zÞ is a generalized hypergeometric function and defined as the following form,   1 X ða1 Þn    ap n zn    p Fq a1 ; a2 ; . . .; ap ; b1 ; b2 ; . . .; bq ; z ¼ n! n¼0 ðb1 Þn    bq n 



Here 1 F1 ða; b; zÞ ¼

1 X ð aÞ n¼0

n

ð bÞ n



1 zn X C ð a þ nÞ CðbÞ zn   ¼ Cðb þ nÞ n! n! n¼0 CðaÞ

where ð Þn ¼

Cð þ nÞ CðÞ

Note that by the ratio test the series is convergent. In this example, using a ¼ 0:5 the exact solution is xðtÞ ¼ ð0; 0:5; 1Þ  sinðaptÞ

5.4 Fuzzy Fractional Euler with Caputo gH-Derivative

221

And the solution has a switching point at the point t ¼ 1:463 and the switching point is type I, before the point it is i  gH differentiable and after it is ii  gH differentiable. In the subintervals ½ tk ; tk þ 1 ; k ¼ 0; 1; . . .; N  1 with assumption that switching point is in the interval tj ; tj þ 1 , the fuzzy Euler’s method is denoted as, (

a

xðtk þ 1 Þ ¼ xðtk Þ Cðahþ 1Þ  f ðtk ; xðtk ÞÞ; k ¼ 1; 2; . . .; j a xðtk þ 1 Þ ¼ xðtk Þ H ð1Þ Cðahþ 1Þ  f ðtk ; xðtk ÞÞ; k ¼ j þ 1; j þ 2; . . .; N

where f ð t k ; xð t k Þ Þ ¼

 

  ptk1a a 1 a 3 a 1  0; ; 1  1 F1 1; 1  ;  ;  p2 tk2 a2 C ð 2  aÞ 2 2 2 2 4

Suppose h ¼ 0:1; k ¼ 0; a ¼ 0:5; t0 ¼ 1; xð1Þ ¼ ð0; 0:5; 1Þ  sin ap. Then xðt1 Þ ¼ ð0; 0:5; 1Þ

h0:5  f ð1; ð0; 0:5; 1ÞÞ Cð0:5 þ 1Þ

where  1   3 5

 1 2 f ð1; ð0; 0:5; 1ÞÞ ¼ Cp0:5 3  0; 2 ; 1 1 F1 1; 4 ; 4 ;  16 p ð 2Þ n 1  3 5

P Cð½34;54Þ ð161 p2 Þ Cð1 þ nÞ 1 2   1 F1 1; 4 ; 4 ;  16 p n! Cð1Þ Cð½34 þ n;54 þ nÞ n¼0 1

n 1 3 1 2 5 1 P Cð1 þ nÞ Cð4Þ ð16p Þ P Cð1 þ nÞ Cð4Þ ð16p2 Þn   ¼  ;  n! n! Cð1Þ Cð1Þ Cð34Þ Cð54Þ n¼0 h n¼0 i 16 16 ¼ 16 16 þ p2 ; 16 þ p2 ¼ 16 þ p2 Thus,   p0:5 1 16 f ðt0 ; xðt0 ÞÞ ¼ f ð1; ð0; 0:5; 1ÞÞ ¼ 3  0; ; 1  2 16 þ p2 C 2 Since

p0:5 16 Cð32Þ 16 þ p2

1:1 and positive then, f ðt0 ; xðt0 ÞÞ ¼ 1:1  ð0; 0:5; 1Þ and in

the level-wise form, fl ðt0 ; xðt0 Þ; r Þ ¼

1:1 1:1 r; fu ðt0 ; xðt0 Þ; r Þ ¼ 1:1  r 2 2

222

5 Numerical Solution of Fuzzy Fractional Differential Equations

and the method for approximate the solution at the second point xðt1 Þ is, 1 h0:5 1:1 r; xu ðt1 ; r Þ  xl ð t 1 ; r Þ ¼ r þ 2 Cð0:5 þ 1Þ 2   0:5 1 h 1:1 r ¼ 1 rþ  1:1  2 2 Cð0:5 þ 1Þ Putting h ¼ 0:1 and in the level for instance r ¼ 0:5, we have xl ðt1 ; 0:5Þ ¼

1 0:50:5 1:1 3 0:50:5 3 þ

0:47; xu ðt1 ; r Þ ¼ þ    1:1 1:41 4 Cð1:5Þ 4 4 Cð1:5Þ 4

In general, for a ¼ 0:5 we have,     ptko:5 0:5 1 1 22  0; ; 1 1 F1 1; ½0:5; 1;  p tk f ðtk ; xðtk ÞÞ ¼ Cð1:5Þ 2 16 And (

0:5

xðtk þ 1 Þ ¼ xðtk Þ Chð1:5Þ  f ðtk ; xðtk ÞÞ; xð t k þ 1 Þ ¼

0:5 xðtk Þ H ð1Þ Chð1:5Þ

 f ðtk ; xðtk ÞÞ;

k ¼ 1; 2; . . .; j k ¼ j þ 1; j þ 2; . . .; N

Table 5.5 The numerical results with order a ¼ 0:75 for step size h ¼ 0:1

5.4 Fuzzy Fractional Euler with Caputo gH-Derivative

223

Fig. 5.5 Exact solution (left) and its C-gH-derivative (right) at the grid points with a ¼ 0:5

The results for h ¼ 0:1 are listed as in Table 5.5. Figure 5.5 show the fuzzy solution and its gH-derivative with a ¼ 0:5.

5.5

ABC-PI Numerical Method with ABC gH-Derivative

In this section, first, the ABC fractional derivative on fuzzy number-valued functions in parametric interval form is defined. Then it is applied for proving the existence and uniqueness of the solution of fuzzy fractional differential equation with ABC fractional derivative. In general, it is shown that the last interval model is as a coupled system of nonlinear equations in interval form. To solve the final system an efficient numerical method called ABC-PI is used.

5.5.1

Definition—ABCGH Fractional Derivative in the Sense of Caputo Derivative

The generalized ABCgH derivative of a fuzzy number valued function xðtÞ on c interval ½t0 ; T  starting at t0 with kernel Ea;l ðk; tÞ where 0\ah1; ReðlÞi0; c 2 a R; k ¼ 1a is defined in the following form, Da;l:c ABCgH xðt Þ

BðaÞ  ¼ 1a

Zt t0

c x0gH ðsÞ  Ea;l ðk; t  sÞds

224

5 Numerical Solution of Fuzzy Fractional Differential Equations

where BðaÞ [ 0 is a normalizing function and is defined as, BðaÞ ¼ 1  a þ

a CðaÞ

With properties Bð0Þ ¼ Bð1Þ ¼ 1. Its corresponding AB fractional integral operator is defined as, I a;l:c AB xðtÞ

¼

c   X c

BðaÞð1  aÞi1

i

i¼0

ai

I a;l;c xðtÞ

Subject to, a;l:c I a;l:c AB DABCgH xðtÞ ¼ xðtÞ gH xðt0 Þ

In case l ¼ c ¼ 1 we get the fractional derivative and integral with order 0\a\1 then the ABCgH derivative that is denoted by DaABCgH is defined on a fuzzy number valued function xðtÞ on interval ½t0 ; T  in the following form, DaABCgH xðtÞ

BðaÞ  ¼ 1a

Zt

 x0gH ðsÞ  Ea 

t0

 a ðt  sÞa ds 1a

Its corresponding fractional integral is defined as, I aAB xðtÞ

1a a  xð t Þ

 ¼ BðaÞ BðaÞCðaÞ

Zt

xðsÞ  ðt  sÞa1 ds

t0

In case a ¼ 0 it recovers initial function and if a ¼ 1 it is ordinary integral. Here we are going to explain its level-wise form regarding the type of differentiability of fuzzy valued function xðtÞ. Case 1 xðtÞ is i  gH differentiable,

DaABCigH xðt; r Þ ¼ DaABC xl ðt; r Þ; DaABC xu ðt; r Þ Case 2 xðtÞ is ii  gH differentiable,

DaABCiigH xðt; r Þ ¼ DaABC xu ðt; r Þ; DaABC xl ðt; r Þ

5.5 ABC-PI Numerical Method with ABC gH-Derivative

225

where DaABC xl ðt; r Þ

DaABC xu ðt; r Þ

BðaÞ ¼ 1a BðaÞ ¼ 1a

Zt

 x0l ðs; r ÞEa 

 a ðt  sÞa ds 1a

 x0u ðs; r ÞEa 

 a ðt  sÞa ds 1a

t0

Zt t0

And its fractional integral, I aAB xl ðtÞ

1a a  xl ð t Þ

 ¼ BðaÞ BðaÞCðaÞ

I aAB xu ðtÞ ¼

1a a  x u ðt Þ

 BðaÞ BðaÞCðaÞ

Zt

xl ðsÞ  ðt  sÞa1 ds

t0

Zt

xu ðsÞ  ðt  sÞa1 ds

t0

Laplace transform of the DaABCgH can be explained as,   BðaÞ sa  LðxðtÞÞ gH sa1  xð0Þ L DaABCgH xðtÞ ¼ a 1a sa þ 1a where Z1 LðxðtÞÞ ¼

est  xðtÞdt;

s[0

0

Based on the definition of gH-difference there are two cases for the Laplace transform, Case 1 i  gH difference   BðaÞ sa  LðxðtÞÞ sa1  xð0Þ H L DaABCgH xðtÞ ¼ a 1a sa þ 1a      

L DaABC xðt; r Þ ¼ L DaABC xl ðt; r Þ ; L DaABC xu ðt; r Þ

226

5 Numerical Solution of Fuzzy Fractional Differential Equations

Case 2 ii  gH difference   BðaÞ H ð1Þsa  LðxðtÞÞ ð1Þsa1  xð0Þ L DaABCgH xðtÞ ¼ a 1a sa þ 1a ¼

BðaÞ sa1  xð0Þ H sa  LðxðtÞÞ a a1 sa þ 1a

     

L DaABC xðt; r Þ ¼ L DaABC xu ðt; r Þ ; L DaABC xl ðt; r Þ where   BðaÞ sa Lðxl ðt; r ÞÞ  sa1 xl ð0; r Þ L DaABC xl ðt; r Þ ¼ a 1a sa þ 1a   BðaÞ sa Lðxu ðt; r ÞÞ  sa1 xu ð0; r Þ L DaABC xu ðt; r Þ ¼ a 1a sa þ 1a

5.5.2

Fuzzy Time Fractional Ordinary Differential Equation

Consider the following fuzzy fractional differential equation, DaABCgH xðtÞ ¼ f ðt; xðtÞÞ;

xðt0 Þ ¼ x0 2 FR ;

t 2 ½t0 ; T ;

0\a\1

By taking the fractional integral I aAB from both sides of the above fuzzy time fractional equation, the fuzzy solution of this equation satisfies its corresponding fuzzy integral equations in the following form, I aAB DaABCgH xðtÞ ¼ I aAB f ðt; xðtÞÞ I aAB DaABCgH xðtÞ ¼ xðtÞ gH xðt0 Þ ¼ I aAB f ðt; xðtÞÞ 1a a  f ðt; xðtÞÞ

 xðtÞ ¼ xðt0 Þ

BðaÞ BðaÞCðaÞ

Zt t0

f ðs; xðsÞÞ  ðt  sÞa1 ds

5.5 ABC-PI Numerical Method with ABC gH-Derivative

227

Case 1 xðtÞ is i  gH differentiable, DaABCigH xðtÞ ¼ f ðt; xðtÞÞ 1a a  f ðt; xðtÞÞ

 xðtÞ ¼ xðt0 Þ

BðaÞ BðaÞCðaÞ

Zt

f ðs; xðsÞÞðt  sÞa1 ds

t0



DaABCigH xðt; r Þ ¼ DaABC xl ðt; r Þ; DaABC xu ðt; r Þ ¼ ½fl ðt; xl ðt; r Þ; xu ðt; r ÞÞ; fu ðt; xl ðt; r Þ; xu ðt; r ÞÞ DaABC xl ðt; r Þ ¼ fl ðt; xl ðt; r Þ; xu ðt; r ÞÞ; DaABC xu ðt; r Þ ¼ fu ðt; xl ðt; r Þ; xu ðt; r ÞÞ Case 2 xðtÞ is ii  gH differentiable, DaABCiigH xðtÞ ¼ f ðt; xðtÞÞ xðtÞ ¼ xðt0 Þ H ð1Þ Zt 

1a a  f ðt; xðtÞÞ H ð1Þ BðaÞ BðaÞCðaÞ

f ðs; xðsÞÞ  ðt  sÞa1 ds

t0



DaABCiigH xðt; r Þ ¼ DaABC xu ðt; r Þ; DaABC xl ðt; r Þ ¼ ½fl ðt; xl ðt; r Þ; xu ðt; r ÞÞ; fu ðt; xl ðt; r Þ; xu ðt; r ÞÞ DaABC xl ðt; r Þ ¼ fu ðt; xl ðt; r Þ; xu ðt; r ÞÞ; DaABC xl ðt; r Þ ¼ fu ðt; xl ðt; r Þ; xu ðt; r ÞÞ where 1a fl ðt; xl ðt; r Þ; xu ðt; r ÞÞ BðaÞ Zt a þ fl ðs; xl ðs; r Þ; xu ðs; r ÞÞðt  sÞa1 ds BðaÞCðaÞ

xl ðt; r Þ ¼ xl ð0; r Þ þ

t0

1a fu ðt; xl ðt; r Þ; xu ðt; r ÞÞ BðaÞ Zt a þ fu ðs; xl ðs; r Þ; xu ðs; r ÞÞðt  sÞa1 ds BðaÞCðaÞ

xu ðt; r Þ ¼ xu ð0; r Þ þ

t0

Indeed, in case 2, we have a system of ABC fractional differential equations and usually solving this system is not straight forward and the solution should be

228

5 Numerical Solution of Fuzzy Fractional Differential Equations

obtained by numerical methods. Now we are going to investigate the conditions for the existence of a unique solution of the fuzzy ABC fractional differential equation. To this end, consider the following equations, DaABCgH xðtÞ ¼ f ðt; xðtÞÞ;

xðt0 Þ ¼ x0 2 FR ;

t 2 ½t0 ; T ;

0\a\1

where xðtÞ is a fuzzy continuous function on ½t0 ; T .

