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Table of contents :
Applied Mathematics and
Fractional Calculus
ISBN 978-3-0365-5148-7 (Hbk)
ISBN 978-3-0365-5147-0 (PDF)
Contents
Saima Rashid, Aasma Khalid, Sobia Sultana, Zakia Hammouch, Rasool Shah and Abdullah
M. Alsharif
About the Editors
Francisco Martinez Gonzalez
Mohammed K. A. Kaabar
Preface to ”Applied Mathematics and Fractional
Calculus”
Francisco Martinez Gonzalez and Mohammed K. A. Kaabar
Existence of Solutions for a Singular Fractional q-Differential
Equations under Riemann-Liouville Integral Boundary Condition
Mohammad Esmael Samei 1,2Ф, Rezvan Ghaffari 1, Shao-Wen Yao 3,* , Mohammed K. A. Kaabar 4'1 * * * 5®, Francisco Martrnez 6 and Mustafa Inc 7,8
2. Essential Preliminaries
3. Main Results
4. Illustrative Examples with Application
5. Conclusions
Appendix A. Supporting Information
References
Spectrum of Fractional and Fractional Prabhakar Sturm-Liouville
Problems with Homogeneous Dirichlet Boundary Conditions
Malgorzata Klimek ©
2. Preliminaries
4. Results on the Spectrum of Integral and Differential Fractional and Fractional Prabhakar Sturm-Liouville Problems
5. Discussion
6. Conclusions
Appendix A
References
General Fractional Integrals and Derivatives of Arbitrary Order
Yuri Luchko
1. Introduction
2. General Fractional Integrals and Derivatives with the Sonine Kernels
3. General Fractional Integrals and Derivatives of Arbitrary Order
4. Conclusions
References
The Existence, Uniqueness, and Stability Analysis of the Discrete Fractional Three-Point Boundary Value Problem for the Elastic Beam Equation
Jehad Alzabut 1,2,*,+ , a. George Maria Selvam 3,+ , R. Dhineshbabu 4,+ and Mohammed K. A. Kaabar 5,6,*,+©
2. Essential Preliminaries
3. EB Existence and Uniqueness
4. EB Stability Analysis
5. Applications
6. Conclusions
References
Uniqueness of Abel’s Integral Equations of the Second Kind with Variable Coefficients
Chenkuan Li * and Joshua Beaudin ©
1. Introduction
2. The Main Results
3. Conclusions
References
Condensing Functions and Approximate Endpoint Criterion for the Existence Analysis of Quantum Integro-Difference FBVPs
Shahram Rezapour 1,2 , Atika Imran 3, Azhar Hussain 3 , Francisco Martinez ^* , Sina Etemad 2 and Mohammed K. A. Kaabar 5©
2. Fundamental Preliminaries
4: end for
6: end if
5: end for
3. Main Results
4. Numerical Examples
5. Conclusions
References
Non-Trivial Solutions of Non-Autonomous Nabla Fractional
Difference Boundary Value Problems
Alberto Cabada k*,+©, Nikolay D. Dimitrov 2,+ and Jagan Mohan Jonnalagadda 3,+
1. Introduction
2. Preliminaries
3. Green’s Function and Its Properties
4. Existence of Solutions
5. Examples
References
Regularity Criteria for the 3D Magneto-Hydrodynamics
Equations in Anisotropic Lorentz Spaces
Maria Alessandra Ragusa 1,2,^ and Fan Wu 3
1. Introduction
3. Conclusions
References
Aboodh Transform Iterative Method for Solving Fractional
Partial Differential Equation with Mittag-Leffler Kernel
Michael A. Awuya * and Dervis Subasi
1. Introduction
2. Preliminaries
3. Aboodh Transform Iterative Method
4. Applications
5. Conclusions
References
Analysis and Optimal Control of ф-Hilfer Fractional Semilinear Equations Involving Nonlocal Impulsive Conditions
Sarra Guechi1 , Rajesh Dhayal 2 , Amar Debbouche ^* and Muslim Malik 3©
4. Existence and Uniqueness
5. Existence of Optimal Controls
6. Example
7. Discussion
References
Bilateral Tempered Fractional Derivatives
Manuel Duarte Ortigueira 1,s and Gabriel Bengochea 2©
1. Introduction
2. Preliminaries
3. Riesz-Feller Tempered Derivatives
Theorem 1.
4. Bilateral Tempered Fractional Derivatives
5. Conclusions
Abbreviations
References
