Frontiers in Fractional Calculus

Book Series: Current Developments in Mathematical Sciences

Volume 1

by

Sachin Bhalekar

DOI: 10.2174/97816810859991180101
eISBN: 978-1-68108-599-9, 2018
ISBN: 978-1-68108-600-2
ISSN: 2589-2711 (Print)
ISSN: 2589-272X (Online)



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This book brings together eleven topics on different aspects of fractional calculus in a single volume. It provides re...[view complete introduction]
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Table of Contents

Foreword

- Pp. i
Francesco Mainardi
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Preface

- Pp. II-III (2)
Sachin Bhalekar
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List of Contributors

- Pp. iv-v (2)
Sachin Bhalekar
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Solving Fractional Diffusion Equation by Wavelet Methods

- Pp. 1-32 (32)
Zhijiang Zhang and Weihua Deng
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A Survey on the Recent Results Regarding Maximum Principles for the Time-Fractional Diffusion Equations

- Pp. 33-69 (37)
Yuri Luchko and Masahiro Yamamoto
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Integral-Balance Solution to Nonlinear Subdiffusion Equation

- Pp. 70-105 (36)
Jordan Hristov
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Analysis of Solution and System Identification of Coupled Fractional Delay Differential Equation by Shifted Jacobi Polynomials

- Pp. 106-131 (26)
B. Ganesh Priya, P. Muthukumar and P. Balasubramaniam
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Dynamics of Fractional Order Modified Bhalekar- Gejji System

- Pp. 159-182 (24)
Sachin Bhalekar
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Grunwald-Letnikov Derivative: Analysis in Range of First Order

- Pp. 183-198 (16)
Radosław Cioc
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GPU Computing of Special Mathematical Functions used in Fractional Calculus

- Pp. 199-232 (34)
Parag Patil, Navin Singhaniya, Chaitanya Jage, Vishwesh A. Vyawahare, Mukesh D. Patil and P.S.V. Nataraj
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New Iterative Method: A Review

- Pp. 233-268 (36)
Varsha Daftardar-Gejji and Manoj Kumar
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Derivatives with Non-Singular Kernels from the Caputo-Fabrizio Definition and Beyond: Appraising Analysis with Emphasis on Diffusion Models

- Pp. 269-341 (73)
Jordan Hristov
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Fractional Order Nonlinear Systems: Some Open Problems in Numerical Computations and Chaos Theory

- Pp. 342-347 (6)
Sachin Bhalekar
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Subject Index

- Pp. 348-359 (12)
Sachin Bhalekar
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Foreword

Fractional calculus, in allowing integrals and derivatives of any positive order (the term fractional is kept only for historical reasons), can be considered a branch of mathematical physics that deals with integro-differential equations, where integrals are of convolution type and exhibit mainly singular kernels of power law or logarithm type.

It is a subject that has gained considerably popularity and importance in the past few decades in diverse fields of science and engineering.

The purpose of this book is to establish a collection of articles that reflect some mathematical and conceptual developments in the field of fractional calculus and explore the scope for applications in applied sciences. The book is divided in 4 sections as follows where the authors are outlined.

I congratulate with the Editor, PhD Sachin Bhalekar who was able to collect so a variety of topics of which the reader can evaluate the relevance.

Francesco Mainardi
Professor of Mathematical Physics
University of Bologna
Italy


Preface

Fractional Calculus is very popular subject among the Mathematicians and Applied Scientists. The nonlocal operators in fractional calculus provided challenging research topics for Mathematicians. On the other hand, the applied scientists found these operators more useful than classical integer order ones.

It is indeed a happy moment to present this book which contains eleven chapters on different aspects of Fractional Calculus.

The first part of this book contains three chapters on Fractional Diffusion Equations. In Chapter 1, Zhang and Deng discussed the solutions of fractional diffusion equations using wavelet methods. Maximum principle for time fractional diffusion equations is proposed by Luchko and Yamamoto in Chapter 2. The Chapter 3, by Hristov is about the nonlinear sub-diffusion equations.