5.5.3

Remark—Uniqueness

Suppose that the fuzzy function f ðt; xðtÞÞ is a continuous function and satisfies the Lipschitz condition, for 8x8y9M [ 0 subject to, DH ðf ðt; xðtÞÞ; f ðt; yðtÞÞÞ  MDH ðxðtÞ; yðtÞÞ Then the above-mentioned differential equation has a unique solution if the following condition is satisfied, 1a a Mþ MT a þ 1 \1 BðaÞ BðaÞCðaÞ The proof is easy to show and it is expressed by using the contraction operator. Let us consider the operator, F ð xð t Þ Þ ¼ xð t 0 Þ

1a a  f ðt; xðtÞÞ

 B ð aÞ BðaÞCðaÞ

DH ðF ðxðtÞÞ; F ðyðtÞÞÞ  DH ðxðt0 Þ; yðt0 ÞÞ þ a þ BðaÞCðaÞ

Zt

Zt

f ðt; xðtÞÞ  ðt  sÞa1 ds

t0

1a DH ðf ðt; xðtÞÞ; f ðt; yðtÞÞÞ BðaÞ

DH ðf ðs; xðsÞÞ; f ðs; yðsÞÞÞðt  sÞa1 ds

t0

Since DH ðxðt0 Þ; yðt0 ÞÞ ¼ 0 then DH ðF ðxðtÞÞ; F ðyðtÞÞÞ  a þ BðaÞCðaÞ

Zt t0

1a DH ðf ðt; xðtÞÞ; f ðt; yðtÞÞÞ BðaÞ

DH ðf ðs; xðsÞÞ; f ðs; yðsÞÞÞðt  sÞa1 ds

5.5 ABC-PI Numerical Method with ABC gH-Derivative

229

Based on Lipschitz condition, DH ðf ðt; xðtÞÞ; f ðt; yðtÞÞÞ  MDH ðxðtÞ; yðtÞÞ We have, 1a MDH ðxðtÞ; yðtÞÞ BðaÞ Zt MaDH ðxðtÞ; yðtÞÞ þ ðt  sÞa1 ds BðaÞCðaÞ

DH ðF ðxðtÞÞ; F ðyðtÞÞÞ 

t0

Since,

Rt

ðt  sÞa1 ds\T a , we have,

t0

DH ðF ðxðtÞÞ; F ðyðtÞÞÞ  B1a MDH ðxðtÞ; yðtÞÞ þ h iðaÞ Ma a ¼ B1a ðaÞ M þ BðaÞCðaÞ T DH ðxðt Þ; yðtÞÞ

MaDH ðxðtÞ;yðtÞÞ a T BðaÞCðaÞ

Applying the assumption of the remark, DH ðF ðxðtÞÞ; F ðyðtÞÞÞ  DH ðxðtÞ; yðtÞÞ It means the operator F is a contraction operator and therefor based on the Banach fixed point theorem our problem does have a unique solution in the mentioned form.

5.5.4

An Efficient Numerical Method for ABC Fractional Problems

As we mentioned, the system of fractional differential equations in the level-wise form are as, DaABC xl ðt; r Þ ¼ fl ðt; xl ðt; r Þ; xu ðt; r ÞÞ; DaABC xu ðt; r Þ ¼ fu ðt; xl ðt; r Þ; xu ðt; r ÞÞ DaABC xl ðt; r Þ ¼ fu ðt; xl ðt; r Þ; xu ðt; r ÞÞ; DaABC xu ðt; r Þ ¼ fl ðt; xl ðt; r Þ; xu ðt; r ÞÞ In general, these equations can be shown as the following form, 

DaABC xl ðt; r Þ ¼ fl ðt; xl ðt; r Þ; xu ðt; r ÞÞ DaABC xu ðt; r Þ ¼ fu ðt; xl ðt; r Þ; xu ðt; r ÞÞ

230

5 Numerical Solution of Fuzzy Fractional Differential Equations

where R1 ; R2 are two continuous functions. Applying AB fractional integral I aAB both sides, we get the following equations, 1a fl ðt; xl ðt; r Þ; xu ðt; r ÞÞ BðaÞ Zt a þ fl ðs; xl ðs; r Þ; xu ðs; r ÞÞðt  sÞa1 ds BðaÞCðaÞ

xl ðt; r Þ ¼ xl ðt0 ; r Þ þ

t0

1a fu ðt; xl ðt; r Þ; xu ðt; r ÞÞ xu ðt; r Þ ¼ xu ðt0 ; r Þ þ BðaÞ Zt a fu ðs; xl ðs; r Þ; xu ðs; r ÞÞðt  sÞa1 ds þ BðaÞCðaÞ t0

0 Taking t ¼ tn ¼ t0 þ nh; h ¼ Tt n the discretized recursive implicit equations are obtained,

1a fl ðtn ; xl ðtn ; r Þ; xu ðtn ; r ÞÞ BðaÞ ti þ 1 n1 Z X a þ fl ðtn ; xl ðs; r Þ; xu ðs; r ÞÞðtn  sÞa1 ds BðaÞCðaÞ i¼0

xl ðtn ; r Þ ¼ xl ðt0 ; r Þ þ

ti

1a fu ðtn ; xl ðtn ; r Þ; xu ðtn ; r ÞÞ BðaÞ ti þ 1 n1 Z X a þ fu ðtn ; xl ðs; r Þ; xu ðs; r ÞÞðtn  sÞa1 ds BðaÞCðaÞ i¼0

xu ðtn ; r Þ ¼ xu ðt0 ; r Þ þ

ti

For 0  r  1. The functions f ðtn ; xl ðs; r Þ; xu ðs; r ÞÞ;  2 fl; ug can be approximated by Lagrange interpolation on ½ti ; ti þ 1  and it is, f ðtn ; xl ðs; r Þ; xu ðs; r ÞÞ f ðti þ 1 ; xl ðti þ 1 ; r Þ; xu ðti þ 1 ; r ÞÞ s  ti þ 1 þ ðf ðti þ 1 ; xl ðti þ 1 ; r Þ; xu ðti þ 1 ; r ÞÞ  f ðti ; xl ðti ; r Þ; xu ðti ; r ÞÞÞ h By substituting and algebraic manipulating, xl ð t n ; r Þ ¼ xl ð t 0 ; r Þ n X aha nn fl ðt0 ; xl ðt0 ; r Þ; xu ðtn ; r ÞÞ þ lni fl ðti ; xl ðti ; r Þ; xu ðti ; r ÞÞ þ BðaÞ i¼1

!

5.5 ABC-PI Numerical Method with ABC gH-Derivative

231

xu ðtn ; r Þ ¼ xu ðt0 ; r Þ n X aha nn fu ðt0 ; xl ðt0 ; r Þ; xu ðtn ; r ÞÞ þ lni fu ðti ; xl ðti ; r Þ; xu ðti ; r ÞÞ þ BðaÞ i¼1

where nn ¼ ( lj ¼

ðn  1Þa þ 1 na ðn  a  1Þ ; Cða þ 2Þ

1 1a Cða þ 2Þ þ aha ; aþ1 ðj1Þ 2ja þ 1 þ ðj þ 1Þa þ 1 Cða þ 2Þ

j¼0 ; j ¼ 1; 2; . . .; n  1

Numerical example Suppose DaABCgH xðtÞ ¼ k  xðtÞ;

xð0; r Þ ¼ ðr; 3  2r Þ;

t 2 ½0; 1;

0\a\1

• k  0 we have i  gH solution

kð 1  aÞ ka xðtÞ ¼ ðr; 3  2r Þ

xðtÞ

 B ð aÞ BðaÞCðaÞ

Zt

xðsÞ  ðt  sÞa1 ds

t0

In the level-wise form we have,

DaABCigH xðt; r Þ ¼ DaABC xl ðt; r Þ; DaABC xu ðt; r Þ ¼ ½kxl ðt; r Þ; kxu ðt; r Þ DaABC xl ðt; r Þ ¼ kxl ðt; r Þ;

DaABC xu ðt; r Þ ¼ kxu ðt; r Þ

And the numerical scheme is as, n X kaha xl ð t n ; r Þ ¼ r þ nn r þ lni xl ðti ; r Þ B ð aÞ i¼1

!

n X kaha xu ðtn ; r Þ ¼ 3  2r þ nn ð3  2r Þ þ lni xu ðti ; r Þ BðaÞ i¼1

!

!

232

5 Numerical Solution of Fuzzy Fractional Differential Equations

Fig. 5.6 i  gH solution for k ¼ 1; a ¼ 0:9

Fig. 5.7 i  gH solution for k ¼ 0:5; a ¼ 0:95

In Figures 5.6 and 5.7, the numerical method has been used for approximating the fuzzy solution of our problem. To this end, first the level 0  r  1 is fixed and then the algorithm is applied for two values of k and a. It is seen that the solutions are fuzzy number valued functions because in all figures the triangular fuzzy number at each point is observed. • k\0 we have ii  gH solution kð 1  aÞ ka xðtÞ ¼ ð3  2r; r Þ H ð1Þ xðtÞ H ð1Þ  BðaÞ BðaÞCðaÞ  ðt  sÞa1 ds

Zt xð sÞ t0

5.5 ABC-PI Numerical Method with ABC gH-Derivative

233

And in the level-wise for,

DaABCiigH xðt; r Þ ¼ DaABC xu ðt; r Þ; DaABC xl ðt; r Þ ¼ ½kxu ðt; r Þ; kxl ðt; r Þ DaABC xl ðt; r Þ ¼ kxu ðt; r Þ; DaABC xu ðt; r Þ ¼ kxl ðt; r Þ And the numerical solution is also obtained as, ! n X kaha xl ðtn ; r Þ ¼ 3  2r þ n ð3  2r Þ þ lni xu ðti ; r Þ B ð aÞ n i¼1 n X kaha xu ð t n ; r Þ ¼ r þ nn r þ lni xl ðti ; r Þ BðaÞ i¼1

!

Here are ii  gH solutions for some values of k; a (Figs. 5.8 and 5.9). Numerical example Consider the following fuzzy fractional differential equation,   DaABCgH xðtÞ ¼ t  xðtÞ; xð1; r Þ ¼ r 2 þ 4; 5:5  0:5r ; t 2 ½1; 1 Here we also have two cases for t 2 ½1; 1, • t 2 ½1; 0 we have ii  gH solution   t ð 1  aÞ xð t Þ xðtÞ ¼ 5:5  0:5r; r 2 þ 4 H ð1Þ BðaÞ Zt a H ð1Þ  s  xðsÞ  ðt  sÞa1 ds BðaÞCðaÞ t0

Fig. 5.8 ii  gH solution for k ¼ 0:8; a ¼ 0:85

234

5 Numerical Solution of Fuzzy Fractional Differential Equations

Fig. 5.9 ii  gH solution for k ¼ 0:5; a ¼ 0:95

In the level-wise form we have, DaABC xl ðt; r Þ ¼ txu ðt; r Þ;

DaABC xu ðt; r Þ ¼ txl ðt; r Þ

The numerical scheme is as, n X aha nn ð5:5  0:5r Þ þ lni ti xu ðti ; r Þ xl ðtn ; r Þ ¼ 5:5  0:5r þ BðaÞ i¼1 n   X aha xu ðtn ; r Þ ¼ r þ 4 þ nn r 2 þ 4 þ lni ti xl ðti ; r Þ BðaÞ i¼1

!

!

2

• t 2 ½0; 1 we have i  gH solution





t ð 1  aÞ a xðtÞ ¼ r þ 4; 5:5  0:5r

xð t Þ

 B ð aÞ BðaÞCðaÞ 2

 ðt  sÞa1 ds

Zt s  xð sÞ t0

In the level-wise form we have, DaABC xl ðt; r Þ ¼ txl ðt; r Þ;

DaABC xu ðt; r Þ ¼ txu ðt; r Þ

5.5 ABC-PI Numerical Method with ABC gH-Derivative

235

The numerical scheme is as, n   X aha nn r 2 þ 4 þ xl ðtn ; r Þ ¼ r þ 4 þ lni ti xl ðti ; r Þ BðaÞ i¼1

!

2

n X aha xu ðtn ; r Þ ¼ 5:5  0:5r þ nn ð5:5  0:5r Þ þ lni ti xu ðti ; r Þ BðaÞ i¼1

!