A-Interval of Triple Positive Solutions for the Perturbed Gelfand Problem
Shugui Kang 1, Youmin Lu 2 and Wenying Feng 3,*
1. Introduction
2. Preliminary
3. Uniqueness for a < 4
4. Upper, Lower Bounds and Order Sequence of Solutions
5. A-Interval for Triple Positive Solutions
References
Approximate Solutions of an Extended Multi-Order Boundary
Value Problem by Implementing Two Numerical Algorithms
Surang Sitho 1,+, Sina Etemad 2,+ , Brahim Tellab 3,+ , Shahram Rezapour 2,*,+© and Sotiris K. Ntouyas 4,5,+ and Jessada Tariboon 6'*,+©
1. Introduction
2. Basic Concepts
3. Results of the Existence Criterion
4. Approximation of Solutions via DGJIM and ADM Methods
5. Application
6. Conclusions
References
Initial Value Problems of Linear Equations with the
Dzhrbashyan-Nersesyan Derivative in Banach Spaces
Vladimir E. Fedorov 1,2,* , Marina V. Plekhanova 1,s© and Elizaveta M. Izhberdeeva 1©
1. Introduction
2. Homogeneous Equation with the Dzhrbashyan-Nersesyan Fractional Derivative
3. Inhomogeneous Equation
4. Degenerate Equation
5. Application to a Class of Initial Boundary Value Problems
References
Bessel Collocation Method for Solving Fredholm-Volterra Integro-Fractional Differential Equations of Multi-High Order in the Caputo Sense
Shazad Shawki Ahmed * and Shabaz Jalil MohammedFaeq *
1. Introduction
2. Preliminary Considerations
3. Fundamental Matrix Relations
5. Numerical Examples
G=
6. Conclusions
References
Hermite Cubic Spline Collocation Method for Nonlinear Fractional Differential Equations with Variable-Order
Tinggang Zhao 1,*,+ and Yujiang Wu 2,+
1. Introduction
2. Preliminaries
3. Hermite Cubic Spline Collocation Method(HCSCM)
4. Order of Convergence of the Approximation with HCSCM
6. Conclusions
Abbreviations
References
The Comparative Study for Solving Fractional-Order Fornberg-Whitham Equation via p-Laplace Transform
Pongsakorn Sunthrayuth 1, Ahmed M. Zidan 2'3©, Shao-Wen Yao 4'*©, Rasool Shah 5 and Mustafa Inc 6'7'8'*
2. Basic Definitions
3. The General Methodology of p-LDM
4. Convergence Analysis
5. The General Methodology of p-Laplace Variational Iteration Method
6. Implementation of Techniques
7. Conclusions
References
New Variational Problems with an Action Depending on Generalized Fractional Derivatives, the Free Endpoint Conditions, and a Real Parameter
Ricardo Almeida and Natalia Martins ©
1. Introduction
2. Preliminaries
3. Main Results
4. Illustrative Examples
5. Concluding Remarks
References
1. Introduction
2. Preliminaries
5. Analysis of the Convergence
6. Conservation Laws
7. Concluding Remarks
References
A Comparative Analysis of Fractional-Order Kaup-Kupershmidt Equation within Different Operators
Nehad Ali Shah 1®, Yasser S. Hamed 2®, Khadijah M. Abualnaja 2®, Jae-Dong Chung 1,* , Rasool Shah 3® and Adnan Khan 3
1. Introduction
2. Basic Preliminaries
4. Convergence Analysis
5. Numerical Examples
6. Results and Discussion
7. Conclusions
References
Midpoint Inequalities in Fractional Calculus Defined Using
Positive Weighted Symmetry Function Kernels
Pshtiwan Othman Mohammed h*©, Hassen Aydi 23'4'* , Artion Kashuri 5@, Y. S. Hamed 6
and Khadijah M. Abualnaja 6©
1. Introduction
2. Auxiliary Results
Proof.
3. Main Results
4. Concluding Remarks
Abbreviations
References
A Novel Analytical View of Time-Fractional Korteweg-De Vries Equations via a New Integral Transform
Saima Rashid 1 * * * * , Aasma Khalid 2, Sobia Sultana 3, Zakia Hammouch 4,5*, Rasool Shah 6 and Abdullah M. Alsharif 7
2. Preliminaries
4. Basic Formulation of the SITM
5. Existence and Uniqueness Results for Shehu Decomposition Method
6. Evaluation of the Fractional KdV Model
$ S[ 2 %
7. Conclusions
References
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symmetry

