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

Contents
Preface to the Second Edition
Preface to the First Edition
Acknowledgments
List of Figures
List of Tables
1. Essentials of Fractional Calculus
1.1 The Fractional Integral with Support in R+
1.2 The Fractional Derivative with Support in R+
1.3 Fractional Relaxation Equations
1.4 The Law of Exponents
1.5 Fractional Integral Equations of Abel Type
1.5.1 Abel integral equation of the first kind
1.5.2 Abel integral equation of the second kind
1.6 Fractional Relaxation Equations of Distributed Order
1.6.1 The two forms for fractional relaxation
1.6.2 The fundamental solutions
1.6.3 Examples
1.6.3.1 The double-order fractional relaxation
1.6.3.2 The uniformly distributed order fractional relaxation
1.7 Fractional Integrals and Derivatives with Support in R
1.8 Tables and Plots of Riemann–Liouville Fractional Integrals and Derivatives
1.9 Notes
2. Essentials of Linear Viscoelasticity
2.1 Introduction
2.2 History in R+: The Laplace Transform Approach
2.3 The Four Types of Viscoelasticity
2.4 The Classical Mechanical Models
2.5 The General Mechanical Models
2.6 Remark on the Initial Conditions
2.7 The Time — and Frequency — Spectral Functions
2.8 History in R: The Fourier Transform Approach and the Dynamic Functions
2.8.1 Storage and dissipation of energy: The loss tangent
2.8.2 The dynamic functions for mechanical models
2.9 The Viscoelastic Models of Becker and Lomnitz
2.9.1 The creep laws
2.9.2 The relaxation laws
2.9.3 The retardation spectra
2.10 Beyond the Jeffreys–Lomnitz Model of Linear Viscoelasticity
2.10.1 The extended Jeffreys–Lomnitz laws of creep and relaxation
2.10.2 Retardation spectrum for the extended Jeffreys–Lomnitz law of creep
2.11 The Mean Relaxation and Retardation Times in Linear Viscoelasticity
2.11.1 The fluids and the mean relaxation time
2.11.2 The solids and the mean retardation time
2.12 Notes
3. Fractional Viscoelastic Models
3.1 The Fractional Calculus in Linear Viscoelasticity
3.1.1 Complex modulus, effective modulus and effective viscosity
3.1.2 Power-law creep and the Scott-Blair model
3.2 The Fractional Operator Equation and the Correspondence Principle
3.2.1 Fractional Kelvin–Voigt model
3.2.2 Fractional Maxwell model
3.2.3 Fractional Zener model
3.2.4 Fractional anti-Zener model
3.2.5 Fractional Burgers model
3.3 Fractional Viscoelasticity with History Since −∞ and the Dynamic Functions
3.4 Analysis of the Fractional Zener Model
3.4.1 The material and the spectral functions
3.4.2 Dissipation: Theoretical considerations
3.4.3 Dissipation: Experimental checks
3.4.4 The physical interpretation of the fractional Zener model via fractional diffusion
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 The Energy Velocity
4.5.1 Introduction
4.5.2 The energy velocity for linear viscoelastic waves
4.6 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 Matching Between the Wave-Front Solution and the Long-Time Solution
5.4 The Saddle-Point Approximation
5.4.1 Generalities
5.4.2 The Lee–Kanter problem for the Maxwell model
5.4.3 The Jeffreys problem for the Zener model
5.5 The Matching Between the Wave-Front and the Saddle-Point Approximations
5.6 Notes
6. Diffusion and Waves via Time 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 Green Functions as Probability Density Functions
6.5 The Signaling Problem in a Viscoelastic Solid with a Power-law Creep
6.6 Locations and Velocities for the Fractional Diffusive Waves
6.6.1 Location and evolution of the maximum of the Green functions
6.6.2 Centers of gravity and medians of the Green functions Gc and Gs
6.6.3 Medians of the Green functions for the Cauchy and signaling problems
6.7 Box Evolution of Fractional Diffusive Waves
6.8 Notes
7. Diffusion and Waves via Space–Time Fractional Calculus
7.1 Fourier and Mellin Transforms
7.1.1 The Fourier transform and pseudo differential operators
7.1.2 The Mellin transform
7.2 The Riesz–Feller Space–Fractional Derivative
7.3 The Space–Time Fractional Diffusion-Wave Equation
7.3.1 Scaling and similarity properties of the Green function
7.4 Particular Cases of the Space–Time Fractional Diffusion-Wave Equation
7.5 Composition Rules for the Green Function with 0 < β ≤ 1
7.6 Mellin–Barnes Integral Representations for the Space–Time Fractional Diffusion–Wave Equation
7.7 Computational Representations for the Green Function
7.8 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
A.5 Notes
A.6 Exercises
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
B.5 Notes
B.6 Exercises
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
C.6 Notes
C.7 Exercises
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
D.5 The Volterra Functions
D.6 Notes
D.7 Exercises
Appendix E. The Mittag-Leffler Functions
E.1 The Classical Mittag-Leffler Function Eα(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 Asymptotic Approximations to the Mittag-Lefler Function eα(t);= Eα(−tα) for 0 < α < 1
E.7.1 The two common asymptotic approximations
E.8 Notes
E.9 Exercises
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 Transform Pairs
F.4.1 The four sisters
F.5 The M-Wright Functions in Probability
F.6 The IM-Wright Function in Two Variables
F.7 Notes
F.8 Exercises
Appendix G. Complete Monotone and Bernstein Functions
G.1 Basic Definitions and Properties for Completely Monotone and Bernstein Functions
G.2 Some Basic Examples of Completely Monotone and Bernstein Functions
G.3 Notes
G.4 Exercises
Bibliography
Index