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Local Fractional Integral Transforms and Their Applications

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Local Fractional Integral Transforms and Their Applications 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 and Institute of Space Sciences, Magurele-Bucharest, Romania H. M. Srivastava Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an imprint of Elsevier

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Academic Press is an imprint of Elsevier 32 Jamestown Road, London NW1 7BY, UK The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK 225 Wyman Street, Waltham, MA 02451, USA 525 B Street, Suite 1800, San Diego, CA 92101-4495, USA Copyright © 2016 Xiao-Jun Yang, Dumitru Baleanu and Hari M. Srivastava. Published by Elsevier Ltd. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this ﬁeld are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. ISBN: 978-0-12-804002-7 Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library For information on all Academic Press publications visit our website at http://store.elsevier.com/

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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 = ln 2/ ln 3 3 Fig. 1.3 The chart of (μ) when ω = 1 and ε = ln 2/ ln 3 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 = ln 2/ ln 3 16 Fig. 1.6 The comparisons of the nondifferentiable functions (1.94) and (1.95) when ε = ln 2/ ln 3 17 Fig. 2.1 The local fractional Fourier series representation of fractal signal ψ (τ) when ε = ln 2/ ln 3, k = 0, k = 1, k = 3, and k = 5 80 Fig. 2.2 The local fractional Fourier series representation of fractal signal ψ (τ) when ε = ln 2/ ln 3, k = 1, k = 2, k = 3, k = 4, and k = 5 82 Fig. 2.3 The plot of fractal signal ψ (τ) is shown when ε = ln 2/ ln 3 82 Fig. 2.4 The plot of fractal signal ψ (τ) is shown when ε = ln 2/ ln(3 ) 84 1 Fig. 3.1 The plots of a family of good kernels: (a) the plot of ε 4π , τ with fractal dimension ε = ln 2/ ln 3 and (b) the plot of ε (1, τ ) with fractal dimension ε = ln 2/ ln 3 126 Fig. 3.2 The graphs of analogous rectangular pulse and its local fractional Fourier transform: (a) the graph of rectε (τ ) and (b) the graph of (ℑ rectε) (ω) 131 Fig. 3.3 The graphs of the analogous triangle function and its local fractional Fourier ( ) transform: (a) the plot of triangε (τ ) and (b) the plot of ℑ triangε (ω) 132 Fig. 3.4 The graphs of (μ) when ε = ln 2/ ln 3, p = 1, p = 2, and p = 3 140 Fig. 3.5 The graph of (μ) when ε = ln 2/ ln 3 141 Fig. 3.6 The graph of (μ) when ε = ln 2/ ln 3 142 Fig. 4.1 The graph of θ (τ ) when ε = ln 2/ ln 3 163 Fig. 4.2 The graph of θ (τ ) when ε = ln 2/ ln 3 163 Fig. 4.3 The graph of θ (τ ) when ε = ln 2/ ln 3 164 Fig. 4.4 The graph of θ (τ ) when ε = ln 2/ ln 3 165 Fig. 4.5 The graph of θ (τ ) when ε = ln 2/ ln 3 166 Fig. 4.6 The graph of θ (τ ) when ε = ln 2/ ln 3 169 Fig. 5.1 The plot of (μ, τ ) in fractal dimension ε = ln 2/ ln 3 182 Fig. 5.2 The plot of (η, μ) in fractal dimension ε = ln 2/ ln 3 184 Fig. 5.3 The plot of (η, μ) in fractal dimension ε = ln 2/ ln 3 190 Fig. 5.4 The plot of (η, μ) in fractal dimension ε = ln 2/ ln 3 192

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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

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Preface 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 prob- lems 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 system- atically 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 coor- dinate 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

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xii Preface 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 dif- ferential 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. Xiao-Jun Yang Dumitru Baleanu H.M. Srivastava