Applied Mathematics and Fractional Calculus Edited by

Francisco Martmez Gonzalez and Mohammed K. A. Kaabar Printed Edition of the Special Issue Published in Symmetry

www.mdpi.com/journal/symmetry

Applied Mathematics and Fractional Calculus

Applied Mathematics and Fractional Calculus

Editors Francisco Martinez Gonzalez Mohammed K. A. Kaabar

MDPI • Basel• Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj* Tianjin

MDPI

Editors Francisco Martinez Gonzalez

Mohammed K. A. Kaabar

Departamento de Matematica

Institute of Mathematical

Aplicada y Estadistica

Sciences, Faculty of Science

Universidad Politecnica de

Universiti Malaya,

Cartagena

Kuala Lumpur 50603

Cartagena

Malaysia

Spain

Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland

This is a reprint of articles from the Special Issue published online in the open access journal

Symmetry (ISSN 2073-8994)

(available at:

www.mdpi.com/journal/symmetry/speciaLissues/

Applied_Mathematics_Fractional_Calculus).

For citation purposes, cite each article independently as indicated on the article page online and as indicated below:

LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year, Volume Number,

Page Range.

ISBN 978-3-0365-5148-7 (Hbk) ISBN 978-3-0365-5147-0 (PDF)

© 2022 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon

published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND.

Contents About the Editors ...............................................................................................................................................

ix

Preface to ”Applied Mathematics and Fractional Calculus” ................................................................

xi

Mohammad Esmael Samei, Rezvan Ghaffari, Shao-Wen Yao, Mohammed K. A. Kaabar, Francisco Martinez and Mustafa Inc Existence of Solutions for a Singular Fractional q-Differential [-25]Equations under Riemann-Liouville Integral Boundary Condition Reprinted from: Symmetry 2021, 13, 1235, doi:10.3390/sym13071235 ...................................................

1

Malgorzata Klimek Spectrum of Fractional and Fractional Prabhakar Sturm-Liouville Problems with Homogeneous Dirichlet Boundary Conditions Reprinted from: Symmetry 2021, 13, 2265, doi:10.3390/sym13122265 ...................................................

21

Yuri Luchko General Fractional Integrals and Derivatives of Arbitrary Order Reprinted from: Symmetry 2021, 13, 755, doi:10.3390/sym13050755

...................................................

43

Jehad Alzabut, A. George Maria Selvam, R. Dhineshbabu and Mohammed K. A. Kaabar The Existence, Uniqueness, and Stability Analysis of the Discrete Fractional Three-Point Boundary Value Problem for the Elastic Beam Equation Reprinted from: Symmetry 2021, 13, 789, doi:10.3390/sym13050789 ...................................................

57

Chenkuan Li and Joshua Beaudin Uniqueness of Abel’s Integral Equations of the Second Kind with Variable Coefficients Reprinted from: Symmetry 2021, 13, 1064, doi:10.3390/sym13061064 ...................................................

75

Shahram Rezapour, Atika Imran, Azhar Hussain, Francisco Martinez, Sina Etemad and Mohammed K. A. Kaabar Condensing Functions and Approximate Endpoint Criterion for the Existence Analysis of Quantum Integro-Difference FBVPs Reprinted from: Symmetry 2021, 13, 469, doi:10.3390/sym13030469 ...................................................

87

Alberto Cabada, Nikolay D. Dimitrov and Jagan Mohan Jonnalagadda Non-Trivial Solutions of Non-Autonomous Nabla Fractional Difference Boundary Value Problems Reprinted from: Symmetry 2021, 13, 1101, doi:10.3390/sym13061101 ................................................... 109 Maria Alessandra Ragusa and Fan Wu Regularity Criteria for the 3D Magneto-Hydrodynamics Equations in Anisotropic Lorentz Spaces Reprinted from: Symmetry 2021, 13, 625, doi:10.3390/sym13040625 ................................................... 125 Michael A. Awuya and Dervis Subasi Aboodh Transform Iterative Method for Solving Fractional Partial Differential Equation with Mittag-Leffler Kernel Reprinted from: Symmetry 2021, 13, 2055, doi:10.3390/sym13112055 ................................................... 135 Sarra Guechi, Rajesh Dhayal, Amar Debbouche and Muslim Malik Analysis and Optimal Control of ф-Hilfer Fractional Semilinear Equations Involving Nonlocal Impulsive Conditions Reprinted from: Symmetry 2021, 13, 2084, doi:10.3390/sym13112084 ................................................... 155 v

Manuel Duarte Ortigueira and Gabriel Bengochea Bilateral Tempered Fractional Derivatives Reprinted from: Symmetry 2021, 13, 823, doi:10.3390/sym13050823

173

Shugui Kang, Youmin Lu and Wenying Feng А-Interval of Triple Positive Solutions for the Perturbed Gelfand Problem Reprinted from: Symmetry 2021, 13, 1606, doi:10.3390/sym13091606 ................................................... 187 Surang Sitho, Sina Etemad, Brahim Tellab, Shahram Rezapour, Sotiris K. Ntouyas and Jessada Tariboon Approximate Solutions of an Extended Multi-Order Boundary Value Problem by Implementing Two Numerical Algorithms Reprinted from: Symmetry 2021, 13, 1341, doi:10.3390/sym13081341 199

