Fractional Calculus with Applications in Mechanics E-Book

Teodor M. Atanackovic

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 348

Veröffentlichungsjahr: 2014

Beschreibung

This book contains mathematical preliminaries in which basic definitions of fractional derivatives and spaces are presented. The central part of the book contains various applications in classical mechanics including fields such as: viscoelasticity, heat conduction, wave propagation and variational Hamilton-type principles. Mathematical rigor will be observed in the applications. The authors provide some problems formulated in the classical setting and some in the distributional setting. The solutions to these problems are presented in analytical form and these solutions are then analyzed numerically. Theorems on the existence of solutions will be presented for all examples discussed. In using various constitutive equations the restrictions following from the second law of thermodynamics will be implemented. Finally, the physical implications of obtained solutions will be discussed in detail.

Leseprobe

Table of Contents

Preface

Part 1. Mathematical Preliminaries, Definitions And Properties Of Fractional Integrals And Derivatives

Chapter 1. Mathematical Preliminaries

1.1. Notation and definitions

1.2. Laplace transform of a function

1.3. Spaces of distributions

1.4. Fundamental solution

1.5. Some special functions

Chapter 2. Basic Definitions and Properties of Fractional Integrals and Derivatives

2.1. Definitions of fractional integrals and derivatives

2.2. Some additional properties of fractional derivatives

2.3. Fractional derivatives in distributional setting

Part 2. Mechanical Systems

Chapter 3. Restrictions Following from the Thermodynamics for Fractional Derivative Models of a Viscoelastic Body

3.1. Method based on the Fourier transform

3.2. Thermodynamical restrictions via the internal variable theory

Chapter 4. Vibrations with Fractional Dissipation

4.1. Linear vibrations with fractional dissipation

4.2. Bagley–Torvik equation

4.3. Nonlinear vibrations with symmetrized fractional dissipation

4.4. Nonlinear vibrations with distributed-order fractional dissipation

Chapter 5. Lateral Vibrations and Stability of Viscoelastic Rods

5.1. Lateral vibrations and creep of a fractional-type viscoelastic rod

5.2. Stability of Beck’s column on viscoelastic foundation

5.3. Compressible elastic rod on a viscoelastic foundation

Chapter 6. Fractional Diffusion-Wave Equations

6.1. Nonlinear fractional diffusion-wave and fractional Burgers/Korteweg–de Vries equations

6.2. Fractional telegraph equation

6.3. Distributed-order diffusion-wave equation

6.4. Maximum principle for fractional diffusion-wave-type equations

Chapter 7. Fractional Heat Conduction Equations

7.1. Cattaneo-type space–time fractional heat conduction equation

7.2. Fractional Jeffreys-type heat conduction equation

Bibliography

Index

First published 2014 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

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© ISTE Ltd 2014The rights of Teodor M. Atanacković, Stevan Pilipović, Bogoljub Stanković and Dušan Zorica to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2013955271