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Introduction to local fractional derivative and integral operators 1 1.1 Introduction 1.1.1 Deﬁnitions of local fractional derivatives The concept of local fractional calculus (also called fractal calculus), which was ﬁrst proposed by Kolwankar and Gangal [1, 2] based on the Riemann–Liouville fractional derivative [3–6], was applied to deal with nondifferentiable problems from science and engineering [7–16]. Several other points of fractal calculus were presented, such as the fractal derivative via Hausdorff measure [1, 17, 18], fractal derivative using fractal geometry [1, 19, 20], and local fractional derivative using the fractal geometry [1, 21–25]. Here, in this chapter, we present the logical extensions of the deﬁnitions to the subject of local derivative on fractals. Let us recall the basic deﬁnitions as follows. Local fractional derivative of (μ) of order ε (0 < ε ≤ 1) deﬁned in [1, 2, 7–16] is given by D(ε) (μ) = dε dμ(εμ) ∣ μ=μ0 = μl→imμ0 dε [[d (μ)−−μ0)(]με 0)] , (1.1) where the term dε [ (μ)] / [d (μ − μ0)]ε is the Riemann–Liouville fractional deriva- tive of order ε of (μ). Local fractional (fractal) derivative of (μ) of order ε (0 < ε ≤ 1) via Hausdorff measure με deﬁned in [1, 17, 18] is given by D(ε) (μ) = dε dμ(εμ) ∣ μ=μ0 = μl→imμ0 (μμ)ε−− μ0ε(μ0) , (1.2) where με is a fractal measure. Local fractional (fractal) derivative using fractal geometry of (μ) of order ε (0 < ε ≤ 1) deﬁned in [1, 19, 20] is written as ∣ D(ε) (μ) = d dμ(εμ) ∣ μ=μ0 = d dσ(μ) = μ→limμ 0 (μB)ϒ−η0ε (μA) , (1.3) where dσ = ϒη0ε with geometric parameter ϒ and measure scale η0 is shown in Figure 1.1. Local Fractional Integral Transforms and Their Applications. http://dx.doi.org/10.1016/B978-0-12-804002-7.00001-2 Copyright © 2016 Xiao-Jun Yang, Dumitru Baleanu and Hari M. Srivastava. Published by Elsevier Ltd. All rights reserved.

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2 Local Fractional Integral Transforms and Their Applications B A h 0 Figure 1.1 The distance between two points of A and B in a discontinuous space-time. The local fractional derivative using the fractal geometry (μ) of order ε (0 < ε ≤ 1) deﬁned in [1, 21–25] has the following form: ∣ (ε) dε (μ) ∣ ε [ (μ) − (μ0)] D (μ) = dμε ∣ μ=μ0 = μl→imμ0 (μ − μ0)ε , (1.4) ε ∼ where [ (μ) − (μ ∫ 0)] = Ŵ (1 + ε) [ (μ) − (μ0)] with the Euler’s Gamma ∞ ε−1 function Ŵ (1 + ε) = : μ exp (−μ) dμ. 0 Following (1.4), we deﬁne ε (0 < ε ≤ 1)-dimensional Hausdorff measure given by [1–25] ε ε H [ ∩ (μ0, μ)] = (μ − μ0) , (1.5) and its plot when ε = ln 2/ ln 3 is the dimension of the fractal set and μ0 = 0 is shown in Figure 1.2. 1.1.2 Comparisons of fractal relaxation equation in fractal kernel functions The fractal relaxation equation with the help of (1.1) is given as (ε) D (μ) + ω (μ) = 0, (1.6)

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Introduction to local fractional derivative and integral operators 3 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 m Figure 1.2 The curve of ε-dimensional Hausdorff measure with ε = ln 2/ ln 3. where (0) = 1. Its solution is written as follows: (μ) = exp (−ωFc (μ)) , (1.7) ε where Fc (μ) is a Lebesgue–Cantor function and Fc (μ) ∼ μ . The fractal relaxation equation with the help of (1.2) is given as follows [26]: (ε) D (μ) + ω (μ) = 0, (1.8) where (0) = 1, and its solution is given by ( ) ε (μ) = exp −ωμ . (1.9) The fractal relaxation equation by using (1.3) (see [19]): (ε) D (μ) + ω (μ) = 0, (1.10) with (0) = 1 that has the solution ( ) ε−1 (μ) = exp −ωϒη μ , (1.11) 0 ε−1 where σ = ϒη μ. 0 The fractal relaxation equation based on (1.4) is given as follows (see [26]): (ε) D (μ) + ω (μ) = 0, (1.12) and its solution is presented as

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