Vladimir E. Fedorov, Marina V. Plekhanova and Elizaveta M. Izhberdeeva Initial Value Problems of Linear Equations with the Dzhrbashyan-Nersesyan Derivative in Banach Spaces Reprinted from: Symmetry 2021, 13, 1058, doi:10.3390/sym13061058 225 Shazad Shawki Ahmed and Shabaz Jalil MohammedFaeq Bessel Collocation Method for Solving Fredholm-Volterra Integro-Fractional Differential Equations of Multi-High Order in the Caputo Sense Reprinted from: Symmetry 2021, 13, 2354, doi:10.3390/sym13122354 239

Tinggang Zhao and Yujiang Wu Hermite Cubic Spline Collocation Method for Nonlinear Fractional Differential Equations with Variable-Order Reprinted from: Symmetry 2021, 13, 872, doi:10.3390/sym13050872 267

Pongsakorn Sunthrayuth, Ahmed M. Zidan, Shao-Wen Yao, Rasool Shah and Mustafa Inc The Comparative Study for Solving Fractional-Order Fornberg-Whitham Equation via p-Laplace Transform Reprinted from: Symmetry 2021, 13, 784, doi:10.3390/sym13050784 297 Ricardo Almeida and Natalia Martins New Variational Problems with an Action Depending on Generalized Fractional Derivatives, the Free Endpoint Conditions, and a Real Parameter Reprinted from: Symmetry 2021, 13, 592, doi:10.3390/sym13040592 313

Sachin Kumar, Baljinder Kour, Shao-Wen Yao, Mustafa Inc and Mohamed S. Osman Invariance Analysis, Exact Solution and Conservation Laws of (2 + 1) Dim Fractional Kadomtsev-Petviashvili (KP) System Reprinted from: Symmetry 2021, 13, 477, doi:10.3390/sym13030477 331 Nehad Ali Shah, Yasser S. Hamed, Khadijah M. Abualnaja, Jae-Dong Chung, Rasool Shah and Adnan Khan A Comparative Analysis of Fractional-Order Kaup-Kupershmidt Equation within Different Operators Reprinted from: Symmetry 2022, 14, 986, doi:10.3390/sym14050986 347

Pshtiwan Othman Mohammed, Hassen Aydi, Artion Kashuri, Y. S. Hamed and Khadijah M. Abualnaja Midpoint Inequalities in Fractional Calculus Defined Using Positive Weighted Symmetry Function Kernels Reprinted from: Symmetry 2021, 13, 550, doi:10.3390/sym13040550 371 vi

Saima Rashid, Aasma Khalid, Sobia Sultana, Zakia Hammouch, Rasool Shah and Abdullah M. Alsharif A Novel Analytical View of Time-Fractional Korteweg-De Vries Equations via a New Integral Transform Reprinted from: Symmetry 2021, 13, 1254, doi:10.3390/sym13071254 393

vii

About the Editors Francisco Martinez Gonzalez Francisco Martinez is a tenured associate professor at the Universidad Politecnica de Cartagena, Spain. He received his PhD degree in Physics from Universidad de Murcia in 1992. His research interests include nonlinear dynamics methods and their applications, fractional calculus, fractional

differential equations, multivariate calculus or special functions, and the divulgation of mathematics.

Mohammed K. A. Kaabar Mohammed K. A. Kaabar received all his undergraduate and graduate degrees in Applied and Theoretical Mathematics from Washington State University (WSU), Pullman, WA, USA. Prof.

Kaabar has diverse experience in teaching, globally, and has worked as Adjunct Full Professor of Mathematics, Math Lab Instructor, and Lecturer at various US institutions such as Moreno Valley

College, California, USA, Washington State University, Washington, USA, and Colorado Early Colleges, Colorado, USA. Prof. Kaabar is an Elected Foreign Member of the Academy of Engineering

Sciences of Ukraine and Ukrainian School of Mining Engineering, Senior Member of the Hong Kong Chemical, Biological & Environmental Engineering Society (HKCBEES), and Council Member of the International Engineering and Technology Institute (IETI). He has published more than 100 research papers indexed by Scopus and Web of Science. He has authored two math textbooks, and he served

as an invited referee for more than 300 Science, Technology, Engineering, and Mathematics (STEM) international conferences and journals all over the world. He served as an editor for the American Mathematical Society (AMS) Graduate Student Blog and full editor for an educational program

(Mathematics and Statistics Section) at California State University, Long Beach, CA, USA. Prof.

