Local Fractional Integral Transforms and Their Applications

By Xiao-Jun Yang, Dumitru Baleanu and Hari M Srivastava

Local Fractional Integral Transforms and Their Applications provides information on how local fractional calculus has been successfully applied to describe the numerous widespread real-world phenomena in the fields of physical sciences and engineering sciences that involve non-differentiable behaviors. The methods of integral transforms via local fractional calculus have been used to solve various local fractional ordinary and local fractional partial differential equations and also to figure out the presence of the fractal phenomenon. The book presents the basics of the local fractional derivative operators and investigates some new results in the area of local integral transforms.

• Provides applications of local fractional Fourier Series
• Discusses definitions for local fractional Laplace transforms
• Explains local fractional Laplace transforms coupled with analytical methods

• Language: English
• Publisher: Academic Press
• Release date: Oct 22, 2015
• ISBN: 9780128040324

Xiao-Jun Yang

Dr. Xiao-Jun Yang is a full professor of China University of Mining and Technology, China. He was awarded the 2019 Obada-Prize, the Young Scientist Prize (Turkey), and Springer's Distinguished Researcher Award. His scientific interests include: Viscoelasticity, Mathematical Physics, Fractional Calculus and Applications, Fractals, Analytic Number Theory, and Special Functions. He has published over 160 journal articles and 4 monographs, 1 edited volume, and 10 chapters. He is currently an editor of several scientific journals, such as Fractals, Applied Numerical Mathematics, Mathematical Methods in the Applied Sciences, Mathematical Modelling and Analysis, Journal of Thermal Stresses, and Thermal Science, and an associate editor of Journal of Thermal Analysis and Calorimetry, Alexandria Engineering Journal, and IEEE Access.

Book preview

Local Fractional Integral Transforms and Their Applications

First Edition

Xiao Jun Yang

Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou, China

Dumitru Baleanu

Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, Ankara, Turkey

Institute of Space Sciences, Magurele-Bucharest, Romania

H.M. Srivastava

Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada

Table of Contents

Cover image

Title page

Copyright

List of figures

List of tables

Preface

1: Introduction to local fractional derivative and integral operators

Abstract

1.1 Introduction

1.2 Definitions and properties of local fractional continuity

1.3 Definitions and properties of local fractional derivative

1.4 Definitions and properties of local fractional integral

1.5 Local fractional partial differential equations in mathematical physics

2: Local fractional Fourier series

Abstract

2.1 Introduction

2.2 Definitions and properties

2.3 Applications to signal analysis

2.4 Solving local fractional differential equations

3: Local fractional Fourier transform and applications

Abstract

3.1 Introduction

3.2 Definitions and properties

3.3 Applications to signal analysis

3.4 Solving local fractional differential equations

4: Local fractional Laplace transform and applications

Abstract

4.1 Introduction

4.2 Definitions and properties

4.3 Applications to signal analysis

4.4 Solving local fractional differential equations

5: Coupling the local fractional Laplace transform with analytic methods

Abstract

5.1 Introduction

5.2 Variational iteration method of the local fractional operator

5.3 Decomposition method of the local fractional operator

5.4 Coupling the Laplace transform with variational iteration method of the local fractional operator

5.5 Coupling the Laplace transform with decomposition method of the local fractional operator

Appendix A: The analogues of trigonometric functions defined on Cantor sets

Appendix B: Local fractional derivatives of elementary functions

Appendix C: Local fractional Maclaurin’s series of elementary functions

Appendix D: Coordinate systems of Cantor-type cylindrical and Cantor-type spherical coordinates

Appendix E: Tables of local fractional Fourier transform operators

Appendix F: Tables of local fractional Laplace transform operators

Bibliography

Index

Copyright

Academic Press is an imprint of Elsevier

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Copyright © 2016 Xiao-Jun Yang, Dumitru Baleanu and Hari M. Srivastava. Published by Elsevier Ltd. All rights reserved.

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Notices

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ISBN: 978-0-12-804002-7

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List of figures

Fig. 1.1 The distance between two points of A and B in a discontinuous space-time 2

Fig. 1.2 The curve of ε-dimensional Hausdorff measure with ε = ln2/ln3 3

when ω = 1 and ε = ln2/ln3 4

Fig. 1.4 The concentration-distance curves for nondifferentiable source (see [23]) 6

Fig. 1.5 The comparisons of the nondifferentiable functions (1.89)–(1.93) when β = 2 and ε = ln2/ln3 16

Fig. 1.6 The comparisons of the nondifferentiable functions (1.94) and (1.95) when ε = ln2/ln3 17

when ε = ln2/ln3, k = 0, k = 1, k = 3, and k = 5 80

when ε = ln2/ln3, k = 1, k = 2, k = 3, k = 4, and k = 5 82

is shown when ε = ln2/ln3 82

is shown when ε = ln2/ln3 84

with fractal dimension ε with fractal dimension ε = ln2/ln3 126

131

132

when ε = ln2/ln3, p = 1, p = 2, and p = 3 140

when ε = ln2/ln3 141

when ε = ln2/ln3 142

when ε = ln2/ln3 163

when ε = ln2/ln3 163

when ε = ln2/ln3 164

when ε = ln2/ln3 165

when ε = ln2/ln3 166

when ε = ln2/ln3 169

in fractal dimension ε = ln2/ln3 182

in fractal dimension ε = ln2/ln3 184

in fractal dimension ε = ln2/ln3 190

in fractal dimension ε = ln2/ln3 192

References

[23] Yang X.-J., Baleanu D., Srivastava H.M. Local fractional similarity solution for the diffusion equation defined on Cantor sets. Appl. Math. Lett. 2015;47:54–60.

