Fractional Calculus with Applications in Mechanics E-Book

Teodor M. Atanackovic

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 411

Veröffentlichungsjahr: 2014

Beschreibung

The books Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes and Fractional Calculus with Applications in Mechanics: Wave Propagation, Impact and Variational Principles contain various applications of fractional calculus to the fields of classical mechanics. Namely, the books study problems in fields such as viscoelasticity of fractional order, lateral vibrations of a rod of fractional order type, lateral vibrations of a rod positioned on fractional order viscoelastic foundations, diffusion-wave phenomena, heat conduction, wave propagation, forced oscillations of a body attached to a rod, impact and variational principles of a Hamiltonian type. The books will be useful for graduate students in mechanics and applied mathematics, as well as for researchers in these fields. Part 1 of this book presents an introduction to fractional calculus. Chapter 1 briefly gives definitions and notions that are needed later in the book and Chapter 2 presents definitions and some of the properties of fractional integrals and derivatives. Part 2 is the central part of the book. Chapter 3 presents the analysis of waves in fractional viscoelastic materials in infinite and finite spatial domains. In Chapter 4, the problem of oscillations of a translatory moving rigid body, attached to a heavy, or light viscoelastic rod of fractional order type, is studied in detail. In Chapter 5, the authors analyze a specific engineering problem of the impact of a viscoelastic rod against a rigid wall. Finally, in Chapter 6, some results for the optimization of a functional containing fractional derivatives of constant and variable order are presented.

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. Waves in Viscoelastic Materials of Fractional-Order Type

3.1. Time-fractional wave equation on unbounded domain

3.2. Wave equation of the fractional Eringen-type

3.3. Space-fractional wave equation on unbounded domain

3.4. Stress relaxation, creep and forced oscillations of a viscoelastic rod

Chapter 4. Forced Oscillations of a System: Viscoelastic Rod and Body

4.1. Heavy viscoelastic rod – body system

4.2. Light viscoelastic rod – body system

Chapter 5. Impact of Viscoelastic Body Against the Rigid Wall

5.1. Rigid block with viscoelastic rod attached slides without friction

5.2. Rigid block with viscoelastic rod attached slides in the presence of dry friction

Chapter 6. Variational Problems with Fractional Derivatives

6.1. Euler–Lagrange equations

6.2. Linear fractional differential equations as Euler–Lagrange equations

6.3. Constrained variational principles

6.4. Approximation of Euler–Lagrange equations

6.5. Invariance properties of J: Nöther’s theorem

6.6. Complementary fractional variational principles

6.7. Generalizations of Hamilton’s principle

Bibliography

Index

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

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Preface

The subject of this book is the application of the 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., 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 of the above-mentioned monographs have had a great influence in the development of the 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 the 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, an 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 discuss 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 consider 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 the 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 Applications in Mechanics: Vibration and Diffusion Processes [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: Wave Propagation, Impact and Variational Principles, has a total of six chapters.

It begins with Part 1, entitled “Mathematical Preliminaries, Definitions and Properties of Fractional 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 Integral and Derivatives”, presents definitions and some of the properties of fractional integral and derivatives. We give references where the presented results are proved. Some of the results, which are required 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 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 described. 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 formulation. 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.

The related book Fractional Calculus Applications in Mechanics: Vibration and Diffusion Processes has a total of seven chapters. It begins with Part 1, entitled “Mathematical Preliminaries, Definitions and Properties of Fractional Derivatives”, which presents an introduction to fractional calculus. It contains 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 Integral and Derivatives”, presents definitions and some of the properties of fractional integral 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 the book. It consists of five chapters. In Chapter 3, entitled “Restrictions Following from the Thermodynamics for Fractional Derivative Order 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 treated 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 discussed. 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 contains 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 discussed. Special attention is given to the stability conditions. The case of Beck’s column positioned on a fractional type of viscoelastic foundation is analyzed, 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 the generalized Burgers/Korteweg–deVries 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 presented.

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 Žigi who worked with us on some problems presented in this book.

We acknowledge the support of the Serbian Ministry of Science and Education, project numbers 174005 and 174024, that we had 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

Let Ω be open in n. Then, we consider Ck(Ω) ⊂ C(Ω): the space of functions having all the derivatives, up to order k ∈ 0, continuous. The Fréchet topology is defined by the semi-norms

[1.1]

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)).