Kaabar is currently serving as an editor for 21 international scientific journals in applied mathematics and engineering. He is an invited keynote speaker in scientific conferences in Hong Kong, France,

Ukraine, Turkey, China, Malaysia, India, Romania, USA, Singapore, and Italy. His research interests

are fractional calculus, applied analysis, fractal calculus, applied mathematics, mathematical physics, mathematical modelling of infectious diseases, deep learning, and nonlinear optimization.

ix

Preface to ”Applied Mathematics and Fractional Calculus” In the last three decades, fractional calculus has broken into the field of mathematical analysis,

both at the theoretical level and at the level of its applications. In essence, the fractional calculus theory is a mathematical analysis tool applied to the study of integrals and derivatives of arbitrary

order, which unifies and g eneralizes the c lassical n otions of d ifferentiation and i ntegration. These

fractional and derivative integrals, which until not many years ago had been used in purely mathematical contexts, have been revealed as instruments with great potential to model problems

in various scientific fi elds, su ch as : flu id mec hanics, vis coelasticity, phy sics, bio logy, chemistry, dynamical systems, signal processing or entropy theory. Since the differential and integral operators

of fractional order are nonlinear operators, fractional calculus theory provides a tool for modeling physical processes, which in many cases is more useful than classical formulations. This is why

the application of fractional calculus theory has become a focus of international academic research. This Special Issue “Applied Mathematics and Fractional Calculus“ has published excellent research studies in the field of a pplied m athematics and f ractional c alculus, a uthored by many well-known mathematicians and scientists from diverse countries worldwide such as China, USA, Canada,

Germany, Mexico, Spain, Poland, Portugal, Iran, Tunisia, South Africa, Albania, Thailand, Iraq,

Egypt, Italy, India, Russia, Pakistan, Taiwan, Korea, Turkey, and Saudi Arabia. Francisco Martinez Gonzalez and Mohammed K. A. Kaabar

Editors

xi

88 symmetry Article

Existence of Solutions for a Singular Fractional q-Differential Equations under Riemann-Liouville Integral Boundary Condition 3,

1

Mohammad Esmael Samei 1,2Ф, Rezvan Ghaffari , Shao-Wen Yao * , Mohammed K. A. Kaabar Francisco Martrnez 6 and Mustafa Inc 7,8 1

2 3 4 5 6

7 8 *

4'1 *5®,

Department of Mathematics, Faculty of Basic Science, Bu-Ali Sina University, Hamedan 65178, Iran; [email protected] (M.E.S.); [email protected] (R.G.) Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China Department of Mathematics and Statistics, Washington State University, Pullman, WA 99163, USA; [email protected] Institute of Mathematical Sciences, University of Malaya, Kuala Lumpur 50603, Malaysia Department of Applied Mathematics and Statistics, Technological University of Cartagena, 30203 Cartagena, Spain; [email protected] Department of Computer Engineering, Biruni University, Istanbul 34025, Turkey; [email protected] Department of Mathematics, Science Faculty, Firat University, Elazig 23119, Turkey Correspondence: [email protected]

Abstract: We investigate the existence of solutions for a system of m-singular sum fractional q-

differential equations in this work under some integral boundary conditions in the sense of Caputo fractional q-derivatives. By means of a fixed point Arzela-Ascoli theorem, the existence of positive Citation: Samei, M.E.; Ghaffari, R.; Yao, S.-W.; Kaabar, M.K.A.; Martrnez,

F.; Inc, M. Existence of Solutions for a

solutions is obtained. By providing examples involving graphs, tables, and algorithms, our funda­ mental result about the endpoint is illustrated with some given computational results. In general,

symmetry and q-difference equations have a common correlation between each other. In Lie algebra,

Singular Fractional q-Differential

Equations under Riemann-Liouville

q-deformations can be constructed with the help of the symmetry concept.

Integral Boundary Condition.

Symmetry 2021, 13, 1235. https://

Keywords: Caputo q-derivative; singular sum fractional q-differential; fixed point; equations;

doi.org/10.3390/sym13071235

Riemann-Liouville q-integral

Academic Editors: Sergei D. Odintsov

MSC: 34A08; 34B16; 39A13

and Sun Young Cho

Received: 2 April 2021 Accepted: 26 May 2021

1. Introduction

Published: 9 July 2021

There are many definitions of fractional derivatives that have been formulated ac­ cording to two basic conceptions: one of a global (classical) nature and the other of a local nature. Under the first formulation, the fractional derivative is defined as an integral, Fourier, or Mellin transformation, which provides its non-local property with memory. The second conception is based on a local definition through certain incremental ratios. This global conception is associated with the appearance of the fractional calculus itself and dates back to the pioneering works of important mathematicians, such as Euler, Laplace, Lacroix, Fourier, Abel, and Liouville, until the establishment of the classical definitions of Riemann-Liouville and Caputo. Until relatively recently, the study of these fractional integrals and derivatives was limited to a purely mathematical context; however, in recent decades, their applications in various fields of natural Sciences and technology, such as fluid mechanics, biology, physics, image processing, or entropy theory, have revealed the great potential of these fractional integrals and derivatives [1-9]. Furthermore, the study from the theoretical and practical point of view of the elements of fractional differential equations has become a focus for interested researchers [10-15].