British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-84821-417-0

Preface

The subject of this book is the application of fractional calculus in mechanics. It is written so as to make fractional calculus acceptable to the engineering scientific community as well as to applied mathematicians who intend to use this calculus in their own research. The application of fractional calculus is mostly directed towards various areas of physics, engineering and biology. There are a number of monographs and a huge number of papers that cover various problems in fractional calculus. The list is large and is growing rapidly. The monograph by Oldham and Spanier [OLD 74], published in 1974, had a great influence on the subject. This was the first monograph devoted to fractional operators and their applications in problems of mass and heat transfer. Another important monograph was written by Miller and Ross in 1993 [MIL 93]. The encyclopedic treatise by Samko et al. is, up to now, the most prominent book in the field. This book [SAM 87] was first published in Russian in 1987, and the English translation appeared in 1993 [SAM 93]. The monograph by Kilbas et al. [KIL 06] contains a detailed introduction to the theory and application of fractional differential equations, mostly given in references. It treats, in a mathematically sound way, the fractional differential equations. Kiryakova in [KIR 94] introduces a generalized fractional calculus. Diethelm in [DIE 10] gives a well-written introduction to fractional calculus before the main exposition on the Caputo-type fractional differential equations. This book has mathematically sound theory and relevant applications. In Russian, besides [SAM 87], we mention monographs by Nahushev [NAH 03], Pshu [PSH 05] and Uchaikin [UCH 08]. Variational calculus with fractional derivatives is analyzed by Klimek in [KLI 09] and by Malinowska and Torres in [MAL 12]. All the above-mentioned monographs have had a great influence in the development of fractional calculus. Also there are other influential books and articles in the field that are worth mentioning. The article by Gorenflo and Mainardi [GOR 97b] had a significant impact in the field of applications of fractional calculus in physics and mechanics. The book by Oustaloup [OUS 95] presents an application of fractional calculus in the control theory. The book edited by Hilfer [HIL 00] and the book by Hermann [HER 11] contain applications of fractional calculus in physics. The article by Butzer and Westphal [BUT 00] contains a complete introduction to fractional calculus. Podlubny’s work [POD 99], which has become a standard reference in the field, contains applications of fractional calculus to various problems of mechanics, physics and engineering. Application of fractional calculus in bioengineering is presented by Magin in [MAG 06]. Baleanu et al. in [BAL 12a] present the models in which fractional calculus is used, together with the numerical procedures that are used for the solutions. It also contains an extensive review of the relevant literature. Tarasov in [TAR 11] presents, among other topics, application of fractional calculus in statistical and condensed matter physics, as well as in quantum dynamics. Various applications, together with theoretical developments, are presented by Ortigueira in [ORT 11], Baleanu et al. in [BAL 12b], Petráš in [PET 11] and Sabatier et al. in [SAB 07]. The book by Mainardi [MAI 10] is the standard reference for the application of fractional calculus in viscoelasticity and for the study of wave motion. Finally, the book by Uchaikin [UCH 13] gives detailed motivation for fractional-order differential equations in various branches of physics. It also contains an introduction to the theory of fractional calculus.

Our book is devoted to the application of fractional calculus to (classical) mechanics. We have chosen to concentrate on more sophisticated constitutive equations that complement fundamental physical and geometrical principles. It is assumed that the reader has some basic knowledge of fractional calculus, i.e. calculus of integrals and derivatives of arbitrary real order. The main objective of this book is to complement, in a certain sense, the contents of the other books treating the theory of fractional calculus mentioned above. We will treat non-local elasticity, viscoelasticity, heat conduction (diffusion) problems, elastic and viscoelastic rod theory, waves in viscoelastic rods and the impact of a viscoelastic rod against a rigid wall. The mathematical framework of the problems that we analyze falls into different levels of abstraction. In our papers, on which the most part of the presentation is based, we use an approach to fractional calculus based on the functional analysis. In this way, we are able to use a well-developed method and techniques of the theory of generalized functions, especially of Schwartz space of tempered distributions and the space of exponential distributions, supported by [0, ∞). The use of the generalized function setting only gives us flexibility in proving our results. If one deals with functions in , equal zero on (−∞, 0), of polynomial or exponential growth, then their Laplace transform is the classical one. The same holds for the convolution of such functions. Namely, the convolution of such functions is again the locally integrable function, equal zero on (−∞, 0), that is of polynomial or exponential growth. When we deal with the Fourier transform, our framework is S′. The deep connection with the real mechanical models is not lost. Even better explanations and correctness of the proofs can help the reader to understand mathematical models of the discussed problems. Our aim is not to make the book too complicated for the readers with less theoretical background in the quoted mathematical sense. The presentation, for the most part, is intended to avoid unnecessary details and state only major results that a detailed, and often abstract, analysis gives.

The related book Fractional Calculus with Applications in Mechanics: Wave Propagation, Impact and Variational Principles [ATA 14] is complementary to this book, and, indeed, they could have been presented together. However, for practical reasons, it has proved more convenient to present the book separately. There are 13 chapters in the two books combined.

This book, Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes, has a total of seven chapters.

