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

Contents......Page 14
Preface......Page 8
Acknowledgements......Page 12
List of Figures......Page 18
1. Essentials of Fractional Calculus......Page 22
1.1 The fractional integral with support in IR+......Page 23
1.2 The fractional derivative with support in IR+......Page 26
1.3 Fractional relaxation equations in IR+......Page 32
1.4 Fractional integrals and derivatives with support in IR......Page 36
1.5 Notes......Page 38
2.1 Introduction......Page 44
2.2 History in IR+: the Laplace transform approach......Page 47
2.3 The four types of viscoelasticity......Page 49
2.4 The classical mechanical models......Page 51
2.5 The time - and frequency - spectral functions......Page 62
2.6 History in IR: the Fourier transform approach and the dynamic functions......Page 66
2.7 Storage and dissipation of energy: the loss tangent......Page 67
2.8 The dynamic functions for the mechanical models......Page 72
2.9 Notes......Page 75
3.1.1 Power-Law creep and the Scott-Blair model......Page 78
3.1.2 The correspondence principle......Page 80
3.1.3 The fractional mechanical models......Page 82
3.2.1 The material and the spectral functions......Page 84
3.2.2 Dissipation: theoretical considerations......Page 87
3.2.3 Dissipation: experimental checks......Page 90
3.3 The physical interpretation of the fractional Zener model via fractional diffusion......Page 92
3.4 Which type of fractional derivative? Caputo or Riemann-Liouville?......Page 94
3.5 Notes......Page 95
4.1 Introduction......Page 98
4.2.1 Statement of the problem by Laplace transforms......Page 99
4.2.2 The structure of wave equations in the space-time domain......Page 103
4.2.3 Evolution equations for the mechanical models......Page 104
4.3.1 Generalities......Page 106
4.3.2 Dispersion: phase velocity and group velocity......Page 109
4.3.3 Dissipation: the attenuation coefficient and the specific dissipation function......Page 111
4.3.4 Dispersion and attenuation for the Zener and the Maxwell models......Page 112
4.3.5 Dispersion and attenuation for the fractional Zener model......Page 113
4.3.6 The Klein-Gordon equation with dissipation......Page 115
4.4.1 Generalities......Page 119
4.4.2 Signal velocity via steepest–descent path......Page 121
4.5 Notes......Page 128
5.1 The regular wave–front expansion......Page 130
5.2 The singular wave–front expansion......Page 137
5.3.1 Generalities......Page 147
5.3.2 The Lee-Kanter problem for the Maxwell model......Page 148
5.3.3 The Jeffreys problem for the Zener model......Page 152
5.4 The matching between the wave–front and the saddle–point approximations......Page 154
6.1 Introduction......Page 158
6.2 Derivation of the fundamental solutions......Page 161
6.3 Basic properties and plots of the Green functions......Page 166
6.4 The Signalling problem in a viscoelastic solid with a power-law creep......Page 172
6.5 Notes......Page 174
A.1 The Gamma function: Γ(z)......Page 176
A.2 The Beta function: B(p, q)......Page 186
A.3 Logarithmic derivative of the Gamma function......Page 190
A.4 The incomplete Gamma functions......Page 192
B.1 The standard Bessel functions......Page 194
B.2 The modified Bessel functions......Page 201
B.3 Integral representations and Laplace transforms......Page 205
B.4 The Airy functions......Page 208
C.1 The two standard Error functions......Page 212
C.2 Laplace transformpairs......Page 214
C.3 Repeated integrals of the Error functions......Page 216
C.4 The Erfi function and the Dawson integral......Page 218
C.5 The Fresnel integrals......Page 219
D.1 The classical Exponential integrals Ei (z),  1(z)......Page 224
D.2 The modi.ed Exponential integral Ein (z)......Page 225
D.3 Asymptotics for the Exponential integrals......Page 227
D.4 Laplace transform pairs for Exponential integrals......Page 228
E.1 The classical Mittag-Leffler function Ea(z)......Page 232
E.2 The Mittag-Leffler function with two parameters......Page 237
E.3 Other functions of the Mittag-Leffler type......Page 241
E.4 The Laplace transformpairs......Page 243
E.5 Derivatives of the Mittag-Leffler functions......Page 248
E.6 Summation and integration of Mittag-Leffler functions......Page 249
E.7 Applications of the Mittag-Leffler functions to the Abel integral equations......Page 251
E.8 Notes......Page 253
F.1 The Wright function Wλ,µ(z)......Page 258
F.2 The auxiliary functions Fν (z) and Mν(z) in C......Page 261
F.3 The auxiliary functions Fν (x) and Mν (x) in IR......Page 263
F.4 The Laplace transformpairs......Page 266
F.5 The Wright M-functions in probability......Page 271
F.6 Notes......Page 279
Bibliography......Page 282
Index......Page 364