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Attribution (CC BY) license (https:// creativecommons.org/licenses/by/

4.0/).

1

Symmetry 2021, 13, 1235

The q-difference equations (qDifEqs) were first proposed by Jackson in 1910 [16]. After that, qDifEqs were investigated in various studies [17-24]. On the contrary, integro­ differential equations (InDifEqs) have been recently studied via various fractional deriva­ tives and formulations based on the original idea of qDifEqs (see [25-32]). The concept of symmetry and q-difference equations are connected to each other while theoretically investigating the differential equation symmetries. The solution existence and uniqueness for the fractional qDifEqs were investigated in 2012 by Ahmad et al. as: cDa [u](t) = T(t,u(t)) with boundary conditions (B.Cs):

aiu(0) - в 1Dq [u](0) = Y1 u(П1),

a2u(1) - в2Dq[u](1) = Y2u(П2),

where a G (1,2], ai, ei, Yi, ni are real numbers, for i = 1,2 and T G C(J x R, R) [20]. The q-integral problem was studied in in 2013 by Zhao et al. as: Dqa[u](t)+ f(t,u(t)) = 0,

with B.Cs: u(1) = цIq[u](у) and u(0) = 0 almost V t G (0,1), where q G (0,1), a G (1,2], G G (0,2], n G (0,1), ц is positive real number, and Da is the q-derivative of RiemannLiouville (RL) and the real values continuous map u defined on I x [0, те) [24]. The problem: cdG (cDY + Л)[ u ](t) = pf (t, u (t))+ k11 [g](t, u (t))

was investigated in 2014 by Ahmad et al. with B.Cs:

a 1 u(0) - в 1(t(1—Y)Dq[u](0))|t=0 = G1 u(n 1) and

a 2 u (1) + в 2Dq [ u ](1) = G2 u (n 2), where t, q G [0,1], cDq is the Caputo fractional q-derivative (CpFqDr), 0 < в, Y < 1,1G(•) represents the RL integral with G G (0,1), f and g are given continuous functions, Л and p, k are real constants, ai, вi, &i G R and ni G (0,1) for i = 1,2 [19]. The solutions’ existence was studied in 2019 by Samei et al. for some multi-term q-integro-differential equations with non-separated and initial B.Cs ([23]). Inspired by all previous works, we investigate in this work the positive solutions for the singular fractional q-differential equation (SFqDEqs) as follows:

cDqa[u](t) + h(t, u(t)) = 0,

(1)

with the B.Cs: u(0) = 0, cu(1) = IY[u](1) and u11 (0) = • • • = u(n—1)(0) = 0, where t G J = (0, 1), IqY[u] is the RL q-integral of order Y for the given function: u, here q G J, c > 1, n = [a] + 1, a > 3, y G [1, те), 2Гq(y) > Гq(a), h : (0,1] x [0, те) ^ [0, те) is continuous, limt ,0- h(t,.) = +те that is, h is singular at t = 0, and cDa represents the CpFqDr of order a, q G J. This work is divided into the following: some essential notions and basic results of q-calculus are reviewed in Section 2. Our original important results are stated in Section 3. In Section 4, illustrative numerical examples are provided to validate the applicability of our main results. 2. Essential Preliminaries

Assume that q G (0,1) and a G R. Define [a]q = 1——q- [16]. The power function:

(x - y)qn with n G N0 is written as: n-1

(x—y ) : = П(x—yqk) Y)

k=0

2

Symmetry 2021, 13, 1235

for n > 1 and (x — y)(q0) = 1, where x and y are real numbers and N0 := {0} U N ([17]). In addition, for a G R and a = 0, we obtain: TO

(x — У)

= xa

qa)

x — yqk

П x — yq

a+k'

k=0

If y = 0, then it is obvious that x(a) = xa. The q-Gamma function is expressed by

)

(

Г (z) = 1 — q (z—1) 1 q z) (1 — q) z-1 ,

(

where z G R\{0, —1, —2, • • •} ([16]). We know that Гq(z + 1) = [z]qГq(z). The value of the q-Gamma function, Гq (z), for input values q and z with counting the sentences’ number n in summation by simplification analysis. A pseudo-code is constructed for estimating q-Gamma function of order n. The q-derivative of function w, is expressed as:

( d \

„ г i/ \

](

w(x) — w(qx)

(

Dq1w x >=[ dxjqw x )=

(1 — q) x

and Dq[w](0) = limx.0 Dq[w](x) ([17]). In addition, the higher order q-derivative of a function w is defined by Dqn [w](x) = DqDqn—1 [w](x) for all n > 1, where Dq0[w](x) = w(x)

([17,18]). The q-integral of a function f defined on [0, b] is expressed as: Iq[w](x)= x w(s) dqS = x(1 — q)

0

£ q w(xqk), k

k=0

for 0 < x < b, provided that the series is absolutely convergent ([17,18]). If a in [0, b], then we have:

£

w w (u) dqu = I-q [w](b) — I-q [w](a) = (1 — q) qk\bw (bqk) — aw (aqk)], a k=0

if the series exists. The operator Iqn is given by Iq0[w](x) = w(x) and Iqn [w](x) = IqIqn—1 [w](x) for n > 1 and g G C([0, b]) ([17,18]). It is proven that DqIq [w](x) = w(x) and IqDq [w](x) = w(x) — w(0) whenever w is continuous at x = 0 ([17,18]). The fractional

RL type q-integral of the function w on J for a > 0 is defined by Iq0[w](t) = w(t), and 1

[ ](

Ia w t) =

1t

v ia /0(t—qs)(

a—1) w (s)d qs

q

= ta (1 — q)a

TO

k

k=0

k

1 (1 — ■ w(tqk), £ qk ПП ——K 1 — q+1) +i)

for t G J and a > 0 ([22,33]). In addition, the CpFqDr of a function w is expressed as: cDqa [w](t) = Iq[a]-a cDq[a] [w] (t)

)

[ ](

■. /0t(t -qs ([a|—a 1,ccDqa| w s)dqs

= =

1

( — q)

ta 1

a

£

qkПk—1 (1 — qi—a) -w (tq) £ Пk—11(1 — ql+1) И ,

where t G J and a > 0 ([22]). It is proven that

7"^ "7"qa [Г777] a+ cnd cD T~)qa ГIT"qa[Гтл] —w 7n( I q [I w] 1 (x) — =I q +в [w](x) and w] 1 ((xvi) = (xvi),

3

(2)

Symmetry 2021, 13, 1235

where а, в > 0 ([22]). Some essential notions and lemmas are now presented as follows: In our work, L1 (J) and Cr(J) are denoted by L and B, respectively, where J = [0,1].

Lemma 1 ([34]). IfxEB(1L with DaxEBttL, then n

Ia Dq x (t ) = x (t) +

£ct

i a - i,

i=1

where n is the smallest integer > a, and ci is some real number. Here, we restate the well-known Arzela-Ascoli theorem. Assume that S = {sn } n>1 is a sequence of bounded and equicontinuous real valued functions on [a, b]. Then, S has a uniformly convergent subsequence. We need the following fixed point theorem in our main result:

Lemma 2 ([35]). Assume that A is a Banach space, P C A is a cone, and O1, 02 are two bounded open balls of A centered at the origin with O1 c 02. Assume that П : P П (02\O1) ^ P is a completely continuous operator such that either ||O(a)|| < ||a || for all a G P П dO1 and ||П( a )|| > || a || for all a G P П d 02, or ||O( a )|| > || a || for each a G P П d O1 and ЦП a || < || a || for a G P П d 02. Then, П has a fixed point in P П (02\O1). 3. Main Results Differential Equation

Let us now present our fundamental lemma as follows: Lemma 3. The u0 is a solution for the q-differential equation Dqa [u](t) + g(t) = 0 with the B.Cs:

u (0) = 0, cu (1) = IY u (1) andu" (0) = ••• = u (n-1)(0) = 0 ifu0 is a solution for the q-integral equation

u(t) = 0 Gq(t, s)f (s) dqs, where -(t - qs)(a 1)

(

Гq a ) t 12Гq(Y + 3) [aГq(a +

+t

Gq(t, s) =

2Гq(Y

'

s < t,

)(1 - qs)(a-1) - Гq(a)(1 - qs)(c+Y-1)]

y

(3)

Гq (a)Гq (a + Y)[cГq (Y + 3) - 2Гq (Y)] + 3) [cГq(a + Y)(1 - qs)(a-1) - Гq(a)(1 - qs)(c+Y-1)]

)

Гq (a Гq (a + Y)[cГq (Y

t < s,

+ 3) - 2Г (Y)] q

for s, t G J,n = [ a ] + 1, the function g G B, a > 3 and y G [1, те) with 2Г q ( y ) > Г q (a). Proof. Let us first assume that u0 is a solution for the equation Dqau(t) + g(t) = 0 with the B.Cs. By using Lemma 1, we obtain: u0(t) = -Iqa [g](t) + c0 + c1t + c2t2 + . . . cn-1 tn-1

and by using the condition u0(0) = u0(0) = • • • = u0n

1)(0)

u0(t) = -Iqa[g](t) + c2t2.