[Hölder inequality]

A real-valued function f defined on [a, b] ⊂ is said to be absolutely continuous on [a, b], if for given ε > 0, there is a δ > 0 such that

for every finite collection of non-overlapping intervals with

ACn([a, b]), n ∈ , n ≥ 2, is the space of functions f, which have continuous derivatives up to the order n 1 on [a, b] and f(n–1) ∈ AC([a, b]). Notation means that the function f ∈ ACn([0, b]), for every b > 0.

A function f on [a, b] is Hölder continuous at x0 ∈ [a, b] if there exist A > 0 and λ > 0, such that

in a neighborhood of x0. Hölder-type spaces on an interval [a, b] are defined as subspaces of integrable functions on this interval with the following properties:

1.2. Laplace transform of a function

[1.2]

is absolutely convergent for any s ∈ such that Re s > Re s1.

It is clear that exists (absolutely exists) for every s ∈ , Re s > ae (s ∈ , Re s > aa). It is an analytic function in the half-plane Re s > ae, since, by partial integration, it can be represented as an absolutely convergent Laplace transform.

[1.3]

We denote by Lexp ([0, ∞)) the space of such functions. The growth order r is greater or equal than the abscissa of absolute convergence, r ≥ aa.

The Laplace transform is a linear operation on the space of exponentially bounded functions. If a function and its derivatives on [0, ∞) up to order k are of exponential growth, then

for suitable c > 0. Let us mention several useful properties of the Laplace transform, based on appropriate assumptions and on corresponding domains

where the convolution of two locally integrable functions on [0, ∞) is defined by

The inverse Laplace transform is defined by

where p > r (see [1.3]).

1.3. Spaces of distributions

The reader of this book has to have a knowledge of the theory of the generalized functions; here we call them distributions, as they are commonly known. This theory is a powerful tool used in mathematical theory and applications. Apart from books that discuss the basic theory, for example [SCH 51, VLA 73], there are a number of application-oriented textbooks such as [DUI 10].

We refer to [SCH 51, VLA 73] for the material of this section. By , the well-known Schwartz spaces are denoted. Norms in the space of smooth functions supported by K are

while in are

Then, is the inductive limit

The corresponding duals, spaces of continuous linear functionals, and its subspace , with the strong topologies, are the space of distributions and the space of tempered distributions. The space of compactly supported distributions is denoted by . It is the dual space for the Fréchet space see section 1.1.

Operations of multiplication and differentiation in are defined in a usual way

We note that contains regular elements defined by f ∈ Lloc (); they are denoted by freg and defined by

We can see that φn → 0 in implies 〈freg, φn〉 → 0, n → ∞.

Polynomially bounded and locally integrable functions on define, in the same way, regular tempered distributions. We will usually denote, by the same symbol f, a function and a corresponding distribution freg. Only if we want to explain in detail the relation between them do we use the symbol freg.

defines the Fourier transform of a tempered distribution. The Fourier transform is an isomorphism on S′. The inverse Fourier transform of φ ∈ S is defined by

If φ, ψ ∈ S, their convolution is defined by

We know

and, as a consequence,

Sobolev space Wk,p(), p ∈ [1, ∞], k ∈ 0, is defined as the space of Lp-functions f with the property that all the distributional derivatives of f up to order k are elements of Lp(). It is a Banach space with the norm

Clearly, Wk,p() ⊂ S′

is a space of smooth functions with all derivatives belonging to Lp. Note is a subspace of , defined as follows: if and only if for every α ∈ 0.

is the dual space of is the dual of and is denoted by B′ (see [SCH 51])

denotes a subspace of tempered distributions consisting of distributions with supports in [0, ∞). Note that is a convolution algebra.

[1.4]

where the derivative is taken in the sense of distributions.

Let f ∈ . Its Laplace transform is defined by

[1.5]

where we assume that f is of the form [1.4]. Clearly, is a holomorphic function for Re s > 0. We will often consider equations, with solutions u determining the tempered distributions, by the use of the Laplace transform. If we assume that u is of exponential growth, then we have , Re s > s0, for some s0 > 0.