The second part Analysis contains four chapters. The shifted Jacobi polynomials are used by Ganesh Priya, Muthukumar and Balasubramaniam in Chapter 4 to analyze the solution and system identification of coupled fractional delay differential equations. In Chapter 5, Kucche presented the monotone iteration principle in the theory of Hadamard fractional delay differential equations. Bhalekar, in Chapter 6 analyzed the dynamics of fractional order modified Bhalekar-Gejji System. Cioc analyzed the Grunwald-Letnikov derivatives in Chapter 7.

Third part of this book is Computational Techniques. It contains the chapter 8 by the research group: P. Patil, N. Singhaniya, C. Jage, V. Vyawahare, M. Patil and P S V Nataraj. They presented the GPU computing of special mathematical functions used in fractional calculus.

The last part of the book is Review which contains three chapters. Daftardar-Gejji and Kumar presented a review on the popular iterative method NIM in chapter 9. A review on fractional derivative with non-singular kernels is taken by Hristov in Chapter 10. Some open problems in fractional order nonlinear system are discussed by Bhalekar in Chapter 11.

I am very thankful to all the contributors of this book for their valuable work. The foreword for this book is written by Prof. Fransesco Mainardi. I am indebted to him for his guidance and support. Mr. Omer Shafi of Bentham Science Publishers helped me in the publication process. I am thankful to the Bentham Science for publishing this book.

I wish that the book will provide the readers a basic knowledge, introduce higher topics and present applications of this subject.

Sachin Bhalekar
Department of Mathematics
Shivaji University, Kolhapur
India

List of Contributors

Editor(s):
Sachin Bhalekar
Department of Mathematics
Shivaji University
Kolhapur 416004 India





Contributor(s):
B. Ganesh Priya
Department of Mathematics
The Gandhigram Rural Institute-Deemed University
Gandhigram 624 302
India


Chaitanya Jage
Department of Electronics Engineering
Ramrao Adik Institute of Technology
D. Y. Patil Vidyanagar, Nerul
Navi Mumbai 400 706
India


Jordan Hristov
Dept.Chem.Eng
UCTM, Sofia 1756, 8 Kl.Ohridsky
Blvd
Bulgaria


Kishor D. Kucche
Department of Mathematics
Shivaji University
Kolhapur 416004 India


Manoj Kumar
Department of Mathematics
Savitribai Phule Pune University
Pune - 411007, India
National Defense Academy
Khadakwasala Pune- 411023
India


Masahiro Yamamoto
Graduate School of Mathematical Sciences
The University of Tokyo
Japan


Mukesh D. Patil
Department of Electronics and Telecommunication Engineering
Ramrao Adik Institute of Technology
D. Y. Patil Vidyanagar, Nerul
Navi Mumbai 400 706
India


Navin Singhaniya
Department of Electronics Engineering
Ramrao Adik Institute of Technology
D. Y. Patil Vidyanagar, Nerul
Navi Mumbai 400 706
India


P. Balasubramaniam
Department of Mathematics
The Gandhigram Rural Institute-Deemed University
Gandhigram 624 302
India


P. Muthukumar
Department of Mathematics
The Gandhigram Rural Institute-Deemed University
Gandhigram 624 302
India


P. S. V. Nataraj
IDP in Systems and Control Engineering
Indian Institute of Technology Bombay
Mumbai 400 076
India


Parag Patil
Department of Electronics Engineering
Ramrao Adik Institute of Technology
D. Y. Patil Vidyanagar, Nerul
Navi Mumbai 400 706
India


Radosław Cioć
Faculty of Transport and Electrical Engineering
Kazimierz Pulaski University of Technology and Humanities in Radom Malczewskiego Str. 29
Radom 26-600
Poland


Sachin Bhalekar
Department of Mathematics
Shivaji University
Kolhapur 416004 India


Varsha Daftardar-Gejji
Department of Mathematics
Savitribai Phule Pune University
Pune - 411007
India


Vishwesh A. Vyawahare
Department of Electronics Engineering
Ramrao Adik Institute of Technology
D. Y. Patil Vidyanagar, Nerul
Navi Mumbai 400 706
India


Weihua Deng
School of Mathematics and Statistics
Gansu Key Laboratory of Applied Mathematics and Complex Systems
Lanzhou University
Lanzhou 730000
P.R. China


Yuri Luchko
Department of Mathematics
TU of Applied Sciences Berlin
Germany


Zhijiang Zhang
School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems
Lanzhou University
Lanzhou 730000
P.R. China




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