In this example the point t ¼ 0 is a switching point of the fuzzy solution. The next example has a non-linear function in the left side (Figs. 5.10 and 5.11). Example Consider the following non-linear fuzzy fractional differential equation, DaABCgH xðtÞ ¼ sin t  xðtÞ t2 ;

xð1; r Þ ¼ ððr  1Þet ; ð1  r Þet Þ;

1  t  1

• For 1  t  0, sin t\0; DaABC xl ðt; r Þ ¼ sin t  xu ðt; r Þ þ t2 ;

xl ð1; r Þ ¼ ðr  1Þet ;

0r1

DaABC xu ðt; r Þ ¼ sin t  xu ðt; r Þ þ t2 ;

xu ð1; r Þ ¼ ð1  r Þet ;

0r1

Fig. 5.10 The solution on ½1; 1 for a ¼ 0:9 (left) and a ¼ 0:85 (right)

Fig. 5.11 The solution on ½1; 1 for a ¼ 0:6 (left) and a ¼ 0:4 (right)

236

5 Numerical Solution of Fuzzy Fractional Differential Equations

Fig. 5.13 The solution on ½1; 1 for a ¼ 0:5 (left) and a ¼ 0:4 (right)

Fig. 5.12 The solution on ½1; 1 for a ¼ 0:95 (left) and a ¼ 0:7 (right)

• For 0  t  1, sin t [ 0; (Figs. 5.12 and 5.13)

5.6

DaABC xl ðt; r Þ ¼ sin t  xl ðt; r Þ þ t2 ;

xl ð1; r Þ ¼ ðr  1Þet ;

0r1

DaABC xu ðt; r Þ ¼ sin t  xu ðt; r Þ þ t2 ;

xu ð1; r Þ ¼ ð1  r Þet ;

0r1

Numerical Method for Fuzzy Fractional Impulsive Differential Equations

In this section, the combination of reproducing kernel Hilbert space method (RKHSM) and fractional differential transformation method (FDTM) is used to solve the fuzzy impulsive fractional differential equations with the help of the concept of generalized Hukuhara differentiability. In Sect. 4.5 of Chap. 4, we discussed the fuzzy impulsive differential equations with Caputo-Katugampola generalized fractional differentiability, now it is explained in case Caputo differentiability with p ¼ 1 in the following form,

5.6 Numerical Method for Fuzzy Fractional Impulsive Differential Equations

8 1 < c Da yðtÞ ¼ f ðt; yðtÞÞ; t 2 ½0; T ; t 6¼ tk ; DyðtÞjt¼tk ¼ Ik y tk ðtÞ : y ð 0Þ ¼ y 0

237

m  1\ 1a \m; m 2 N

1

where k ¼ 1; 2; . . .; m, and c Da denotes the Caputo fractional generalized derivative of order 1a, and yðtÞ is an unknown fuzzy function of real variable t. Also suppose that f : ½0; T  FR ! FR ; is continuous fuzzy function, and Ik : FR ! FR , is a fuzzy difference transform with fuzzy number initial value y0 2 FR . As it was explained in chapter four, the fuzzy solution is a peace-wise continuous fuzzy function and is defined on the subintervals, 0 ¼ t0 \t1 \. . .\tm \tm þ 1 ¼ T. The fuzzy difference map is defined as,         Djt¼tk ¼ y tkþ gH y tk ; y tkþ ¼ limh!0 þ yðtk þ hÞ; y tk ¼ limh!0 yðtk þ hÞ     The values y tkþ and y tk represent the right and left limits of yðtÞ at t ¼ tk . In Sect. 4.5, it was shown that the solution of fuzzy impulsive differential equation satisfies the following fuzzy integral equations, 1 yðtÞ ¼ y0 1 C a

Zt

ðt  sÞa1 f ðs; yðsÞÞds; 1

t 2 ½0; t1 

0

whenever yðtÞ is i  gH differentiable, and 1 yðtÞ ¼ y0 H ð1Þ 1 C a

Zt

ðt  sÞa1 f ðs; yðsÞÞds; 1

t 2 ½0; t1 

0

whenever yðtÞ is ii  gH differentiable, and finally if there is a switching point like t1 , 8 Rt 1 > 1 > y

ðt  sÞa1 f ðs; yðsÞÞds; t 2 ½0; t1  > 0 1 > CðaÞ > 0 > > < m Rtk P 1 ðtk  sÞa1 f ðs; yðsÞÞds ð1Þ yðtÞ ¼ y0 H ð1Þ Cð11Þ a k¼1 t > k1 > > > t m    R > P 1 > 1 1 a > Ik y tk ; t 2 ðt1 ; tk þ 1  : Cð1Þ ðt  sÞ f ðs; yðsÞÞds H ð1Þ a t k¼1 k if there exists a point t1 2 ð0; tk þ 1 Þ such that yðtÞ is i  gH differentiable on ½0; t1  and ii  gH differentiable on ðt1 ; tk þ 1 Þ.

238

5 Numerical Solution of Fuzzy Fractional Differential Equations

5.6.1

Remark—Uniqueness

Let us assume, the fuzzy functions f and Ik satisfy the Lipschitz condition, H1. There exists a constant 0  L1 such that DH ðf ðt; y1 ðtÞÞ; f ðt; y2 ðtÞÞÞ  L1 DH ðy1 ðtÞ; y2 ðtÞÞ; for each t 2 ½0; T , and any y1 ; y2 2 FR . H2. There exists a constant 0  L2 such that DH ðIk ðy1 ðtÞÞ; Ik ðy2 ðtÞÞÞ  L2 DH ðy1 ðtÞ; y2 ðtÞÞ; for each y1 ; y2 2 FR , and k ¼ 1; 2; . . .; m: If "

# L1 1   ðm þ 1ÞT a þ mL2 \1 C 1a

Such that T is very small numbers the fuzzy fractional impulsive differential equation has a unique solution on ½0; T . Proof By using the distance, it is easy to show, to this end, suppose the function F is defined as, t m Zk 1 1 X   ðtk  sÞa1 f ðs; yðsÞÞds F ðyðtÞÞ ¼ y0 H ð1Þ 1 C a k¼1 tk1

1 H ð1Þ 1 C a

DH ðF ðy1 ðtÞÞ; F ðy2 ðtÞÞÞ 

þ

1  C 1a

Zt tk

C

Zt

ðt  sÞa1 f ðs; yðsÞÞds H ð1Þ

k¼1

Ztk

ðtk  sÞa1 DH ðf ðs; y1 ðsÞÞ; f ðs; y2 ðsÞÞÞds 1

tk1

ðt  sÞa1 DH ðf ðs; y1 ðsÞÞ; f ðs; y2 ðsÞÞÞds þ 1

m X    Ik y tk k¼1

tk

m 1 X 1 a

1

m X k¼1

       DH Ik y1 tk ; Ik y2 tk

5.6 Numerical Method for Fuzzy Fractional Impulsive Differential Equations

DH ðF ðy1 ðtÞÞ; F ðy2 ðtÞÞÞ 

þ

1  C 1a

Zt

C

m 1 X 1 a

k¼1

Ztk

239

ðtk  sÞa1 L1 DH ðy1 ðtÞ; y2 ðtÞÞds 1

tk1

ðt  sÞa1 L1 DH ðy1 ðtÞ; y2 ðtÞÞds þ mL2 DH ðy1 ðtÞ; y2 ðtÞÞ 1

tk

t m Zk X 1 1 DH ðF ðy1 ðtÞÞ; F ðy2 ðtÞÞÞ  1 L1 DH ðy1 ðtÞ; y2 ðtÞÞ ðtk  sÞa1 ds C a k¼1 tk1

1 þ 1 L1 DH ðy1 ðtÞ; y2 ðtÞÞ C a

Since

Rt tk

Zt

ðt  sÞa1 ds þ mL2 DH ðy1 ðtÞ; y2 ðtÞÞ 1

tk

ðt  sÞa1 ds  T a , 1

1

DH ðF ðy1 ðtÞÞ; F ðy2 ðtÞÞÞ 

1 1 1 L1 DH ðy1 ðtÞ; y2 ðtÞÞmT a C a

1 1 þ 1 L1 DH ðy1 ðtÞ; y2 ðtÞÞT a þ mL2 DH ðy1 ðtÞ; y2 ðtÞÞ C a "

#

L1 1 DH ðF ðy1 ðtÞÞ; F ðy2 ðtÞÞÞ  DH ðy1 ðtÞ; y2 ðtÞÞ 1 ðm þ 1ÞT a þ mL2 C a The proof is completed if the map F is a fixed point Banach contraction map and this means, DH ðF ðy1 ðtÞÞ; F ðy2 ðtÞÞÞ  DH ðy1 ðtÞ; y2 ðtÞÞ For this purpose, we should have, "

5.6.2

# L1 1   ðm þ 1ÞT a þ mL2 \1 C 1a

Reproducing Kernel Hilbert Space Method (RKHSM)

This method is a semi analytical method to find the functional approximation of our problem. Here the RKHSM is explained very shortly,

240

5 Numerical Solution of Fuzzy Fractional Differential Equations

Definition—Hilbert space A Hilbert space is a complete infinite-dimensional inner-product space. The elements of this space can be functions defined on a set T. In particular, the abstract (RKHS), H, is a Hilbert space of functions defined on a set T such that there exists a unique function, Rðt; yÞ, defined on T T with the following properties: ðIÞ: Ry ðtÞ ¼ Rðt; yÞ 2 H 8t 2 T ðIIÞ: f ðtÞ; Ry ðtÞ ¼ f ð yÞ 8t 2 T 8f 2 H The function Rðt; yÞ is called the reproducing kernel of the abstract RKHSM. Definition—Kernel Let / be a mapping from T into the space H such that /i ¼ Rðt; ti Þ. A function R: T T ! R such that Ry ðtÞ ¼ Rðt; yÞ ¼ h/ðtÞ; /ð yÞi, for all t; y 2 T is called a kernel. Definition—Inner product and norm For an absolutely continuous real valued function xðm1Þ ðtÞ; on ½a; b and n W2m ½a; b ¼ xðtÞjxðmÞ ðtÞ 2 L2 ½0; 1;

x ð aÞ ¼ x ð bÞ ¼ 0

o

n R o b where L2 ½a; b ¼ xj a x2 dt\1 . The inner product and norm of the functions x; y 2 in W2m ½a; b are given respectively by hxðtÞ; yðtÞi ¼

m1 X

ðiÞ

Zb

ðiÞ

x ðaÞy ðaÞ þ

i¼0

and k xðtÞ kW2m ½a;b ¼

xðmÞ ðtÞxðmÞ ðtÞdt

a

pffiffiffiffiffiffiffiffiffiffi hx; xiW m ½a;b 2

Note In the special case of definition, in case m ¼ 1; ½a; b ¼ ½0; 1; xðtÞ is an absolutely continuous fuzzy valued function on ½0; 1, and n W21 ½0; 1 ¼ xðtÞjx0gH 2 L2 ½0; 1;

xð0Þ ¼ xð1Þ ¼ 0

o

The inner product and norm in W21 ½0; 1 are given respectively by Z1 hxðtÞ; yðtÞi ¼ xð0Þ  yð0Þ

0

x0gH ðtÞ  y0gH ðtÞdx

5.6 Numerical Method for Fuzzy Fractional Impulsive Differential Equations

and k x kW ¼

241

pffiffiffiffiffiffiffiffiffiffi hx; xiW 1 2

W21 ½0; 1 is a reproducing kernel space with reproducing kernel Rðt; yÞ that is defined as  Rðt; yÞ ¼

1 þ y; y  t 1 þ t; y [ t

Also another special case for m ¼ 2; n W22 ½0; 1 ¼ xðtÞjx00gH 2 L2 ½0; 1;

xð0Þ ¼ xð1Þ ¼ 0

o

where x0gH ðtÞ are absolutely continuous real valued functions on ½0; 1. The inner product and norm in W22 ½0; 1 are given respectively by hxðtÞ; yðtÞi ¼ xð0Þ  yð0Þ

x0gH ð0Þ



y0gH ð0Þ

Z1

x00gH ðtÞ  y00gH ðtÞdt

0

Note In all above mentioned relations all H-differences and gH-derivatives should be defined properly (exist) and the integration can be defined in level-wise form. The representation of the reproducing kernel function Rt ð yÞ is provided by R t ð yÞ ¼

5.6.3

1

2 2 6 ðy  aÞð2a  y þ 3tð2 þ yÞ  að6 þ 3t þ yÞÞ; 1 2 2 6 ðt  aÞð2a  t þ 3yð2 þ t Þ  að6 þ 3y þ tÞÞ;

y  t; y [ t:

Numerical Solving Fuzzy Fractional Impulsive Differential Equation in W22 ½0; 1

We show how RKHSM is applied to solve integral equation. Thus 8 Rt 1 > > y0 C 11 ðt  sÞa1  f ðs; yðsÞÞds; t 2 ½0; t1  > > ð Þ > a 0 > > < m Rtk P 1 ðtk  sÞa1  f ðs; yðsÞÞds ð1Þ yðtÞ ¼ y0 H ð1Þ Cð11Þ a k¼1 t > k1 > > > m    Rt > P 1 > 1 1 > Ik y tk ; t 2 ðt1 ; tk þ 1  : Cð1Þ ðt  sÞa  f ðs; yðsÞÞds H ð1Þ a

tk

k¼1

242

5 Numerical Solution of Fuzzy Fractional Differential Equations

if there exists a point t1 2 ð0; tk þ 1 Þ such that yðtÞ is i  gH differentiable on ½0; t1  and ii  gH differentiable on ðt1 ; tk þ 1 Þ. By impulsive effect, we have: DyðtÞ ¼ yð0 þ Þ gH yð0 Þ ¼ I0 ðyð0 ÞÞ and based on the definition of gH-difference, 

yð0 þ Þ ¼ I0 ðyð0 ÞÞ yð0 Þ; yð0 Þ ¼ yð0 þ Þ ð1ÞI0 ðyð0 ÞÞ;

t 2 ½0; t1 ; t 2 ðt1 ; tk þ 1 :

By substituting we have, 8 Rt 1 >   1 > ð Þ

I ð y ð 0 Þ Þ

ðt  sÞa1  f ðs; yðsÞÞds; t 2 ½0; t1 ; y 0 > 0 1 > CðaÞ > 0 > > < m Rtk P 1   ðtk  sÞa1  f ðs; yðsÞÞds yðtÞ ¼ yð0 Þ H ð1ÞI0 ðyð0 ÞÞ H ð1Þ Cð11Þ a > k¼1 tk1 > > > m Rt > P 1 > 1 1 > Ik ðyðtk ÞÞ; t 2 ðt1 ; tk þ 1 ; : H ð1Þ Cð1Þ ðt  sÞa  f ðs; yðsÞÞds H ð1Þ a t k¼1 k

Based on the assumptions,   y tkþ ¼ limh!0 þ yðtk þ hÞ;

  y tk ¼ limh!0 yðtk þ hÞ

we define yðt Þ ¼ yðtÞ. Thus 8 Rt k > > I0 ðyð0ÞÞ yð0Þ C 11 ðt  sÞa1  f ðs; yðsÞÞds; t 2 ½0; t1 ; > > ð Þ > a 0 > > < m Rtk P 1 ðtk  sÞa1  f ðs; yðsÞÞds yðtÞ ¼ yð0Þ ð1ÞI0 ðyð0ÞÞ H ð1Þ Cð11Þ a k¼1 t > k1 > > > m Rt > P 1 > 1 1 a > Ik ðyðtk ÞÞ; t 2 ðt1 ; tk þ 1 ; : H ð1Þ Cð1Þ ðt  sÞ  f ðs; yðsÞÞds ð1Þ a t k¼1 k

