TY  - BOOK
AU  - Mainardi,F.
ED  - Imperial College Press.
TI  - Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models 
SN  - 9781848163294
AV  - QA314 .M35 2010
PY  - 2010///]
CY  - London, Hackensack, NJ
PB  - Imperial College Press
KW  - Fractional calculus
KW  - Waves
KW  - Mathematical models
KW  - Viscoelasticity
N1  - 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
ER  -