Fractional Calculus and Waves in Linear Viscoelasticity : An Introduction to Mathematical Models

by Mainardi, Francesco

ISBN13: 9781848163294

ISBN10: 1848163290

Format: Hardcover

Pub. Date: 2010-05-31

Publisher(s): World Scientific Pub Co Inc

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Summary

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.

Preface	p. vii
Acknowledgements	p. xi
List of Figures	p. xvii
Essentials of Fractional Calculus	p. 1
The fractional integral with support in IR⁺	p. 2
The fractional derivative with support in IR⁺	p. 5
Fractional relaxation equations in IR⁺	p. 11
Fractional integrals and derivatives with support in IR	p. 15
Notes	p. 17
Essentials of Linear Viscoelasticity	p. 23
Introduction	p. 23
History in IR⁺: the Laplace Transform approach	p. 26
The four types of viscoelasticity	p. 28
The Classical mechanical models	p. 30
The time - and frequency - spectral functions	p. 41
History in IR: the Fourier transform approach and the dynamic functions	p. 45
Storage and dissipation of energy: the loss tangent	p. 46
The dynamic functions for the mechanical models	p. 51
Notes	p. 54
Fractional Viscoelastic Models	p. 57
The fractional calculus in the mechanical models	p. 57
Power-Law creep and the Scott-Blair model	p. 57
The correspondence principle	p. 59
The fractional mechanical models	p. 61
Analysis of the fractional Zener model	p. 63
The material and the spectral functions	p. 63
Dissipation: theoretical considerations	p. 66
Dissipation: experimental checks	p. 69
The physical interpretation of the fractional Zener model via fractional diffusion	p. 71
Which type of fractional derivative? Caputo or Riemann-Liouville?	p. 73
Notes	p. 74
Waves in Linear Viscoelastic Media: Dispersion and Dissipation	p. 77
Introduction	p. 77
Impact waves in linear viscoelasticity	p. 78
Statement of the problem by Laplace transforms	p. 78
The structure of wave equations in the space-time domain	p. 82
Evolution equations for the mechanical models	p. 83
Dispersion relation and complex refraction index	p. 85
Generalities	p. 85
Dispersion: phase velocity and group velocity	p. 88
Dissipation: the attenuation coefficient and the specific dissipation function	p. 90
Dispersion and attenuation for the Zener and the Maxwell models	p. 91
Dispersion and attenuation for the fractional Zener model	p. 92
The Klein-Gordon equation with dissipation	p. 94
The Brillouin signal velocity	p. 98
Generalities	p. 98
Signal velocity via steepest-descent path	p. 100
Notes	p. 107
Waves in Linear Viscoelastic Media: Asymptotic Representations	p. 109
The regular wave-front expansion	p. 109
The singular wave-front expansion	p. 116
The saddle-point approximation	p. 126
Generalities	p. 126
The Lee-Kanter problem for the Maxwell model	p. 127
The Jeffreys problem for the Zener model	p. 131
The matching between the wave-front and the saddle-point approximations	p. 133
Diffusion and Wave-Propagation via Fractional Calculus	p. 137
Introduction	p. 137
Derivation of the fundamental solutions	p. 140
Basic properties and plots of the Green functions	p. 145
The Signalling problem in a viscoelastic solid with a power-law creep	p. 151
Notes	p. 153
The Eulerian Functions	p. 155
The Gamma function: ¿(z)	p. 155
The Beta function: B(p,q)	p. 165
Logarithmic derivative of the Gamma function	p. 169
The incomplete Gamma functions	p. 171
The Bessel Functions	p. 173
The standard Bessel functions	p. 173
The modified Bessel functions	p. 180
Integral representations and Laplace transforms	p. 184
The Airy functions	p. 187
The Error Functions	p. 191
The two standard Error functions	p. 191
Laplace transform pairs	p. 193
Repeated integrals of the Error functions	p. 195
The Erfi function and the Dawson integral	p. 197
The Fresnel integrals	p. 198
The Exponential Integral Functions	p. 203
The classical Exponential integrals Ei(z), ¿₁(z)	p. 203
The modified Exponential integral Ein(z)	p. 204
Asymptotics for the Exponential integrals	p. 206
Laplace transform pairs for Exponential integrals	p. 207
The Mittag-Leffler Functions	p. 211
The classical Mittag-Leffler function E_¿(z)	p. 211
The Mittag-Leffler function with two parameters	p. 216
Other functions of the Mittag-Leffler type	p. 220
The Laplace transform pairs	p. 222
Derivatives of the Mittag-Leffler functions	p. 227
Summation and integration of Mittag-Leffler functions	p. 228
Applications of the Mittag-Leffler functions to the Abel integral equations	p. 230
Notes	p. 232
The Wright Functions	p. 237
The Wright functions W_¿,¿(z)	p. 237
The auxiliary functions F_¿(z) and M_¿(z) in C	p. 240
The auxiliary functions F_¿(x) and M_¿(x) in IR	p. 242
The Laplace transform pairs	p. 245
The Wright M-functions in probability	p. 250
Notes	p. 258
Bibliography	p. 261
Index	p. 343
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Fractional Calculus and Waves in Linear Viscoelasticity : An Introduction to Mathematical Models

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