4

= 0, we have

Symmetry 2021, 13, 1235

Indeed, y^{

г

1 (t\

]( + r (

1/м । 2Г q Y) i+2~2 [g t ) c 2 q (Y + 3) tY ,

а++Yr

](

Iq [ u 0 t ) = -Iq

and thus

(

]( ) +

]( )

TYr,, l/i\ _ _T(a+YhfflM-l- 2 2rq Y) Iq [uо 1 = Iq [g t c2 г (y । 3).

Note that cu0(1) = — cIq[g](1) + cc2 and

Y

c2 c -

гq^(YY Y+'Y'3) l

(1) - aI Y (I1d )\ cг (a + Y) I г (a ) I + ,1. = г (a + Y) I [g](1) - W I [g](1) ''7-Л I d \

++ q

) = cIag

q

q

a 7

q

q

q

q

Г1 c г q (a + Y )(1 — qs)(a -1) — г q (a )(1 — qs(a+Y -1))

(

= Л --------------------- гq (а )гq (a + Y)---------------------- g s) d s On the other hand,

2Г q ( y )

_

( + 3)

C

= c г q (Y + 3) — 2г q (Y)

( + 3)

гq Y



гq Y

Hence,

л гq(y + 3) [cгq(a + c2

)(1 — qs)(a—1) — гq(a)(1 — qs)(a+Y—1)]

y

(

г q (а )г q (a + y )[ c г q (Y + 3) — 2г q (Y)]



g s) d qs'

Therefore, we have

u о (t ) = —Ia [g](t) 2

1 г(Y + 3) [cгq(a + Y)(1 — qs)(a—1) — гq(a)(1 — qs)(a+Y—1)]

+t Л

гq(а)гq(a + y)[cгq(Y + 3) — 2гq(Y)]

(

g s) dqs

= о1 Gq(s, t)g(s) dqs, where

Gq(t, s) =

— (t — qs)(a 1)

(

г q a) 2гq(y + 3) [cгq(a + y)(1 — qs)(a 1) — гq(a)(1 — qs)c+Y

() ( +

гq a гq a

'

( + 3) — 2г (Y)]

) [cгq Y

y

q

whenever 0 < s < t < 1 and 2гq(y + 3) [cгq(a + y)(1 — qs)(a—1) — гq(a)(1 — qs)(c+Y—1)]

1

( ) ( + Y) [cг (Y + 3) — 2г (Y)]

гq a гq a

q

q

whenever 0 < t < s < 1. Hence, u0 is an integral equation’s solution. By simple review, we can see that u0 is a solution for the equation Da u (t) + g(t) = 0 with the B.Cs whenever u0 is an integral equation’s solution. □

5

Symmetry 2021, 13, 1235

Remark 1. By applying some simple calculations, one can show that Gq (t, s) > 0 for each s, t e J. Now, let us define the operator О on the Banach space B by

O(u(t)) =

Gq(t,s)h(s,u(s)) dqs.

0

It is easy to check that u 0 is a fixed point of the operator О if u 0 is a solution for Equation (1). Consider B together the supremum norm and cone, P is the set of all u e B such that u (t) > 0 V t e J. Suppose that h : (0,1] x [0, те) ^ [0, те) is the singular function at t = 0 in the Equation (1) and Gq(t, s) is the q-Green function in Lemma 3. Now, define the self operator О on P by

O(u(t)) =

Gq(t,s)h(s,u(s)) dqs,

0

for all t e J .At present, we can provide our first main result on the solution’s existence for problem (1) under some assumptions.

Theorem 1. Problem (1) has a unique solution if the following conditions hold. I.

There exists a continuous function h : (0,1] x [0, те) ^ [0, те) such that

lim h(t, s) = те,

t -0-

II.

for s e [0, те). There exists L > 0, в e J and positive constant k such that kcГq(Y + 3) < (cГq(Y + 3) - 2Гq(Y)), | teh (t,0)| < L for each t e J and

|teh(t,u(t)) - teh(t,v(t))| < k||u — v||,

for each u, v belang to P. Proof. Note that,

+

c Г„ (Y + 3)

[ ]( ( ))

u