It begins with Part 1, entitled “Mathematical Preliminaries, Definitions and Properties of Fractional Integrals and Derivatives”, which presents an introduction to fractional calculus. It comprises two chapters. Chapter 1, entitled “Mathematical Preliminaries”, is brief and gives definitions and notions that are used in later parts of the book. Chapter 2, entitled “Basic Definitions and Properties of Fractional Integrals and Derivatives”, presents definitions and some of the properties of fractional integrals and derivatives. We give references where the presented results are proved. Some of the results, which are needed in application, of our own research, such as expansion formulas for fractional derivatives and functional dependence of the fractional derivative on the order of derivative, are also presented in this chapter.

Part 2, entitled “Mechanical Systems”, is the central part of the book. It comprises five chapters. In Chapter 3, entitled “Restrictions Following from the Thermodynamics for Fractional Derivative Models of a Viscoelastic Body”, the analysis of constitutive equations of fractional-order viscoelasticity is presented. The constitutive equations must satisfy two principles: the principle of material frame-indifference (objectivity), which asserts that the response of a material is the same for all observers, and the second law of thermodynamics, which in the case of isothermal processes reduces to dissipation inequality. Sometimes dissipation inequality is called the Volterra theorem for hereditary systems (see [UCH 13]). We analyze, in detail, various constitutive equations and determine the restrictions of the material functions or constants so that the dissipation inequality is satisfied. Material objectivity is discussed for non-local elastic materials in section 3.3, Chapter 3 of the companion book Fractional Calculus with Applications in Mechanics: Wave Propagation, Impact and Variational Principles [ATA 14], for the simple one-dimensional spatial case. In Chapter 4, entitled “Vibrations with Fractional Dissipation”, we analyze various vibration problems with single degree of freedom. Problems with single and multiple dissipation terms are analyzed. Also, nonlinear vibrations with symmetrized fractional dissipation is analyzed in detail. Finally, the case of nonlinear vibrations with distributed-order fractional dissipation is analyzed. Existence uniqueness and regularity of solution are examined. Chapter 4 also presents an example from compartmental methods in pharmacokinetics, where the conservation of mass principle is observed. In Chapter 5, entitled “Lateral Vibrations and Stability of Viscoelastic Rods”, we present an analysis of lateral vibrations for several choices of constitutive functions for the rod. Thus, the cases of fractional Kelvin–Voigt, Zener and generalized Zener materials are studied. Special attention is given to the stability conditions. The case of Beck’s column positioned on a fractional type of viscoelastic foundation is discussed, as well as the case of a compressible rod on a fractional type of viscoelastic foundation. In Chapter 6, entitled “Fractional Diffusion-Wave Equations”, we present an analysis of the fractional partial differential equations with the order of time derivatives between 0 and 2. In the case of generalized Burgers/Korteweg–de Vries equation, we consider fractional derivatives with respect to space variable in the range between 2 and 3. By using similarity transformation, we even study some nonlinear cases of generalized heat equations. In Chapter 7, entitled “Fractional Heat Conduction Equations”, Cattaneo-type space-time fractional heat conduction equations and fractional Jeffreys-type heat conduction equations are discussed.

The related book Fractional Calculus with Applications in Mechanics: Wave Propagation, Impact and Variational Principles [ATA 14] has a total of six chapters. It begins with Part 1, entitled “Mathematical Preliminaries, Definitions and Properties of Fractional Integrals and Derivatives”, which presents an introduction to fractional calculus. It consists of two chapters. Chapter 1, entitled “Mathematical Preliminaries”, is brief and gives definitions and notions that are used in later parts of the book. Chapter 2, entitled “Basic Definitions and Properties of Fractional Integrals and Derivatives”, presents definitions and some of the properties of fractional integrals and derivatives. We give references where the presented results are proved. Some of the results, of our own research, which are required in applications, such as expansion formulas for fractional derivatives and functional dependence of the fractional derivative on the order of derivative, are also presented in this chapter.