To use RKHSM we define linear operator, L ¼ W22 ½a; b ! W21 ½a; b;

t 2 ½0; tk þ 1 

5.6 Numerical Method for Fuzzy Fractional Impulsive Differential Equations

243

as follows: 8 Rt k 1 > > > yðtÞ H I0 ðyð0ÞÞ H yð0Þ H Cð11Þ ðt  sÞa  f ðs; yðsÞÞds; t 2 ½0; t1 ; > > a 0 > > < m Rtk P 1 ðtk  sÞa1  f ðs; yðsÞÞds

LyðtÞ ¼ yðtÞ ð1ÞI0 ðyð0ÞÞ H yð0Þ ð1Þ Cð11Þ a k¼1 t > k1 > > > m Rt > P 1 > 1 1 a > Ik ðyðtk ÞÞ; t 2 ðt1 ; tk þ 1 ; : ð1Þ Cð1Þ ðt  sÞ  f ðs; yðsÞÞds ð1Þ a t k¼1 k

It is clear that L is a bounded linear operator so the model changes to the following problems:  LyðtÞ ¼

F ðt; yðtÞ; T1 ðyðtÞÞÞ; F ðt; yðtÞ; T ðyðtÞÞ; SðyðtÞÞÞ;

t 2 ½0; t1  t 2 ðt 1 ; t k þ 1 

Such that F ðt; yðtÞ; T1 ðyðtÞÞÞ ¼ yðtÞ H I0 ðyð0ÞÞ H yð0Þ H T1 ðyðtÞÞ 1 T 1 ð yð t Þ Þ ¼  k  C a

Zt

ðt  sÞa1  f ðs; yðsÞÞds k

0

Subject to all H-differences exist and F ðt; yðtÞ; T ðyðtÞÞ; SðyðtÞÞÞ ¼ yðtÞ ð1ÞI0 ðyð0ÞÞ H yð0Þ ð1Þ

k X

Ik ðyðtk ÞÞ ð1ÞT ðyðtÞÞ ð1ÞSðyðtÞÞ

k¼1

where t m Zk k 1 X T ðyðtÞÞ ¼ k  ðtk  sÞa1  f ðs; yðsÞÞds C a k¼1 tk1

1 SðyðtÞÞ ¼ k  C a

Zt tk

ðt  sÞa1  f ðs; yðsÞÞds k

244

5 Numerical Solution of Fuzzy Fractional Differential Equations

where yðtÞ; LðyðtÞÞ 2 W21 ½a; b; F ðt; yðtÞ; T1 ðyðtÞÞÞ; F ðt; yðtÞ; T ðyðtÞÞ; SðyðtÞÞÞ 2 W21 ½a; b For all t 2 ½a; b. Using the following reproducing kernel function Rt ð yÞ, R t ð yÞ ¼

1

2 2 6 ðy  aÞð2a  y þ 3tð2 þ yÞ  að6 þ 3t þ yÞÞ; 1 2 2 6 ðt  aÞð2a  t þ 3yð2 þ t Þ  að6 þ 3y þ tÞÞ;

y  t; y [ t:

 Put /i ðtÞ ¼ Rðt; ti Þ and wi ðtÞ ¼ L /i ðtÞ where fti g1 i¼1 is dense in ½a; b and L is the adjoint operator of L. It is easy to see that





wi ðtÞ ¼ Ly Rt ð yÞ y¼ti ¼ Ly Rðt; ti Þ y¼ti 8 1 > > > Rðt; ti Þ I0 ðRð0; ti ÞÞÞ H Rð0; ti Þ H 1 > C > > a > > > t Z > > 1 > > > ðt  sÞa1  f ðs; Rðs; ti ÞÞds; > > > > > 0 > > > > Rðt; ti Þ ð1ÞI0 ðRð0; ti ÞÞ Rð0; ti Þ > > > < m X Ik ðRðtk ; ti ÞÞ ð1Þ

ð1Þ > > k¼1 > > > t > m Zk > 1 > 1 X > >   ðtk  sÞa1  f ðs; Rðs; ti ÞÞds > 1 > C > > a k¼1 > tk1 > > > > Zt > > > 1 1 > >

ð1Þ 1 ðt  sÞa1  f ðs; Rðs; ti ÞÞds; > > : C a

t 2 ½0; t1 

t 2 ðt 1 ; t k þ 1 

tk

1 Note If fti g1 i¼1 is dense on ½a; b then fwi gi¼1 is the complete function system of the space W22 ½a; b and wi ðtÞ ¼ ½Lt Rt ð yÞy¼ti where the subscript t in the operator L indicates that the operator L applies to the function of t.

Note wi ðtÞ ¼ hðL /i ÞÞ; Rt ð yÞiW 2 ¼ h/i ; Lt Rt ð yÞiW 2 ¼ Lt Rt ð yÞjy¼ti ; 2

Clearly wi 2 W22 ½a; b.

2

t 2 ½0; tk þ 1 

5.6 Numerical Method for Fuzzy Fractional Impulsive Differential Equations

245

Note For each fixed yðtÞ 2 W22 ½a; b, let hyðtÞ; wi ðtÞiW 2 ½a;b ¼ 0; 2

t 2 ½0; tk þ 1 

i ¼ 1; 2; . . .; it means hyðtÞ; L /i ðtÞiW 2 ½a;b ¼ hLt yðtÞ; /i ðtÞiW 2 ½a;b ¼ Lt yðti Þ ¼ 0; 2

2

t 2 ½0; tk þ 1 

Assume that fti g1 i¼1 is dense on ½a; b and so Lt yðtÞ ¼ 0. It follows that y ¼ 0 from the existence of L1 t . b ðtÞg1 of Definition—Orthonormal system The orthonormal system f w i i¼1 2 W ½a; b can be derived from the Gram-Schmidt orthogonalization process of fwi ðtÞ1 i¼1 g, b ðt Þ ¼ w i

i X

bik wk ðtÞ;

t 2 ½0; tk þ 1 

k¼1

where bik ; i ¼ 1; 2; . . .; k ¼ 1; 2; . . . are coefficients of Gram-Schmidt orthonorb ðtÞg1 is an orthonormal system, could be determined by malizarion and f w i i¼1 solving the following equations. b ¼ Bik ¼ wi ; w i

b ð aÞ þ w 0 ð aÞ w b 0 ð aÞ þ wi ðaÞ w i i i

Zb

b 00 ðtÞdt; w00i ðtÞ w i

t 2 ½0; tk þ 1 

1 ffi; bii ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R b  00 Pi1 2 i 2 2 2 0 ½ðwi ðaÞÞ þ ðwi ðaÞÞ þ a wi ðtÞÞ dt  k¼1 Bik

t 2 ½0; tk þ 1 

a

where

bij ¼ bii 



i1 X

! Bik  bkj

i ¼ 1; 2; . . .;

j ¼ 1; 2; . . .; i  1;

k¼j

k ¼ 1; 2; . . .; i  1 Remark If fti g1 i¼1 is dense on ½a; b and the solution of  LyðtÞ ¼

F ðt; yðtÞ; T1 ðyðtÞÞÞ; F ðt; yðtÞ; T ðyðtÞÞ; SðyðtÞÞÞ;

Is unique the it is in the form of,

t 2 ½0; t1  t 2 ðt 1 ; t k þ 1 

246

5 Numerical Solution of Fuzzy Fractional Differential Equations

uð t Þ ¼

1 X i X

b ðtÞ; bik  yðtk Þ  w i

t 2 ½0; tk þ 1 

i¼1 k¼1

Using b ðt Þ ¼ w i

i X

bik wk ðtÞ;

1 X

uð t Þ ¼

b ðt Þ b ðtÞi 1 w hyðtÞ; w i i W 2

i¼1

k¼1

we have uð t Þ ¼

1 X

b ðt Þ ¼ b ðtÞi 1 w hyðtÞ; w i i W

i¼1

¼

1 X i X

2

1 X i X

1 X i X

i X

1 X i X

2

2

b ðt Þ bik  hyðtÞ; L /k ðtÞiW 1 w i 2

i¼1 k¼1

b ðt Þ ¼ bik  hLyðtÞ; /k ðtÞiW 1 w i 2

b ðtÞ; bik yðtk Þ w i

b ðtÞ bik wk ðtÞiW 1 w i

k¼1

b ðt Þ ¼ bik  hyðtÞ; wk ðtÞiW 1 w i

i¼1 k¼1

¼

hyðtÞ;

i¼1

i¼1 k¼1

¼

1 X

1 X i X

b ðt Þ bik  Lyðtk Þ w i

i¼1 k¼1

t 2 ½0; tk þ 1 

i¼1 k¼1

On the other hand, uðtÞ 2 W22 ½a; b and LuðtÞ ¼ 0; yð t Þ ¼

1 X

b ðtÞ; ai  w i

t 2 ½0; tk þ 1 :

t 2 ½0; tk þ 1 

i¼1

where ai 2 FR ; b ðtÞ; ai ¼ yðtÞ; w i

t 2 ½0; tk þ 1 

b ðtÞg1 on the are the Fourier series expansion about normal orthogonal system f w i i¼1 2 Hilbert space W2 ½a; b. Thus the series, 1 X

b ðtÞ; ai  w i

t 2 ½0; tk þ 1 

i¼1

is convergent in the sense of k  kW22 and the proof would be completed. Now the approximate solution yN ðtÞ can be obtained by the N-term intercept of the exact solution yðtÞ and

5.6 Numerical Method for Fuzzy Fractional Impulsive Differential Equations

yN ð t Þ ¼

N X i X

b ðtÞ; bik  yðtk Þ  w i

247

t 2 ½0; tk þ 1 

i¼1 k¼1

In the sequel, a new iterative method to achieve the solution is also presented. If Ai ¼

i X

bik  yðtk Þ;

t 2 ½0; tk þ 1 

k¼1

The equation can be written as, yðtÞ ¼

1 X

b ðtÞ; Ai  w i

t 2 ½0; tk þ 1 

k¼1

Now suppose, for some ðti ; yðti ÞÞ the value of yðtÞ is known. There is no problem if we assume i ¼ 1. We put y0 ðt1 Þ ¼ yðt1 Þ and define the N-term approximation to yðtÞ by yN ðtÞ ¼

N X

b ðtÞ; ci  w i

t 2 ½0; tk þ 1 

bik  yk1 ðtk Þ;

t 2 ½0; tk þ 1 

k¼1

where ci ¼

k X i¼1

In the following, it would be proven that the approximate solution yN ðtÞ in the iterative is convergent to the exact solution of the following equations uniformly.  LyðtÞ ¼

F ðt; yðtÞ; T1 ðyðtÞÞÞ; F ðt; yðtÞ; T ðyðtÞÞ; SðyðtÞÞÞ;

t 2 ½0; t1  t 2 ðt 1 ; t k þ 1 

Remark Suppose that kyN ðtÞkW 2 is bounded. If fti g1 i¼1 is dense on ½a; b then N2 term approximate solution yN ðtÞ converges to the exact solution yðtÞ of LyðtÞ and yðtÞ ¼ lim

n!1

n X

ci  wi ðtÞ;

t 2 ½0; tk þ 1 

i¼1

where ci are give by ci ¼

k X i¼1

bik  yk1 ðtk Þ;

t 2 ½0; tk þ 1 

248

5 Numerical Solution of Fuzzy Fractional Differential Equations

In level-wise form for all 0  r  1 we have, yl ðt; r Þ ¼ lim

n!1

yu ðt; r Þ ¼ lim

n!1

n X

ci;l ðt; r Þwi ðtÞ;

t 2 ½0; tk þ 1 

ci;u ðt; r Þwi ðtÞ;

t 2 ½0; tk þ 1 

bik  yk1;l ðtk ; r Þ;

t 2 ½0; tk þ 1 

bik  yk1;u ðtk ; r Þ;

t 2 ½0; tk þ 1 

i¼1 n X i¼1

where ci;l ðt; r Þ ¼

k X i¼1

ci;u ðt; r Þ ¼

k X i¼1

Proof. First of all, the convergence of yN ðtÞ is going to be proved. We infer b yn þ 1;l ðt; r Þ ¼ yn;l ðt; r Þ þ cn þ 1;l ðt; r Þ w n þ 1 ðtÞ;

t 2 ½0; tk þ 1 

b yn þ 1;u ðt; r Þ ¼ yn;u ðt; r Þ þ cn þ 1;u ðt; r Þ w n þ 1 ðtÞ;

t 2 ½0; tk þ 1 

  For any arbitrary and fixed r, it is obvious that the sequences yn;l ðtÞW 2 and 2   yn;u ðtÞ 2 are monotonically increasing. Because both are bounded and converW2 P1 2 P1 2 gent. Then i¼1 ci;l and i¼1 ci;u are bounded and this implies that 1 1 2 fci;l gi¼1 ; fci;u gi¼1 2 L . If m [ n then nX þ 1

jjym;l  yn;l jj2W 2 ½a;b ¼ jj 2

nX þ1    yi;l  yi1;l jj2W 2 ½a;b ¼ jj yi;l  yi1;l jj2W 2 ½a;b 2

i¼m

jjym;u  yn;u jj2W 2 ½a;b ¼ jj

nX þ 1

2

2

i¼m

nX þ1    yi;u  yi1;u jj2W 2 ½a;b ¼ jj yi;u  yi1;u jj2W 2 ½a;b 2

i¼m

2

i¼m

    So jj yi;l  yi1;l jj2W 2 ½a;b ¼ c2i;l ; jj yi;u  yi1;u jj2W 2 ½a;b ¼ c2i;u . 2 2 Consequently if n ! 1 jjym;l  yn;l jj2W 2 ½a;b ¼ 2

