Editor: Sachin Bhalekar

Series Title: Current Developments in Mathematical Sciences

Frontiers in Fractional Calculus

Volume 1

eBook: US $49 Special Offer (PDF + Printed Copy): US $147
Printed Copy: US $123
Library License: US $196
ISSN: 2589-2711 (Print)
ISSN: 2589-272X (Online)
ISBN: 978-1-68108-600-2 (Print)
ISBN: 978-1-68108-599-9 (Online)
Year of Publication: 2018
DOI: 10.2174/97816810859991180101

Introduction

This book brings together eleven topics on different aspects of fractional calculus in a single volume. It provides readers the basic knowledge of fractional calculus and introduces advanced topics and applications. The information in the book is presented in four parts:

  1. - Fractional Diffusion Equations: (i) solutions of fractional diffusion equations using wavelet methods, (ii) the maximum principle for time fractional diffusion equations, (iii) nonlinear sub-diffusion equations.
  2. - Mathematical Analysis: (i) shifted Jacobi polynomials for solving and identifying coupled fractional delay differential equations, (ii) the monotone iteration principle in the theory of Hadamard fractional delay differential equations, (iii) dynamics of fractional order modified Bhalekar-Gejji System, (iv) Grunwald-Letnikov derivatives.
  3. - Computational Techniques: GPU computing of special mathematical functions used in fractional calculus.
  4. - Reviews: (i) the popular iterative method NIM, (ii) fractional derivative with non-singular kernels, (iii) some open problems in fractional order nonlinear system

This is a useful reference for researchers and graduate level mathematics students seeking knowledge about of fractional calculus and applied mathematics.

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

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