Part 2, entitled “Mechanical Systems”, is the central part of this book, containing four chapters. In Chapter 3, entitled “Waves in Viscoelastic Materials of Fractional-Order Type”, we present an analysis of waves in fractional viscoelastic materials on infinite domain of space. A wave equation of the fractional Eringen-type is also studied. Stress relaxation, creep and forced oscillations of a viscoelastic of finite length are discussed in detail. In Chapter 4, entitled “Forced Oscillations of a System: Viscoelastic Rod and Body”, the problem of oscillation of a rigid body, attached to viscoelastic rod and moving translatory, is analyzed in detail. The case of a light rod (mass of the rod is negligible with respect to the mass of the attached body) and the case of a heavy rod (mass of the rod is comparable with respect to the mass of the attached body) are discussed separately. Also, constitutive equations for solid-like and fluid-like bodies are distinguished. In Chapter 5, entitled “Impact of Viscoelastic Body against the Rigid Wall”, we analyze a specific engineering problem of a viscoelastic rod impacting against the rigid wall. The case of a light viscoelastic rod sliding without friction is discussed first. Then, the more complicated case of a light viscoelastic rod attached to a rigid block that slides with dry friction is discussed. Finally, the case of a heavy viscoelastic rod attached to a rigid block that slides without friction is presented. In Chapter 6, entitled “Variational Problems with Fractional Derivatives”, we present some results for the optimization of a functional containing fractional derivatives. We formulate the necessary conditions for optimality in standard and generalized problems. We also present dual variational principles for Lagrangians having fractional derivatives. The necessary conditions for optimality are formulated in generalized problems where optimization is performed with respect to the order of the derivative and not only with respect to a given set of admissible functions. Invariance properties of variational principles and Nöther’s theorem are discussed in this chapter, as well as the problem of approximation of Euler–Lagrange equations in two different ways. Also, in this chapter, we propose a constrained minimization problem in which the order of the derivative is considered as a constitutive quantity determined by the state and control variables.

A full bibliography of the two related titles is presented together at the end of each book. The bibliography does not pretend to be complete. It only contains references to the papers and books that we used. In every chapter, in the introductory section, we list the references used in that chapter. In this way, the presentation in every chapter is more readable.

We believe that a reader can find enough information for understanding the presented materials and apply the methods used in the book to his/her own investigations. We hope that this book may be useful for graduate students in mechanics and applied mathematics, as well as researchers in those fields.

We are grateful to our colleagues Nenad Grahovac, Alfio Grillo, Diana Dolićanin, Marko Janev, Sanja Konjik, Ljubica Oparnica, Dragan Spasić and Miodrag Žigić who worked with us on some problems presented in this book.

We acknowledge the Serbian Ministry of Science and Education, project numbers 174005 and 174024, for support while writing this book. We thank Professor Nöel Challamel for suggesting that we write this book.

T. M. ATANACKOVIĆS. PILIPOVIĆB. STANKOVIĆD. ZORICANovi Sad, December 2013

Part 1

Mathematical Preliminaries, Definitions and Properties of Fractional Integrals and Derivatives

Chapter 1

Mathematical Preliminaries

1.1. Notation and definitions

Sets of natural, integer real and complex numbers are denoted, respectively, by , and .

Let Ω be an arbitrary subset of . We denote by Cb(Ω) the set of continuous functions on Ω such that

It is well known that Cb(Ω) is a Banach space. If Ω is open, then we consider compact subsets of Ω, K ⊂⊂ Ω, continuous functions f on Ω and the semi-norms

We can take a sequence of compact sets K1 ⊂ K2 ⊂,…, so that Ω. Then, the sequence of semi-norms defines the Fréchet topology on C (Ω). This topology does not depend on a sequence with the given property. If K is compact, then C (K) always denotes the set of continuous functions on K with the sup-norm over K.

Let Ω be open in . Then, we consider Ck(Ω) ⊂ C(Ω): the space of functions having all the derivatives, up to order continuous. The Fréchet topology is defined by the semi-norms

[1.1]

The same topology is obtained if we again take, for compact sets in [1.1], a sequence of compact sets K1 ⊂ K2 ⊂ …, so that Ω.

Analytic functions on (a, b) ⊂ are smooth functions on (a, b), so that their Taylor series converges in any point a0 of (a, b) on a suitable interval around a0. A space ((a, b)) of such functions is a Fréchet space under the convergence structure from C∞((a, b)).