We consider the family (see [VLA 84])

[1.6]

where the mth derivative is understood in the distributional sense. Family is defined by . The Heaviside function is defined as

Operators are convolution operators

The semi-group property holds for fα

The Laplace transform of fα is

where

while in the classical case,

EXAMPLE 1.2.– Let u(x, t), x ∈ n, t > 0, be a classical solution of the wave equation

that is the second derivative above are locally integrable functions on [0, ∞) × n, equal to zero for t < 0, u0, v0 are locally integrable functions on n and so that it has the classical Laplace transform with respect to t in the domain Re s > 0.

Writing

for the corresponding distributions, we rewrite the wave equation in the space of distributions as

where the last equation is written in the space of distributions. So with the application of the distributional Laplace transform with respect to t, for Re s > 0, we have

The space is the space of smooth functions φ with the property

[1.7]

[1.8]

The Lizorkin space of test functions Φ is introduced so that Riesz integro-differentiation (and therefore symmetrized fractional derivative) is well defined (see [SAM 93]). Let

The space Ψ′ and the space of Lizorkin generalized functions Φ′ are dual spaces of Ψ and Φ, respectively. Recall, for f ∈ Φ′, we have

Let f ∈ C∞( \ {0}) be such that it has all the derivatives bounded by the polynomials in \ {0}. Then, product f · u is defined by

1.4. Fundamental solution

The Cauchy problem for the second-order linear partial integro-differential operator with constant coefficients P is given by

[1.9]

[1.10]

where f is continuous for t ≥ 0, u0 ∈ C1() and v0 ∈ C(). A classical solution u(x, t) to the Cauchy problem [1.9], [1.10] is of class C2 for t > 0 and of class C1 for t ≥ 0, satisfies equation [1.9] for t > 0, and initial conditions [1.10] when t → 0. If functions u and f are continued by zero for t < 0, then the following equation is satisfied in D′(2):

[1.11]

The explanation is given in example 1.2 in the case of the wave equation. The problem of finding generalized solutions (in D′(2)) of equation [1.11] that vanish for t < 0 will be called the generalized Cauchy problem for the operator P. If there is a fundamental solution E of the operator P and if f ∈ D′(2) vanishes for t < 0, then there exists a unique solution to the corresponding generalized Cauchy problem and is given by

if the convolution E*f exits. We refer to [DAU 00, TRE 75, VLA 84] for more details.

1.5. Some special functions

The Euler gamma function is defined by

The gamma function can also be represented by

We refer to [GOR 97b, MAI 00] for the theory of Mittag-Leffler functions presented in this section. The one-parameter Mittag-Leffler function is defined by

[1.12]

The one-parameter Mittag-Leffler function is an entire function of order and type 1. In some special cases of α, the one-parameter Mittag-Leffler function becomes

with erf being the error function.

The asymptotics of [1.12] are as follows:

The two-parameter Mittag-Leffler function is defined by

It is an entire function of order and type 1. In some special cases of α and β, it becomes

We define one- and two-parameter Mittag-Leffler-type functions, respectively, by

In applications, we will often omit the parameter λ. According to [MAI 00], if α ∈ (0, 1) and λ > 0, we have eα ∈ C∞((0, ∞)) ∩ C([0, ∞)) and . Also, eα is a completely monotonic function, i.e. .

The Laplace transforms of eα and eα,β are

respectively.

Functions eα and eα,β admit integral representations given by

Chapter 2

Basic Definitions and Properties of Fractional Integrals and Derivatives

2.1. Definitions of fractional integrals and derivatives

In this section, we review some basic properties of fractional integrals and derivatives, which we will need later in the analysis of concrete problems. This section contains results from various books and papers [ALM 12, ATA 14a, ATA 13a, ATA 07a, ATA 09b, ATA 09d, ATA 08b, BUT 00, CAN 87, CAP 67, CAP 71b, DIE 10, HER 11, KIL 04, KIL 06, KIR 94, NAH 03, ODI 07, POO 12a, POO 12b, POO 13, ROS 93, SAM 95, SAM 93, TAR 06, TRU 99, UCH 08, WES 03].