1 X i¼1

c2i;l ! 0;

jjym;u  yn;u jj2W 2 ½a;b ¼ 2

1 X i¼1

c2i;u ! 0

5.6 Numerical Method for Fuzzy Fractional Impulsive Differential Equations

5.6.4

249

Combination of RKHM and FDTM

One of the applications of fuzzy Taylor expansion is fractional differential transformation method. In fact, in the Taylor series instead of the differentials some functions in terms of time and order of diferential. In the fuzzy fractional Taylor expansion assume that the derivative is defined as, ðk Þ

ygH ðtÞ ¼

dk ygH ðtÞ :¼ /ðt; kÞ; dtk

Such that

k ¼ 0; 1; . . .; m  1;

dk Y ðkÞ :¼ /ðti ; kÞ ¼ ygH ðtÞ dtk

t 2 ½a; b

t¼ti

where Y ðkÞ is called the spectrum of yðtÞ at t ¼ ti and yðtÞ  D1 Y ðkÞ and the symbol “D” denoting the differential transformation process. This transformation can be explained in the fractional derivative in the following form. As it is mentioned, the solution of fuzzy fractional impulsive differential equations, 8 1 < c Da yðtÞ ¼ f ðt; yðtÞÞ; t 2 ½0; T ; DyðtÞjt¼tk ¼ Ik y tk ðtÞ : y ð 0Þ ¼ y 0

t 6¼ tk ;

m  1\ 1a \m;

m2N

Do have a fuzzy peace-wise continuous solution with linear and nonlinear terms. Such that 8 Rt k > 1 > y ð t Þ I ð y ð 0 Þ Þ y ð 0 Þ ðt  sÞa1  f ðs; yðsÞÞds; t 2 ½0; t1 ; > H 0 H H 1 > CðaÞ > 0 > > < m Rtk P 1 ðtk  sÞa1  f ðs; yðsÞÞds

LyðtÞ ¼ yðtÞ ð1ÞI0 ðyð0ÞÞ H yð0Þ ð1Þ Cð11Þ a k¼1 t > k1 > > > m Rt > P 1 > 1 1 > Ik ðyðtk ÞÞ; t 2 ðt1 ; tk þ 1 ; : ð1Þ Cð1Þ ðt  sÞa  f ðs; yðsÞÞds ð1Þ a t k¼1 k

The linear term,  LyðtÞ ¼

F ðt; yðtÞ; T1 ðyðtÞÞÞ; F ðt; yðtÞ; T ðyðtÞÞ; SðyðtÞÞÞ;

t 2 ½0; t1  t 2 ðt 1 ; t k þ 1 

250

5 Numerical Solution of Fuzzy Fractional Differential Equations

Such that F ðt; yðtÞ; T1 ðyðtÞÞÞ ¼ yðtÞ H I0 ðyð0ÞÞ H yð0Þ H T1 ðyðtÞÞ where 1 T 1 ð yð t Þ Þ ¼  k  C a

Zt

ðt  sÞa1  f ðs; yðsÞÞds k

0

Subject to all H-differences exist and F ðt; yðtÞ; T ðyðtÞÞ; SðyðtÞÞÞ ¼ yðtÞ ð1ÞI0 ðyð0ÞÞ H yð0Þ ð1Þ

k X

Ik ðyðtk ÞÞ ð1ÞT ðyðtÞÞ ð1ÞSðyðtÞÞ

k¼1

where t m Zk k 1 X T ðyðtÞÞ ¼ k  ðtk  sÞa1  f ðs; yðsÞÞds C a k¼1 tk1

1 Sð yð t Þ Þ ¼  k  C a

Zt

ðt  sÞa1  f ðs; yðsÞÞds k

tk

By applying L1 to both sides of t  Lt ðyðtÞÞ ¼

t 2 ½0; t1  F ðt; yðtÞ; T1 ðyðtÞÞÞ; F ðt; yðtÞ; T ðyðtÞÞ; SðyðtÞÞÞ; t 2 ðt1 ; tk þ 1 

We have  yð t Þ ¼

L1 t 2 ½0; t1  t ðF ðt; yðtÞ; T1 ðyðt ÞÞÞÞ; L1 ð F ð t; y ð t Þ; T ð y ð t Þ Þ; S ð y ð t Þ Þ Þ Þ; t 2 ðt1 ; tk þ 1  t

The FDTM introduces the exact solution yðtÞ and the nonlinear function N ðt; yðtÞÞ by infinite series Lt yðtÞ

1 X i¼0

! yi ð t Þ

 ¼

F ðt; yðtÞ; T1 ðyðtÞÞÞ N ðt; yðtÞÞ; t 2 ½0; t1  F ðt; yðtÞ; T ðyðtÞÞ; SðyðtÞÞÞ N ðt; yðtÞÞ; t 2 ðt1 ; tk þ 1 

5.6 Numerical Method for Fuzzy Fractional Impulsive Differential Equations

251

where N ðt; yðtÞÞ ¼

1 X k¼0

k

Yak ðk Þta

where Yak ðkÞ; k ¼ 0; 1; . . . are fractional FDTM polynomials for the nonlinear term N ðt; yðtÞÞ and can be found from the following formula, ( Yak ðkÞ ¼

1 Cðak þ 1Þ

h k i DaCgH yðtÞ

t¼t0

0

k a k a

2 Zþ 62 Z þ

Now, we can evaluate yðti Þ; i ¼ 0; 1; . . . as follows,  yi ðtÞ :¼ yðti Þ ¼

ti 2 ½0; t1  L1 t ðF ðti ; yðti Þ; T1 ðyðti ÞÞÞÞ; L1 ð F ð t ; y ð t Þ; T ð y ð t Þ Þ; S ð y ð t Þ Þ Þ Þ; t i 2 ðt 1 ; t k þ 1  i i i i t

Also  yð t Þ ¼

t 2 ½0; t1  L1 t ðF ðt; yðtÞ; T1 ðyðt ÞÞÞÞ; L1 ð F ð t; y ð t Þ; T ð y ð t Þ Þ; S ð y ð t Þ Þ Þ Þ; t 2 ðt1 ; tk þ 1  t

Substituting yields,

yð t Þ

1 X

yi ð t Þ ¼ yð t Þ

i¼0

yð t Þ

1 X

81 P 1 > > < Lt ðF ðti ; yðti Þ; T1 ðyðti ÞÞÞÞ;

t 2 ½0; t1 

> > :

t 2 ðt 1 ; t k þ 1 

i¼0 1 P i¼0

L1 t ðF ðti ; yðti Þ; T ðyðti ÞÞ; Sðyðti ÞÞÞÞ;

yi ð t Þ

i¼0

8 1  P k > 1 1 > Yka ðkÞta ; > < Lt ðF ðt; yðtÞ; T1 ðyðtÞÞÞÞ Lt k¼0 1  ¼ P > k > 1 1 > Lt ðF ðt; yðtÞ; T ðyðtÞÞ; SðyðtÞÞÞÞ Lt Yka ðkÞta ; :

t 2 ½0; t1  t 2 ðt 1 ; t k þ 1 

k¼0

According to the FFDTM, the components yk ðtÞ can be determined as,

252

5 Numerical Solution of Fuzzy Fractional Differential Equations

( Y0 ð0Þt ¼ 0

( 1 a

Y1a ð1Þt ¼ ( 2 a

Y2a ð2Þt ¼

L1 t ðF ðt0 ; yðt0 Þ; T1 ðyðt0 ÞÞÞÞ;

L1 t1 2 ½0; t1  t ðF ðt1 ; yðt1 Þ; T1 ðyðt1 ÞÞÞÞ; 1 Lt ðF ðt1 ; yðt1 Þ; T ðyðt1 ÞÞ; Sðyðt1 ÞÞÞÞ; t1 2 ðt1 ; tk þ 1  L1 t ðF ðt2 ; yðt2 Þ; T1 ðyðt2 ÞÞÞÞ;

Yma ðmÞt ¼

t2 2 ½0; t1 

L1 t ðF ðt2 ; yðt2 Þ; T ðyðt2 ÞÞ; Sðyðt2 ÞÞÞÞ; t2 2 ðt1 ; tk þ 1  (

m a

t0 2 ½0; t1 

L1 t ðF ðt0 ; yðt0 Þ; T ðyðt0 ÞÞ; Sðyðt0 ÞÞÞÞ; t0 2 ðt1 ; tk þ 1 

.. . L1 t ðF ðtm ; yðtm Þ; T1 ðyðtm ÞÞÞÞ;

tm 2 ½0; t1 

L1 t ðF ðtm ; yðtm Þ; T ðyðtm ÞÞ; Sðyðtm ÞÞÞÞ; tm 2 ðt1 ; tk þ 1 

b ðtÞg1 of W 2 ½a; b, Therefore, it can be approximated by f w i i¼1 1 X i h k i X ta b ðtÞ; k  DaCgH yðtÞ ¼ bik  yðtk Þ w i t¼t0 C a þ1 i¼0 k¼1 k¼1

m X

5.6.5

k

t 2 ½0; tk þ 1 

Algorithm

For finding the approximate and exact solutions yN ðt; r Þ and yðt; r Þ for fuzzy impulsive fractional differential equations respectively, we do the following main steps: Step 1. Fix t; y 2 ½a; b, 1:1 1:2 1:3 1:4 1:5 1:6

If y  t set R2t ð yÞ ¼ 16 ðy  aÞð2a2  y2 þ 3tð2 þ yÞ  að6 þ 3t þ yÞÞ Else set R2t ð yÞ ¼ 16 ðt  aÞð2a2  t2 þ 3yð2 þ tÞ  að6 þ 3y þ tÞÞ For i ¼ 1; 2; . . .; n; h ¼ 1; 2; . . .; m and ¼ 1; 2, do the following: i1 Set ti ¼ n1 , h1 Set rh ¼ m1 , Set wi ðti Þ ¼ L1 R2t ð yÞjy¼ti ;

t 2 ½0; tk þ 1 

Output: The orthogonal function system wi ðti Þ. Step 2. b ¼ w ðaÞ w b ð aÞ þ w 0 ð aÞ w b 0 ð aÞ þ Bik ¼ wi ; w i i i i i

Zb a

b 00 ðtÞdt; w00i ðtÞ w i

t 2 ½0; tk þ 1 

5.6 Numerical Method for Fuzzy Fractional Impulsive Differential Equations

1 ffi; bii ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R b  00 Pi1 2 i 2 2 2 0 ½ðwi ðaÞÞ þ ðwi ðaÞÞ þ a wi ðtÞÞ dt  k¼1 Bik bij ¼ bii 



i1 X

253

t 2 ½0; tk þ 1 

! Bik  bkj

i ¼ 1; 2; . . .;

j ¼ 1; 2; . . .; i  1;

k¼j

k ¼ 1; 2; . . .; i  1 Output: The orthogonalization coefficients bik Step 3. b ðtÞ ¼ Pi b w ðtÞ; ðb [ 0; i ¼ 1; 2; . . .Þ: Set w ii i k¼1 ik k b ðt Þ Output: The orthogonal function system w i Step 4. Set y0 ðt1 Þ ¼ yðt1 Þ Step 5. Set n ¼ 1 Step 6. Set cn ¼

n X

bnk yk1 ðtk Þ;

t 2 ð0; tk þ 1 :

k¼1

Step 7. Set Example Let us consider the fuzzy impulsive fractional differential equation, 1

DaCgH yðtÞ ¼

t ; 10ð1 þ tÞ

t 2 ½0; 1;

1 t 6¼ ; 2

1 m  1\ \m; a

m 2 N;

Set Ik ðt Þ ¼

ð3r  1Þt ð3  r Þt ; ; tþ3 tþ3

f ðt; yðtÞÞ ¼

ðr  1Þt ð1  r Þt ; 10ð1 þ tÞ 10ð1 þ tÞ

The singleton zero number is considered as the initial value. Choosing k ¼ 2 and 1a ¼ 12 the results for yl ðt; r Þ are shown in Tables 5.6 and 5.7 for sevel r-levels and t. Table 5.7 shows the results for the upper function of solution yu ðt; r Þ for several values of levels and time. Example Let us consider the fuzzy impulsive fractional equation, k

DaCgH yðtÞ ¼ m2N

ty2 ðtÞ ; ð3 þ tÞð1 þ y2 ðtÞÞ

t 2 J :¼ ½0; 1;

1 t 6¼ ; 2

1 m  1\ \m; a

0.2

−0.00480 −0.00589 −0.00713 −0.00854 −0.01015 −0.01196 −0.01401 −0.01630 −0.01886 −0.02171

0.1

−0.00540 −0.00662 −0.00802 −0.00961 −0.01142 −0.01346 −0.01576 −0.01834 −0.02122 −0.02443

r/t

0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1

−0.00420 −0.00515 −0.00748 −0 .00888 −0.01047 −0.01225 −0.01426 −0.01426 −0.01650 −0.01899

0.3

Table 5.6 The results for yl ðt; r Þ with k ¼ 2 and 1a ¼ 12 0.4 −0.00360 −0.00441 −0.00535 −0.00761 −0.00897 −0.01012 −0.01050 −0.01212 −0.01414 −0.01628

0.5 −0.0030 −0.00368 −0.00445 −0.00534 −0.00634 −0.00747 −0.00875 −0.01018 −0.01178 −0.01365

0.6 −0.00243 −0.00294 −0.00365 −0.00427 −0.00507 −0.00598 −0.00700 −0.00814 −0.00942 −0.01084

0.7 −0.00180 −0.00221 −0.00267 −0.00320 −0.00380 −0.00448 −0.00525 −0.00610 −0.00706 −0.00813

0.8 −0.00120 −0.00147 −0.00178 −0.00213 −0.00254 −0.00350 −0.00407 −0.22024 −0.00471 −0.00542

0.9 −0.00060 −0.00074 −0.00089 −0.00107 −0.00127 −0.00149 −0.00175 −0.00203 −0.00235 −0.00271

