Fractional calculus and waves in linear viscoelasticity : an introduction to mathematical models / Francesco Mainardi

Includes bibliographical references and index

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 transform pairs 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), ε1(z) D.2 The modified 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 transform pairs 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 Fv(z) and Mv(z) in C F.3 The auxiliary functions Fv(x) and Mv(x) in IR F.4 The Laplace transform pairs F.5 The Wright M-functions in probability F.6 Notes Bibliography Index