Fractional Calculus And Waves In Linear Viscoelasticity: An Introduction To Mathematical Models

by Francesco Mainardi

Overview

This monograph provides a comprehensive overview of the author's work on the fields of fractional calculus and waves in linear viscoelastic media, which includes his pioneering contributions on the applications of special functions of the Mittag-Leffler and Wright types.It is intended to serve as a general introduction to the above-mentioned areas of mathematical modeling. The explanations in the book are detailed enough to capture the interest of the curious reader, and complete enough to provide the necessary background material needed to delve further into the subject and explore the research literature given in the huge general bibliography.This book is likely to be of interest to applied scientists and engineers.

Product Details

ISBN-13:	9781848163294
Publisher:	Imperial College Press
Publication date:	05/22/2010
Edition description:	New Edition
Pages:	368
Product dimensions:	6.00(w) x 8.90(h) x 1.00(d)

Table of Contents

Preface vii

Acknowledgements xi

List of Figures xvii

1 Essentials of Fractional Calculus 1

1.1 The fractional integral with support in IR⁺ 2

1.2 The fractional derivative with support in IR⁺ 5

1.3 Fractional relaxation equations in IR⁺ 11

1.4 Fractional integrals and derivatives with support in IR 15

1.5 Notes 17

2 Essentials of Linear Viscoelasticity 23

2.1 Introduction 23

2.2 History in IR⁺: the Laplace Transform approach 26

2.3 The four types of viscoelasticity 28

2.4 The Classical mechanical models 30

2.5 The time - and frequency - spectral functions 41

2.6 History in IR: the Fourier transform approach and the dynamic functions 45

2.7 Storage and dissipation of energy: the loss tangent 46

2.8 The dynamic functions for the mechanical models 51

2.9 Notes 54

3 Fractional Viscoelastic Models 57

3.1 The fractional calculus in the mechanical models 57

3.1.1 Power-Law creep and the Scott-Blair model 57

3.1.2 The correspondence principle 59

3.1.3 The fractional mechanical models 61

3.2 Analysis of the fractional Zener model 63

3.2.1 The material and the spectral functions 63

3.2.2 Dissipation: theoretical considerations 66

3.2.3 Dissipation: experimental checks 69

3.3 The physical interpretation of the fractional Zener model via fractional diffusion 71

3.4 Which type of fractional derivative? Caputo or Riemann-Liouville? 73

3.5 Notes 74

4 Waves in Linear Viscoelastic Media: Dispersion and Dissipation 77

4.1 Introduction 77

4.2 Impact waves in linear viscoelasticity 78

4.2.1 Statement of the problem by Laplace transforms 78

4.2.2 The structure of wave equations in the space-time domain 82

4.2.3 Evolution equations for the mechanical models 83

4.3 Dispersion relation and complex refraction index 85

4.3.1 Generalities 85

4.3.2 Dispersion: phase velocity and group velocity 88

4.3.3 Dissipation: the attenuation coefficient and the specific dissipation function 90

4.3.4 Dispersion and attenuation for the Zener and the Maxwell models 91

4.3.5 Dispersion and attenuation for the fractional Zener model 92

4.3.6 The Klein-Gordon equation with dissipation 94

4.4 The Brillouin signal velocity 98

4.4.1 Generalities 98

4.4.2 Signal velocity via steepest-descent path 100

4.5 Notes 107

5 Waves in Linear Viscoelastic Media: Asymptotic Representations 109

5.1 The regular wave-front expansion 109

5.2 The singular wave-front expansion 116

5.3 The saddle-point approximation 126

5.3.1 Generalities 126

5.3.2 The Lee-Kanter problem for the Maxwell model 127

5.3.3 The Jeffreys problem for the Zener model 131

5.4 The matching between the wave-front and the saddle-point approximations 133

6 Diffusion and Wave-Propagation via Fractional Calculus 137

6.1 Introduction 137

6.2 Derivation of the fundamental solutions 140

6.3 Basic properties and plots of the Green functions 145

6.4 The Signalling problem in a viscoelastic solid with a power-law creep 151

6.5 Notes 153

Appendix A The Eulerian Functions 155

A.1 The Gamma function: Γ(z) 155

A.2 The Beta function: B(p,q) 165

A.3 Logarithmic derivative of the Gamma function 169

A.4 The incomplete Gamma functions 171

Appendix B The Bessel Functions 173

B.1 The standard Bessel functions 173

B.2 The modified Bessel functions 180

B.3 Integral representations and Laplace transforms 184

B.4 The Airy functions 187

Appendix C The Error Functions 191

C.1 The two standard Error functions 191

C.2 Laplace transform pairs 193

C.3 Repeated integrals of the Error functions 195

C.4 The Erfi function and the Dawson integral 197

C.5 The Fresnel integrals 198

Appendix D The Exponential Integral Functions 203

D.1 The classical Exponential integrals Ei(z), ε₁(z) 203

D.2 The modified Exponential integral Ein(z) 204

D.3 Asymptotics for the Exponential integrals 206

D.4 Laplace transform pairs for Exponential integrals 207

Appendix E The Mittag-Leffler Functions 211

E.1 The classical Mittag-Leffler function E_α(z) 211

E.2 The Mittag-Leffler function with two parameters 216

E.3 Other functions of the Mittag-Leffler type 220

E.4 The Laplace transform pairs 222

E.5 Derivatives of the Mittag-Leffler functions 227

E.6 Summation and integration of Mittag-Leffler functions 228

E.7 Applications of the Mittag-Leffler functions to the Abel integral equations 230

E.8 Notes 232

Appendix F The Wright Functions 237

F.1 The Wright functions W_λ,μ(z) 237

F.2 The auxiliary functions F_ν(z) and M_ν(z) in C 240

F.3 The auxiliary functions F_ν(x) and M_ν(x) in IR 242

F.4 The Laplace transform pairs 245

F.5 The Wright M-functions in probability 250

F.6 Notes 258

Bibliography 261

Index 343