1 0 0 0 0 0 0 0 0 0 0

254 5 Numerical Solution of Fuzzy Fractional Differential Equations

0.1

0.00539 0.006608 0.008001 0.009588 0.011386 0.013417 0.015702 0.018264 0.021128 0.024319

r/t

0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.905 0.95 1

0.004792 0.005874 0.007113 0.008524 0.010123 0.011929 0.013960 0.016239 0.018785 0.021623

0.2

0 .004193 0.005141 0.006225 0 .007459 0 .008859 0 .010439 0 .012218 0 .014212 0 .016441 0 .018925

0.3

Table 5.7 The results for yu ðt; r Þ with k ¼ 2 and 1a ¼ 12 0.4 0.003595 0.004407 0.005336 0.006395 0.007595 0.00895 0.010475 0.012184 0.014095 0.016225

0.5 0.002886 0.003673 0.004447 0.005330 0.00633 0 0.00746 0 0.000873 0.010156 0.011749 0.013524

0.002397 0.002939 0.003558 0.004264 0.005065 0.005969 0.006986 0.008126 0.009401 0.018220

0.6 0.001798 0.002204 0.002669 0.003199 0.003799 0.004477 0.005241 0.006096 0.007035 0.008119

0.7

0.001199 0.001470 0.003422 0.001780 0.002533 0.002986 0.003494 0.004065 0.004703 0.005414

0.8

0.000599 0.000735 0.00089 0.001076 0.001267 0.001493 0.001748 0.002033 0.002352 0.002708

0.9

0 0 0 0 0 0 0 0 0 0

1

5.6 Numerical Method for Fuzzy Fractional Impulsive Differential Equations 255

0.1

0.013806 0.027612 0.044673 0.065519 0.090734 0.120964 0.156914 0.199357 0.249129 0.307135

r/t

0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1

0.3

−0.04988 −0.03913 −0.02584 −0.009610 0 .010039 0 .033597 0 .061620 0 .094710 0 .133522 0 .178764

0.2

−0.01804 −0.005760 −0.009418 0.027961 0.050394 0.077291 0.109282 0.147954 0.191353 0.242986

Table 5.8 The results for yl ðt; r Þ with k ¼ 2 and 1a ¼ 12 0.4 −0.00817 −0.07251 −0.06111 −0.04719 −0.03033 −0.01012 0.013927 0.042324 0.075635 0.114469

0.5 −0.11385 −0.10589 −0.09639 −0.08477 −0.07072 −0.05386 −0.03380 −0.01010 0.01769 0.05010

0.6 −0.14543 0.139280 −0.13167 −0.12237 −0.11112 −0.09762 −0.08155 −0.06257 −0.04031 −0.01434

0.7 −0.17729 −0.17267 −0.16696 −0.15998 −0.15153 −0.14140 −0.12933 −0.11509 −0.09836 −0.07886

0.8 −0.20915 −0.20607 −0.20226 −0.19760 −0.19176 −0.18520 −0.17715 −0.22024 −0.15648 −0.14346

0.9 −0.27288 −0.23947 −0.23757 −0.23523 −0.23241 −0.22903 −0.22500 −0.27288 −0.21465 −0.20813

1 0 0 0 0 0 0 0 0 0 0

256 5 Numerical Solution of Fuzzy Fractional Differential Equations

0.1

0.028928 0.020779 0.015366 0.011649 0.009022 0.007121 0.005713 0.004652 0.003838 0.003204

r/t

0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1

0.025721 0.018476 0.013663 0.010357 0.008022 0.006331 0.00508 0.004137 0.003413 0.002849

0.2

0 .022512 0.016171 0.011958 0 .009065 0 .007022 0 .005542 0 .004447 0 .003621 0 .002987 0 .002494

0.3

Table 5.9 The results for yu ðt; r Þ with k ¼ 2 and 1a ¼ 12 0.019302 0.013865 0.011025 0.007773 0.006020 0.007452 0.003813 0.003105 0.002561 0.002138

0.4 0.016089 0.011557 0.008547 0.006479 0.005019 0.003961 0.003178 0.002588 0.002135 0.001783

0.5 0.012875 0.009249 0.006839 0.005185 0.004016 0.003170 0.002543 0.002071 0.001709 0.001427

0.6 0.009659 0.006938 0.005131 0.003890 0.003013 0.002378 0.001908 0.001554 0.001282 0.001070

0.7

0.006441 0.004627 0.003422 0.002594 0.002009 0.001586 0.001273 0.001036 0.000855 0.000714

0.8

0.003221 0.002314 0.001711 0.001297 0.001005 0.000793 0.000636 0.000518 0.000428 0.000357

0.9

0 0 0 0 0 0 0 0 0 0

1

5.6 Numerical Method for Fuzzy Fractional Impulsive Differential Equations 257

258

5 Numerical Solution of Fuzzy Fractional Differential Equations

The singltone zero number is considered as the initial value. Set

t3 ðr  1Þ t 3 ð1  r Þ ; f ðt; yðtÞÞ ¼ ; ð3 þ tÞð1 þ t2 Þ ð3 þ tÞð1 þ t2 Þ

Ik ðtÞ ¼

t ; tþ2

t 2 ½0; 1Þ

Again k ¼ 2 and 1a ¼ 12, the results are shown in Tables 5.8 and 5.9.

Chapter 6

Applications of Fuzzy Fractional Differential Equations

6.1

Introduction

In general, fractional calculus deals with the generalization of differentiation and integration of non-integer orders. In recent years, fractional calculus has played a significant role in several sciences such as physics, chemistry, biology, electronics, and control theory. In this chapter, first the fuzzy fractional optimal control problem is investigated and then the fuzzy fractional diffusion with applications in drug release are explained.

6.2

Fuzzy Fractional Calculus—Preliminaries for Control Problem

Fractional optimal control problems are optimal control problems associated with fractional dynamic systems. As defined by Agrawal, a fractional dynamic system is a system whose dynamics is described by fractional differential equations. It has been demonstrated that fractional order differential equations model dynamic systems and processes more accurately than integer order differential equations. Therefore, the solution of fractional differential equations and the problem containing them with analytical and numerical schemes are of growing interest. The fractional optimal control theory is a novel topic in mathematics. The fractional optimal control problems may be defined in terms of different types of fractional derivatives. But the most important types of fractional derivatives are the Riemann-Liouville and the Caputo fractional derivatives. It is notable to mention that the uncertainty is inherent in most real-world systems. Fuzzy set theory is a powerful tool for modeling uncertainty and for processing vague or subjective information in mathematical models, which has been applied to a wide variety of real problems. Fuzzy fractional optimal control problems are © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. Allahviranloo, Fuzzy Fractional Differential Operators and Equations, Studies in Fuzziness and Soft Computing 397, https://doi.org/10.1007/978-3-030-51272-9_6

259

260

6 Applications of Fuzzy Fractional Differential Equations

fractional optimal control problems with ambiguity, which could appear, for example, in parameter values, functional relationships, or initial conditions. Like integer-order optimal control problems, the formulations stem from Fractional variational calculus (which is extended for fuzzy case). However, to our best knowledge, there are few reports on the necessary optimality conditions for fuzzy fractional optimal control problems. Only limited work has been done in the area of fuzzy variational problems, or more specifically, in the area of fuzzy fractional optimal control. It seems that, it is a new idea to derive the necessary optimality conditions for fuzzy fractional optimal control problems by using the fuzzy generalized Hakahara differentiability concept. To this end, some basic concepts such as generalized Hakahara differentiability (gH-differentiability) and integration for fully fuzzy functions that have been introduced in Chap. 2, are used. Furthermore, some fuzzy fractional theorems such as a formula for fuzzy fractional integration by parts are proved. Then, Euler-Lagrange equations (necessary optimality conditions) for them are derived. Finally, the sufficient conditions are discussed. Here, we reconsider some required and considerable fractional definitions and concepts of fuzzy valued functions. The fuzzy fractional Riemann-Liouville derivative operators (RL) and Caputo are brought (all have been fully discussed in Chap. 2). To remind, the derivatives and integral operators are in the following form. RL—Fuzzy fractional integral I aRL f ð xÞ ¼

1  CðaÞ

Zb

ðt  xÞa1 f ðtÞdt;

0\a\1

x

Level-wise form   I aRL f ðx; r Þ ¼ I aRL fl ðx; r Þ; I aRL fu ðx; r Þ ;

x 2 ½a; b

where I aRL fl ðx; r Þ

1 ¼ CðaÞ

Zx ðx  t Þ

a1

I aRL fu ðx; r Þ

fl ðt; r Þdt;

a

1 ¼ CðaÞ

Zx

ðx  tÞa1 fu ðt; r Þdt

a

RL—Fuzzy fractional derivative DaRLgH f ð xÞ

1 d  ¼ Cð1  aÞ dx

Zx a

ðx  tÞa f ðtÞdt;

0\a\1

6.2 Fuzzy Fractional Calculus—Preliminaries for Control Problem

261

or 0 x 10 Z 1  @ ðx  tÞa f ðtÞdtA ; DaRLgH f ð xÞ ¼ Cð1  aÞ a

0\a\1

gH

In the definition of RL-derivative if the right area is used then 0 b 10 Z 1  @ ðt  xÞa f ðtÞdtA ; DaRLgH f ð xÞ ¼ Cð1  aÞ x

0\a\1

gH

Level-wise form      DaRLgH f ðx; r Þ ¼ min DaRL fl ðx; r Þ; DaRL fu ðx; r Þ ; max DaRL fl ðx; r Þ; DaRL fu ðx; r Þ where DaRL fl ðx; r Þ

1 d ¼ Cð1  aÞ dx

DaRL fu ðt; r Þ ¼

1 d Cð1  aÞ dx

Zx

ðx  tÞa fl ðt; r Þdt;

a

Zx

ðx  tÞa fu ðt; r Þdt

a

Caputo—Fuzzy fractional gH-derivative DaCgH f ð xÞ

1  ¼ Cð1  aÞ

Zx

0 ðx  tÞa fgH ðtÞdt;

0\a\1

a

Level-wise form      DaCgH f ðx; r Þ ¼ min DaC fl ðx; r Þ; DaC fu ðx; r Þ ; max DaC fl ðx; r Þ; DaC fu ðx; r Þ where DaC fl ðx; r Þ ¼ DaC fu ðx; r Þ ¼

1 Cð1  aÞ 1 Cð1  aÞ

Zx

ðx  tÞa fl0 ðt; r Þdt;

a

Zx a

ðx  tÞa fu0 ðt; r Þdt

262

6 Applications of Fuzzy Fractional Differential Equations

Relation between fuzzy RL and CK derivatives If the RL derivative DaRLgH f ð xÞ exists in ½a; b and 0\a\1,   DaCK gH f ð xÞ ¼ DaRLgH f ð xÞgH f ðaÞ We know that,  0 ð1aÞ DaRLgH f ð xÞ ¼ I RL f ð xÞ: gH

where ð1aÞ I RL f ð xÞ

1  ¼ Cð1  aÞ

Zx

ðx  sÞa f ðsÞds

a

And we have, ð1aÞ

I RL f ðaÞ ¼

ðx  aÞ1a  f ð aÞ Cð2  aÞ

If a :! 1 þ a; DaRLgH f ðaÞ ¼ I a RL f ðaÞ ¼

ðx  aÞa  f ð aÞ Cð1  aÞ

Then   DaCK gH f ð xÞ ¼ DaRLgH f ð xÞgH f ðaÞ ¼ DaRLgH f ð xÞgH DaRLgH f ðaÞ ¼ DaRLgH f ð xÞgH

ðx  aÞa  f ð aÞ Cð1  aÞ

So, the relation is, DaCK gH f ð xÞ ¼ DaRLgH f ð xÞgH

ðx  aÞa  f ð aÞ C ð 1  aÞ

Two sides in the level-form and based on the first case of gH difference, we have, Case i − gH difference, DaRLgH f ð xÞ ¼ DaCK gH f ð xÞ 

ðx  aÞa  f ð aÞ Cð1  aÞ

6.2 Fuzzy Fractional Calculus—Preliminaries for Control Problem

263

Case ii − gH difference, DaCK gH f ð xÞ ¼ DaRLgH f ð xÞ  ð1Þ

ðx  aÞa  f ð aÞ Cð1  aÞ

Note Please note that this relation can also be obtained by using the integration by part that is explained in Chap. 2. To prove we need the following relation, 0 B @

ZvðxÞ

10 C gðt; xÞdtA ¼

u ð xÞ

gH

ZvðxÞ

g0gH ðt; xÞdt  v0 ð xÞ  gðvð xÞ; xÞgH u0 ð xÞ  gðvð xÞ; xÞ

u ð xÞ

where uð xÞ; vð xÞ are real valued functions. This can be proved very easily by using the level-wise form. Proof (Relation of Caputo and RL derivative) By integration part and assuming, u ¼ f ðtÞ; ðx  tÞa dt ¼ dv we have, 0 x 10 Z 1 a  @ ðx  tÞ f ðtÞdtA DaRLgH f ð xÞ ¼ Cð1  aÞ 0 ¼

a

gH

a þ 1

1  ðx  t Þ @ Cð1  aÞ a þ 1 0

10

x Zx

ðx  tÞa þ 1

0  f ðtÞ gH  fgH ðtÞdtA

1a a

a þ 1

1 ðx  aÞ  @0gH ¼ Cð1  aÞ 1a

a

 f ðaÞgH a

0

¼

gH

Zx

1 ðx  aÞa þ 1 @  f ðaÞgH Cð1  aÞ 1a

Zx a

a þ 1

 ðx  t Þ 1a

0  fgH ðtÞdtA

10

ðx  tÞa þ 1 0  fgH ðtÞdtA 1a

gH

Since we have, 0

10 Zx a þ 1 a þ 1 ð x  a Þ  ð x  t Þ 0 @  f ðaÞgH  fgH ðtÞdtA 1a 1a a gH 2 0 x 10 3 Z  0 1 6 7 a þ 1 0 ¼ f ðaÞ gH @ ðx  tÞa þ 1 fgH ðtÞdtA 5 4 ðx  aÞ gH 1a a

10

gH

gH

264

6 Applications of Fuzzy Fractional Differential Equations

And also, using 0 @

R

vð xÞ uð xÞ

gðt; xÞdt

0 gH

the second derivative,

10

Zx  ðx  t Þ

a þ 1

Zx

0 fgH ðtÞdtA

a

¼

0 ðx  tÞa fgH ðtÞdt  0gH 0

a

xgH

By substituting, 0

Zx

a þ 1

10

a þ 1

 ðx  t Þ 0 @ ð x  aÞ  f ðaÞgH  fgH ðtÞdtA 1a 1a a gH 2 3 Zx 1 4 0 ðx  aÞa f ðaÞgH ðx  tÞa fgH ¼ ðtÞdt5 1a a

The proof van be completed.

