Contents -- Preface -- Acknowledgements -- List of Figures -- 1. Essentials of Fractional Calculus -- 1.1 The fractional integral with support in IR+ -- 1.2 The fractional derivative with support in IR+ -- 1.3 Fractional relaxation equations in IR+ -- 1.4 Fractional integrals and derivatives with support in IR -- 1.5 Notes -- 2. Essentials of Linear Viscoelasticity -- 2.1 Introduction -- 2.2 History in IR+: the Laplace transform approach -- 2.3 The four types of viscoelasticity -- 2.4 The classical mechanical models -- 2.5 The time - and frequency - spectral functions -- 2.6 History in IR: the Fourier transform approach and the dynamic functions -- 2.7 Storage and dissipation of energy: the loss tangent -- 2.8 The dynamic functions for the mechanical models -- 2.9 Notes -- 3. Fractional Viscoelastic Models -- 3.1 The fractional calculus in the mechanical models -- 3.1.1 Power-Law creep and the Scott-Blair model -- 3.1.2 The correspondence principle -- 3.1.3 The fractional mechanical models -- 3.2 Analysis of the fractional Zener model -- 3.2.1 The material and the spectral functions -- 3.2.2 Dissipation: theoretical considerations -- 3.2.3 Dissipation: experimental checks -- 3.3 The physical interpretation of the fractional Zener model via fractional diffusion -- 3.4 Which type of fractional derivative? Caputo or Riemann-Liouville? -- 3.5 Notes -- 4. Waves in Linear Viscoelastic Media: Dispersion and Dissipation -- 4.1 Introduction -- 4.2 Impact waves in linear viscoelasticity -- 4.2.1 Statement of the problem by Laplace transforms -- 4.2.2 The structure of wave equations in the space-time domain -- 4.2.3 Evolution equations for the mechanical models -- 4.3 Dispersion relation and complex refraction index -- 4.3.1 Generalities -- 4.3.2 Dispersion: phase velocity and group velocity.

4.3.3 Dissipation: the attenuation coefficient and the specific dissipation function -- 4.3.4 Dispersion and attenuation for the Zener and the Maxwell models -- 4.3.5 Dispersion and attenuation for the fractional Zener model -- 4.3.6 The Klein-Gordon equation with dissipation -- 4.4 The Brillouin signal velocity -- 4.4.1 Generalities -- 4.4.2 Signal velocity via steepest-descent path -- 4.5 Notes -- 5. Waves in Linear Viscoelastic Media: Asymptotic Representations -- 5.1 The regular wave-front expansion -- 5.2 The singular wave-front expansion -- 5.3 The saddle-point approximation -- 5.3.1 Generalities -- 5.3.2 The Lee-Kanter problem for the Maxwell model -- 5.3.3 The Jeffreys problem for the Zener model -- 5.4 The matching between the wave-front and the saddle-point approximations -- 6. Diffusion and Wave-Propagation via Fractional Calculus -- 6.1 Introduction -- 6.2 Derivation of the fundamental solutions -- 6.3 Basic properties and plots of the Green functions -- 6.4 The Signalling problem in a viscoelastic solid with a power-law creep -- 6.5 Notes -- Appendix A The Eulerian Functions -- A.1 The Gamma function: Γ(z) -- A.2 The Beta function: B(p, q) -- A.3 Logarithmic derivative of the Gamma function -- A.4 The incomplete Gamma functions -- Appendix B The Bessel Functions -- B.1 The standard Bessel functions -- B.2 The modified Bessel functions -- B.3 Integral representations and Laplace transforms -- B.4 The Airy functions -- Appendix C The Error Functions -- C.1 The two standard Error functions -- C.2 Laplace transformpairs -- C.3 Repeated integrals of the Error functions -- C.4 The Erfi function and the Dawson integral -- C.5 The Fresnel integrals -- Appendix D The Exponential Integral Functions -- D.1 The classical Exponential integrals Ei (z), 1(z) -- D.2 The modi.ed Exponential integral Ein (z).

D.3 Asymptotics for the Exponential integrals -- D.4 Laplace transform pairs for Exponential integrals -- Appendix E The Mittag-Leffler Functions -- E.1 The classical Mittag-Leffler function Ea(z) -- E.2 The Mittag-Leffler function with two parameters -- E.3 Other functions of the Mittag-Leffler type -- E.4 The Laplace transformpairs -- E.5 Derivatives of the Mittag-Leffler functions -- E.6 Summation and integration of Mittag-Leffler functions -- E.7 Applications of the Mittag-Leffler functions to the Abel integral equations -- E.8 Notes -- Appendix F The Wright Functions -- F.1 The Wright function Wλ,µ(z) -- F.2 The auxiliary functions Fν (z) and Mν(z) in C -- F.3 The auxiliary functions Fν (x) and Mν (x) in IR -- F.4 The Laplace transformpairs -- F.5 The Wright M-functions in probability -- F.6 Notes -- Bibliography -- Index.