| 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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