6.2.1

Theorem—Interchanging Operators

Let us continuous f ; g : ½a; b ! FR are two continuous fuzzy functions on ½a; b for 0\a\1 then Zb gð x Þ 

I aRL f ð xÞdx

Zb ¼

a

f ð xÞ  I aRL gð xÞdx

a

It seems that looks like the Fubini theorem for two regular integrals. The left side is, Zb

gð xÞ  I aRL f ð xÞdx ¼

a

Zb a

0 1  gð x Þ  @ CðaÞ

1

Zb ð t  xÞ

a1

x

where I aRL f ð xÞ

1  ¼ CðaÞ

Zb x

ðt  xÞa1 f ðtÞdt

f ðtÞdtAdx

6.2 Fuzzy Fractional Calculus—Preliminaries for Control Problem

Zb gð x Þ 

I aRL f ð xÞdx

a

1  ¼ CðaÞ ¼

1  CðaÞ

1  ¼ CðaÞ Zb ¼ a

Zb

gð x Þ  @

a

t

0

f ðt Þ  @

Zb

1 ðt  xÞa1 gð xÞdxAdt

t

0

I aRL gðtÞdt

Zb

1 ðt  xÞa1 gð xÞdxAdt

t

Zb

f ð xÞ  I aRL gð xÞdx

:¼

a

6.2.2

f ðtÞdtAdx

ðt  xÞa1 gð xÞ  f ðtÞdxdt

1 f ðt Þ  @  CðaÞ f ðt Þ 

ð t  xÞ

a1

x

Zb a

1

Zb

a

Zb Zb

Zb ¼

0

265

a

Theorem—Fuzzy Fractional Integration by Part I

Let us continuous f ; g : ½a; b ! FR are two continuous fuzzy functions on ½a; b for 0\a\1 then Zb gð x Þ 

DaCgH f ð xÞdx

¼ f ð xÞ 

b

ð1aÞ I RL gð xÞ

a

Zb gH ð1Þ

a

f ð xÞ  DaRLgH gð xÞdx

a

By using the definition of fuzzy fractional derivative and mentioned above interchanging theorem, the left side is as, Zb

gð xÞ  DaCgH f ð xÞdx ¼

a

Zb a

0 1  gð x Þ  @ Cð1  aÞ

Zx a

Since we have,  0 ð1aÞ DaRLgH f ð xÞ ¼ I RL f ð xÞ gH

1 0 ðx  tÞa fgH ðtÞdtAdx

266

6 Applications of Fuzzy Fractional Differential Equations

Substituting, Zb gð x Þ 

DaCgH f ð xÞdx

Zb ¼

a

gð x Þ 

ð1aÞ 0 I RL fgH ð xÞdx

Zb ¼

a

ð1aÞ

0 fgH ð xÞ  I RL gð xÞdx

a

Now using integral by part, we set  0 ð1aÞ ð1aÞ 0 u ¼ I RL gð xÞ; dv ¼ fgH ð xÞdx; du ¼ I RL gð xÞ ¼ DaRLgH gð xÞ gH

Therefore, Zb

ð1aÞ

0 fgH ð xÞ  I RL gð xÞdx

a

¼ f ð bÞ 

ð1aÞ I RL gðbÞgH f ðaÞ



ð1aÞ I RL gðaÞgH ð1Þ

Zb

f ð xÞ  DaRLgH gð xÞdx

a

b

ð1aÞ ¼ f ð xÞ  I RL gð xÞ gH ð1Þ

Zb

a

f ð xÞ  DaRLgH gð xÞdx

a

The proof is completed. If the right area or part of RL-derivative is used, then another face of integration by part is designed and denoted by integration by part II where, 0 b 10 Z 1  @ ðt  xÞa f ðtÞdtA DaRLgH f ð xÞ ¼ Cð1  aÞ x

6.2.3

gH

Theorem—Fuzzy Fractional Integration by Part II

Let us continuous f ; g : ½a; b ! FR are two continuous fuzzy functions on ½a; b for 0\a\1 then Zb

gðxÞ  DaRLgH f ðxÞdx

a

Zb

b

ð1aÞ ¼ ð1Þgð xÞ  I RL f ð xÞ gH ð1Þ f ðxÞ  DaCgH gðxÞdx a

a

6.2 Fuzzy Fractional Calculus—Preliminaries for Control Problem

267

By using the definition of fuzzy fractional derivative and mentioned above interchanging theorem, the left side is as, Zb gð x Þ 

DaRLgH f ð xÞdx

Zb ¼

a

a

0 b 10 Z 1 a  @ ðt  xÞ f ðtÞdtA dx gð x Þ  Cð1  aÞ x

gH

We set, 0 b 10 Z 1  @ ðt  xÞa f ðtÞdtA dx; du ¼ g0gH ð xÞ u ¼ gð xÞ; dv ¼ C ð 1  aÞ x

v¼

1  Cð1  aÞ

Zb

gH ð1aÞ

ðt  xÞa f ðtÞdt ¼ ð1ÞI RL f ð xÞ

x

Substituting, Zb

gðxÞ  DaRLgH f ðxÞdx ¼

a

Zb a

0 1  g ð xÞ  @ Cð1  aÞ

Zb

10 ðt  xÞa f ðtÞdtA dx

x

gH

Zb

b

ð1aÞ ð1aÞ ¼ ð1ÞgðxÞ  I RL f ðxÞ gH ð1Þ I RL f ð xÞ  g0gH ð xÞdx a

a

b

ð1aÞ ¼ ð1ÞgðxÞ  I RL f ðxÞ gH ð1Þ

Zb

b

ð1aÞ ¼ ð1ÞgðxÞ  I RL f ðxÞ gH ð1Þ

Zb

a

ð1aÞ

f ð xÞ  IRL g0gH ð xÞdx

a

a

f ð xÞ  DaCgH gð xÞdx

a

The proof is finished.

6.3

Fuzzy Optimal Control Problem

In this section, after stating some necessary definitions and theorems, the main problem is proposed, i.e., Fuzzy fractional optimal control problem. Furthermore, the necessary conditions for this problem are obtained. Note In this problem, all the functions are in terms of the time variable t.

268

6 Applications of Fuzzy Fractional Differential Equations

Definition (Fuzzy functional) The fuzzy rule like J that assigns any fuzzy valued function x 2 FR to a unique fuzzy number is called fuzzy functional. Definition (Increment) If x and x  dx are two fuzzy valued functions the increment of fuzzy functional J is defined and denoted as, DJ ðx; dxÞ ¼ J ðx  dxÞgH J ð xÞ Definition (Variation) The first variation of a fuzzy functional J ð xÞ is denoted by dJ ðx  dxÞ that includes the terms of DJ ðx; dxÞ and it is linear in terms of dx.

6.3.1

Definition—Relative Extremum of Fuzzy Functional Function

A fuzzy functional function J has a relative extremum at x on its domain, if there is an 9 [ 0 such that for all fuzzy valued functions 8x 2 FR DH ðx; x Þ\; • J ðx Þ is relative minimum, if DJ ¼ J ð xÞgH J ðx Þ > > minJ ðuÞ ¼ F ðx; u; tÞdt > < 0 s:t: > a > > > DCgH xðtÞ ¼ Gðx; u; tÞ : x ð 0Þ ¼ x 0 To find the fuzzy optimal control we follow the traditional approach and define a fuzzy modified performance index as Z1   e J ð uÞ ¼ F ðx; u; tÞ  k  DaCgH xðtÞgH Gðx; u; tÞ dt 0

where k is Lagrange multiplier. Now by using the increment, the variation of e J ð uÞ is determined. De J¼e J ðu  duÞgH e J ðuÞ ¼

Z1  0

 F ðx  dx; u  du; tÞ  ðk  dkÞ  DaCgH ðx  dxÞðtÞgH Gðx  dx; u  du; tÞ dt

6.3 Fuzzy Optimal Control Problem

gH

Z1 

271

 F ðx; u; tÞ  k  DaCgH xðtÞgH Gðx; u; tÞ dt

0

Z1 ¼

  F ðx  dx; u  du; tÞgH F ðx; u; tÞ 

0

 ðk  dkÞ  DaCgH ðx  dxÞðtÞgH Gðx  dx; u  du; tÞ gH k i DaCgH ð xÞðtÞgH Gðx; u; tÞ dt Z1 ¼ 0

   F ðx  dx; u  du; tÞgH F ðx; u; tÞ  ðk  dkÞ  DaCgH ðx  dxÞðtÞgH k

DaCgH ð xÞðtÞ gH ðk  dkÞ  Gðx  dx; u  du; tÞgH k  Gðx; u; tÞ

Now using the fuzzy two dimensional first order Taylor expansion for the functions F and G (please see the Chap. 2). We have, de J¼

Z1

  @xgH F  dx  @ugH F  du

0

h  i   k  d DaCgH x  dk  DaCgH x gH k  @GxgH  dx  k  @GugH du  dk  GÞdt where de J is the linear part of De J in terms of dx; du and dk. Moreover dx; du and dk are the variations of x, u and k respectively. Now by using the integration by part I, we get that, Z1 kd



DaCgH x

Z1  dt ¼ k  DaCgH dx dt

0

¼ dx 

0

1

ð1aÞ I RL kðtÞ

0

Z1 gH ð1Þ

dx  DaRLgH kðtÞdt

0

Z1 ¼ 0GH ð1Þ

dx  0

DaRLgH kðtÞdt

Z1 ¼

dx  DaRLgH kðtÞdt

0

Provided that dxð0Þ ð0; 0; 0Þ or kð0Þ ð0; 0; 0Þ and dxð1Þ ð0; 0; 0Þ or kð1Þ ð0; 0; 0Þ. Because xð0Þ is specified and we have dxð0Þ ð0; 0; 0Þ and xð1Þ is not specified, we require that kð1Þ ¼ 0. Now by substituting in de J we have,

272

6 Applications of Fuzzy Fractional Differential Equations

de J¼

Z1 h

 i   @FxgH  DaRLgH k gH k  @GxgH  dx  @FugH gH k  @GugH  du

0

h i  DaCgH xgH G  dk dt

In this equation, if we apply the fundamental theorem of calculus of variation then de J ¼ 0. Also minimization of e J ðuÞ and hence minimization of J ðuÞ requires that the coefficients of dx; du and dk be zero. This leads that, 8  a > @F  D k gH k  @GxgH ¼ 0 > xgH RLgH > < @FugH gH k  @GugH ¼ 0 a > > > DCgH xðtÞgH Gðx; u; tÞ ¼ 0 : xð0Þ ¼ x0 ; kð1Þ ð0; 0; 0Þ These equations represent the Euler-Lagrange equations for the fuzzy fractional optimal control problem. These equations give the necessary conditions for the optimality of the fuzzy fractional optimal control problem that is considered here. 8 R1 > > > minJ ðuÞ ¼ F ðx; u; tÞdt > < 0 s:t: > > > DaCgH xðtÞ ¼ Gðx; u; tÞ > : x ð 0Þ ¼ x 0 They are very similar to the Euler Lagrange equations for crisp fractional optimal control problem. Determination of the optimal control for the fuzzy fractional system requires solution of Euler-Lagrange equations. In the end, we obtain the necessary conditions for a special case of this problems. Now consider the following problem: Find the control uðtÞ that minimizes the quadratic performance index 1 J ð uÞ ¼ 2

Z1

  qðtÞ  x2 ðtÞ  r ðtÞ  u2 dt

0

where qðtÞ 0 and uðtÞ [ 0, and the system whose dynamic is described by the following linear fractional differential equation. DaCgH xðtÞ ¼ aðtÞ  xðtÞ  bðtÞ  u;

x ð 0Þ ¼ 0

6.3 Fuzzy Optimal Control Problem

273

Comparing with the previous one, F ðx; u; tÞ :=

1 1 qðtÞ  x2 ðtÞ  r ðtÞ  u2 2 2

And Gðx; u; tÞ ¼ aðtÞ  xðtÞ  bðtÞ  uðtÞ Now to have the new conditions for our new problem, we need for satisfaction of all the compared conditions. So, @FxgH ¼ qðtÞ  xðtÞ; @FugH ¼ r ðtÞ  uðtÞ; @GxgH ¼ aðtÞ; @GugH ¼ bðtÞ Therefor the conditions are considered as,  8 a > q ð t Þ  x ð t Þ  D k gH k  aðtÞ ¼ 0 > RLgH > < r ðtÞ  uðtÞgH k  bðtÞ ¼ 0 a > D xðtÞ ¼ aðtÞ  xðtÞ  bðtÞ  uðtÞ > > : CgH xð0Þ ¼ x0 ; kð1Þ ð0; 0; 0Þ So far, we have provided a theoretical approach to fuzzy fractional optimal control problems, which involves solving fuzzy fractional differential equations. As it is known, solving such equations is in most cases impossible to do, and numerical methods are used to find approximated solutions for the problem.

