#### 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