List of tables

Table 1.1 Basic operations of local fractional derivative of some of nondifferentiable functions defined on fractal sets 21

Table 1.2 Basic operations of local fractional integral of some of nondifferentiable functions defined on fractal sets 33

Table 1.3 Basic operations of local fractional integral of some of nondifferentiable functions via Mittag–Leffler function defined on fractal sets 33

Table E.1 Tables for local fractional Fourier transform operators 223

Table F.1 Tables for local fractional Laplace transform operators 230

Preface

Xiao-Jun Yang; Dumitru Baleanu; H.M. Srivastava

The purpose of this book is to give a detailed introduction to the local fractional integral transforms and their applications in various fields of science and engineering. The local fractional calculus is utilized to handle various nondifferentiable problems that appear in complex systems of the real-world phenomena. Especially, the nondifferentiability occurring in science and engineering was modeled by the local fractional ordinary or partial differential equations. Thus, these topics are important and interesting for researchers working in such fields as mathematical physics and applied sciences.

In light of the above-mentioned avenues of their potential applications, we systematically present the recent theory of local fractional calculus and its new challenges to describe various phenomena arising in real-world systems. We describe the basic concepts for fractional derivatives and fractional integrals. We then illustrate the new results for local fractional calculus. Specifically, we have clearly stated the basic ideas of local fractional integral transforms and their applications.

The book is divided into five chapters with six appendices.

Chapter 1 points out the recent concepts involving fractional derivatives. We give the properties and theorems associated with the local fractional derivatives and the local fractional integrals. Some of the local fractional differential equations occurring in mathematical physics are discussed. With the help of the Cantor-type circular coordinate system, Cantor-type cylindrical coordinate system, and Cantor-type spherical coordinate system, we also present the local fractional partial differential equations in fractal dimensional space and their forms in the Cantor-type cylindrical symmetry form and in the Cantor-type spherical symmetry form.

In Chapter 2, we address the basic idea of local fractional Fourier series via the analogous trigonometric functions, which is derived from the complex Mittag–Leffler function defined on the fractal set. The properties and theorems of the local fractional Fourier series are discussed in detail. We mainly focus on the Bessel inequality for local fractional Fourier series, the Riemann–Lebesgue theorem for local fractional Fourier series, and convergence theorem for local fractional Fourier series. Some applications to signal analysis, ODEs and PDEs are also presented. We specially discuss the local fractional Fourier solutions of the homogeneous and nonhomogeneous local fractional heat equations in the nondimensional case and the local fractional Laplace equation and the local fractional wave equation in the nondimensional case.

Chapter 3 is devoted to an introduction of the local fractional Fourier transform operator via the Mittag–Leffler function defined on the fractal set, which is derived by approximating the local fractional integral operator of the local fractional Fourier series. The properties and theorems of the local fractional Fourier transform operator are discussed. A particular attention is paid to the logical explanation for the theorems for the local fractional Fourier transform operator and for another version of the local fractional Fourier transform operator (which is called the generalized local fractional Fourier transform operator). Meanwhile, we consider some application of the local fractional Fourier transform operator to signal processing, ODEs, and PDEs with the help of the local fractional differential operator.

Chapter 4 addresses the study of the local fractional Laplace transform operator based on the local fractional calculus. Our attentions are focused on the basic properties and theorems of the local fractional Laplace transform operator and its potential applications, such as those in signal analysis, ODEs, and PDEs involving the local fractional derivative operators. Some typical examples for the PDEs in mathematical physics are also discussed.

Chapter 5 treats the variational iteration and decomposition methods and the coupling methods of the Laplace transform with them involved in the local fractional operators. These techniques are then utilized to solve the local fractional partial differential equations. Their nondifferentiable solutions with graphs are also discussed.

We take this opportunity to thank many friends and colleagues who helped us in our writing of this book. We would also like to express our appreciation to several staff members of Elsevier for their cooperation in the production process of this book.

1

Introduction to local fractional derivative and integral operators

Abstract

We present the recent introduction to fractional derivatives. We give the properties and theorems of local fractional derivatives and local fractional integrals. Specially, we focus upon the local fractional Rolle’s theorem, mean value theorem, Newton–Leibniz formula, integration by parts, and Taylor’s theorem within the local fractional integrals. We also present the heat equation, wave equation, Laplace equation, Klein–Gordon equation, Schrödinger equation, diffusion equation, transport equation, Poisson equation, linear Korteweg–de Vries equation, Tricomi equation, Fokker–Planck equation, Lighthill–Whitham–Richards equation, Helmholtz equation, damped wave equation, dissipative wave equation, Boussinesq equation, nonlinear wave equation, Burgers equation, forced Burgers equation, inviscid Burgers equation, nonlinear Korteweg–de