6.3.4

Sufficient Optimality Conditions

Here, it has been proved that the obtained fuzzy necessary optimality conditions are sufficient by considering some appropriate assumptions for the fuzzy fractional optimal control problem. Theorem (Solution of fuzzy fractional optimal control) Let us consider ðx ; u ; k Þ satisfying the following fuzzy necessary optimality conditions, 8  a > @F  D k gH k  @GxgH ¼ 0 > xgH RLgH > < @FugH gH k  @GugH ¼ 0 > > DaCgH xðtÞgH Gðx; u; tÞ ¼ 0 > : xð0Þ ¼ x0 ; kð1Þ ð0; 0; 0Þ

274

6 Applications of Fuzzy Fractional Differential Equations

And assume that dx ¼ xðtÞgH x ðtÞ and du ¼ uðtÞgH u ðtÞ, moreover suppose that, 1. 2.

F ðxðtÞ; uðtÞ; tÞgH F ðx ðtÞ; u ðtÞ; tÞ< @xgH F ðx ðtÞ; u ðtÞ; tÞ  dxgH @ugH F ðx ðtÞ; u ðtÞ; tÞ  du GðxðtÞ; uðtÞ; tÞgH Gðx ðtÞ; u ðtÞ; tÞ< @xgH Gðx ðtÞ; u ðtÞ; tÞ  dxgH @ugH Gðx ðtÞ; u ðtÞ; tÞ  du

3. For all k 2 ½0; 1 or G is linear in terms of x and u. Then ðx ðtÞ; u ðtÞÞ := ðx ðtÞ; u ðtÞ; tÞ is a fuzzy optimal solution to the fuzzy fractional optimal control problem. Proof For the equation,  @FxgH  DaRLgH k gH k  @GxgH ¼ 0 We deduce that  @FxgH ðx ðtÞ; u ðtÞ; tÞ  DaRLgH k gH k  @GxgH ðx ðtÞ; u ðtÞ; tÞ ¼ 0 Or from the definition of gH-difference,  @FxgH ðx ðtÞ; u ðtÞ; tÞ ¼ k  @GxgH ðx ðtÞ; u ðtÞ; tÞgH DaRLgH k Using the second condition of necessity, @FugH gH k  @GugH ¼ 0 We have, @FugH ðx ðtÞ; u ðtÞ; tÞ ¼ k  @GugH ðx ðtÞ; u ðtÞ; tÞ From the assumptions, dx ¼ xðtÞgH x ðtÞ;

du ¼ uðtÞgH u ðtÞ

xðtÞ ¼ x ðtÞ  dx;

uðtÞ ¼ u ðtÞ  du

Means that,

6.3 Fuzzy Optimal Control Problem

275

Also this means the solution ðx; uÞ should be considered as admissible solution in which satisfies the DaCgH xðtÞ ¼ Gðx; u; tÞ;

xð0Þ ¼ x0

In this case we are going to show that J ðuÞgH J ðu Þ minJðuÞ ¼ u t dt > gH 2 0 Cð1 þ baÞ < s  t: > Da xðtÞ ¼ uðtÞ > > : CgH xð0Þ ¼ ð0:5; 0; 0:5Þ The necessary Euler-Lagrange equations are, 8  a > @F  D k gH k  @GxgH ¼ 0 > xgH RLgH > < @FugH HgH k  @GugH ¼ 0 > > DaCgH xðtÞgH Gðx; u; tÞ ¼ 0 > : xð0Þ ¼ x0 ; kð1Þ ð0; 0; 0Þ

6.3 Fuzzy Optimal Control Problem

277

where @FxgH ¼ 0; @FugH

Gðx; u; tÞ ¼ u; @GugH ¼ 1;

Cð1 þ bÞ ba t ¼ ugH C ð 1 þ b  aÞ

@GxgH ¼ 0;

So, 8 a DRLgH k ¼ 0 > >  > < Cð1 þ bÞ ba k ¼ ugH Cð1 t þ baÞ > > DaCgH xðtÞ ¼ u > : xð0Þ ¼ ð0:5; 0; 0:5Þ; kð1Þ ð0; 0; 0Þ At this point, we encounter following fractional boundary value problem that needs to be solved in order to reach the optimal solution of the example,



Cð1 þ bÞ ba t ugH Cð1 þ b  aÞ kð1Þ ð0; 0; 0Þ

DaRLgH

¼ 0;

xð0Þ ¼ ð0:5; 0; 0:5Þ;

If xðtÞ is found out the control variable uðtÞ can be obtained using equation, DaCgH xðtÞ ¼ u Example Consider the following fuzzy problem: find the fuzzy control uðtÞ which minimizes the fuzzy quadratic performance index, 8 R1 2 min JðuÞ ¼ 12 0 ½1  x2 ðtÞ  u2 ðtÞ dt > > < s:t: a > > DcgH xðtÞ ¼ 1  xðtÞ  2  uðtÞ : xð0Þ ¼ ð0; 1; 2Þ where 1 ¼ ð0; 1; 2Þ; 1 ¼ ð2; 1; 0Þ; 2 ¼ ð1; 2; 3Þ. Now the fuzzy necessary conditions implies that, 1  xðtÞ  DaRLgH kgH k  ð1Þ ¼ 0; ugH k  2 ¼ 0 DaCgH xðtÞ ¼ ð1Þ  x  2  u;

xð0Þ ¼ ð0; 1; 2Þ;

kð1Þ ð0; 0; 0Þ

The state function xðtÞ, the control uðtÞ and the Lagrange multiplier kðtÞ are obtained by solving the fractional differential equations subject to the given conditions. As we mentioned before, we need numerical scheme to obtain approximate solution.

278

6.4

6 Applications of Fuzzy Fractional Differential Equations

Fuzzy Fractional Diffusion Equations

In this section the fuzzy solution of the following fuzzy fractional diffusion equation is investigated,

where 0 6¼ a 2 R; 0\a 2 and

where p  gH differentiability of the solution is defined as,

To discuss the concept, we need some results about the differentiability of the production of functions in different cases. For instance consider two functions in which the one is real valued function and the second is fuzzy valued function. For the real one, the options are: positive, negative, increasing and decreasing. For the second fuzzy function the options are: i  gH and ii  gH differentiability. In the following remark they are brought briefly. All have been fully discussed in Chap. 4. Remark Let suppose that the function is a fuzzy valued function on ½a; b and is a real valued function. Then the following cases are stablished.

1.

is i  gH differentiable and 1:1 If is a positive and increasing function then and

is i  gH differentiable

1:2 If is a positive and decreasing function then and

is i  gH differentiable

6.4 Fuzzy Fractional Diffusion Equations

2.

279

1:3 If is a negative and increasing function then tiable and

is ii  gH differen-

1:4 If is a negative and decreasing function then tiable and

is ii  gH differen-

is ii  gH differentiable and 2:1 If is a positive and increasing function then and

2:2 If is a positive and decreasing function then tiable and

is ii  gH differentiable

is ii  gH differen-

280

6 Applications of Fuzzy Fractional Differential Equations

2:3 If is a negative and increasing function then and

2:4 If is a negative and decreasing function then tiable and

is i  gH differentiable

is i  gH differen-

Remark Let suppose that the function are gH-differentiable fuzzy valued function on ½a; b and is a monotonic and continuous real valued function. Then the following cases are stablished.

1. If and ferentiable and

And

are i  gH differentiable then

is i  gH dif-

6.4 Fuzzy Fractional Diffusion Equations

and 2. If ferentiable and

281

are ii  gH differentiable then

is i  gH dif-

And

3. If

is i  gH differentiable and is ii  gH differentiable and

are ii  gH differentiable then

is ii  gH differentiable and is ii  gH differentiable and

are i  gH differentiable then

And

4. If

And

282

6 Applications of Fuzzy Fractional Differential Equations

is a continuous and gH-differentiable fuzzy function of Remark Consider exponential exp bt without switching point function for t 0. Also suppose is a fuzzy peace-wise continuous and gH-differentiable on ½a; b. Then the Laplace transforms of and are listed in the following form for ReðsÞ [ b.

1.

and

are i  gH differentiable

2.

and

are ii  gH differentiable

3.

is i  gH differentiable and

are ii  gH differentiable

4.

is ii  gH differentiable and

are i  gH differentiable

6.4 Fuzzy Fractional Diffusion Equations

6.4.1

283

Remark—Laplace Transform of Caputo Derivative with Order 0\a\2

are continuous functions on ½0; 1Þ and If continuous Caputo derivative then

is a fuzzy peace-wise

1.

and

are i  gH differentiable

2.

and

are ii  gH differentiable

3.

is i  gH differentiable and

are ii  gH differentiable

4.

is ii  gH differentiable and

are i  gH differentiable

6.4.2

Fundamental Solution of Fuzzy Fractional Diffusion Equation

In this section the fuzzy solution of the following equations is going to be considered. In this equations the order is 0\a 2 generaly. But in each initial condition the order can be different.

where 0 6¼ a 2 R, p0 is a triangular fuzzy number and dð xÞ is the Dirac delta function. The partial fractional derivative with respect to t is also defined as follow such that the Dirac delta function is defined in the measure sense and

284

6 Applications of Fuzzy Fractional Differential Equations

 dð x Þ ¼

1; 0;

x 0 x\0

And

where p  gH differentiability of the solution is defined as,

The fuzzy solution for these equations with fuzzy initial conditions is known as the fundamental solution. This fuzzy solution is derived by the fuzzy Laplace transform of a fuzzy function uðt; xÞ with respect to t,

Also consider fuzzy Fourier transform with respect to x for fixed t [ 0,

Let LF denotes the space of all fuzzy number valued functions such that the fuzzy Laplace transform and the fuzzy Fourier transform exist with the following notation,

6.4 Fuzzy Fractional Diffusion Equations

6.4.3

285

Theorem—Fundamental Solution

and both are i  p  gH or ii  p  gH differentiable with Let respect to t. Then the fuzzy solution is defined as,

where M ða; zÞ :¼ W ða; 1  a; zÞ ¼

1 X

ð1Þz zk k!C½ak þ ð1  aÞ k¼0

The functions M ða; zÞ and W ða; b; zÞ are Mainardi and Wright functions respectively. Proof In case i  p  gH differentiability, by applying the Laplace transform for two sides of

We have,

Since

Thus,

Now we can apply the fuzzy Fourier transform,

where

286

And know,

6 Applications of Fuzzy Fractional Differential Equations

is discovered using the following procedure. We

Suppose

On the other hand,

Because,

Since,

. Finally,

6.4 Fuzzy Fractional Diffusion Equations

287

Now by substituting,

So we have sa  V ðw; sÞH sa1  p0 ¼ ð1Þaw2  V ðw; sÞ   sa1  p0 ¼ sa  V ðw; sÞ  aw2  V ðw; sÞ ¼ sa þ aw2  V ðw; sÞ a1 s V ðw; sÞ ¼ a  p0 s þ aw2 Now by using the inverse Laplace we get,

According to Laplace transform of Mittag-Leffler function, for jlsa j\1, LðEa ðlta ÞÞ ¼

sa1 sa  l

Assuming l ¼ aw2 it is concluded that,

Finally by using inverse Fourier, the solution of our problem

Is found as follows,

288

6 Applications of Fuzzy Fractional Differential Equations

Now, if the order of the inverse fuzzy integral transforms is changed and the inverse Fourier transform is applied first, we have,

By inverse Laplace it is resulted as,

We have,   1  a L1 sa1 expðasa Þ ¼ a M a; a t t So

Finally we get the final form of the solution in the compact form. This is also the fuzzy fundamental solution in case two derivatives and both are i  p  gH or ii  p  gH differentiable. Note In case the type of differentials are different the fuzzy fundamental solution can be also obtained as,

6.4.4

Application of Fuzzy Fractional Diffusion in Drug Release

In this section, the diffusion of an anti-cancer drug in the tumor is considered. The drug is delivered to the patient’s body through intravenous injection. If this diffusion process studies for fractals, we have the following fractional diffusion equation,

where D is the diffusion coefficient, define the diffusivity of the drug in the tumor, C is the speed of drug delivery to the tumor. The term D is the diffusion coefficient, as the property of the tumor is related to a tumor’s resistance to the anti-cancer drug. The real value 0 a 1 demonstrates the Hurst exponent. Measuring the

6.4 Fuzzy Fractional Diffusion Equations

289

initial amount of the anti-cancer drug is an uncertain problem and this vagueness may be appearing in the initial conditions. Therefore the fuzzy initial conditions for equation also are considered as follows Here p0 is a triangular fuzzy number and dð xÞ is Dirac delta function in the sense of measure. Note If a ¼ 0 the equation is defined as non-fractional diffusion equation

If a ¼ 12 then If a ¼ 1 then we have a Poisson equation Based on the mentioned fundamental theorem the fuzzy solution can be found as follows,

Example Let is the diffusion of Temozolomide in the tumor. Temozolomide is an anti-cancer chemotherapy drug. In adults with Anaplastic Astrocytoma, the dosage is based on medical condition, height, weight, and response to treatment. The fuzzy initial dosage of Temozolomide for a patient with 160 cm height and 87 kg weight is ð284:23; 284:93; 285:02Þmg=m2 and it is ð293:2; 293:41; 293:95Þmg=m2 for patient with 175 cm height and 80 kg weight which is prescribed once daily for 5 days. Now, suppose that D ¼ C ¼ 1 and a ¼ 12, then by using the diffusion equation, • For patient with 160 cm height and 87 kg weight we obtain

the solution is as,

• For patient with 175 cm height and 80 kg weight we obtain the solution is as,

290

6 Applications of Fuzzy Fractional Differential Equations

Example Vincristine (VCR) is important for the treatment of acute lymphoblastic leukemia (ALL). ALL is a disease that accounts for approximately one-third of all childhood cancer diagnoses. Let is the diffusion of VCR in the tumor. The fuzzy initial dosage of VCR for patient with 160 cm height and 87 kg weight is ð2:64; 2:66; 2:69Þmg=m2 and it is ð2:70; 2:74; 2:82Þmg=m2 for patient with 175 cm height and 80 kg weight which is prescribed once daily. Now, suppose that D ¼ C ¼ 1 and a ¼ 13, then by using the diffusion equation, • For patient with 160 cm height and 87 kg weight we obtain

the solution is as,

• For patient with 175 cm height and 80 kg weight we obtain the solution is as,

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