Fractional Calculus in Wave Propagation Problems

Fractional calculus, in allowing integrals and derivatives of any positive order (the term "fractional" kept only for historical reasons), can be considered a branch of mathematical physics which mainly deals with integro-differential equations, where integrals are of convolution form with weakly singular kernels of power law type. In recent decades fractional calculus has won more and more interest in applications in several fields of applied sciences. In this lecture we devote our attention to wave propagation problems in linear viscoelastic media. Our purpose is to outline the role of fractional calculus in providing simplest evolution processes which are intermediate between diffusion and wave propagation. The present treatment mainly reflects the research activity and style of the author in the related scientific areas during the last decades.

[1]  Hans Engler,et al.  Similarity solutions for a class of hyperbolic integrodifferential equations , 1997, Differential and Integral Equations.

[2]  Michele Caputo,et al.  A model for the fatigue in elastic materials with frequency independent Q , 1979 .

[3]  G. W. Scott-Blair.  A survey of general and applied rheology , 1951 .

[4]  F. Mainardi,et al.  The fundamental solution of the space-time fractional diffusion equation , 2007, cond-mat/0702419.

[5]  Massimiliano Giona,et al.  Fractional diffusion equation for transport phenomena in random media , 1992 .

[6]  Y. Rossikhin,et al.  Stress waves in a viscoelastic medium with a singular hereditary kernel , 1973 .

[7]  Francesco Mainardi,et al.  Linear models of dissipation in anelastic solids , 1971 .

[8]  Yasuhiro Fujita,et al.  Cauchy problems of fractional order and stable processes , 1990 .

[9]  R. Christensen,et al.  Theory of Viscoelasticity , 1971 .

[10]  Roberto Mecca,et al.  Fractional-order diffusion for image reconstruction , 2012, 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[11]  F. Mainardi,et al.  On a general class of constant-Q solids , 1982 .

[12]  R. Metzler,et al.  Fractional model equation for anomalous diffusion , 1994 .

[13]  F. A. Leckie,et al.  Creep problems in structural members , 1969 .

[14]  Francesco Mainardi,et al.  Seismic pulse propagation with constant Q and stable probability distributions , 1997, 1008.1341.

[15]  H. Kolsky,et al.  LXXI. The propagation of stress pulses in viscoelastic solids , 1956 .

[16]  M. Caputo.  Linear Models of Dissipation whose Q is almost Frequency Independent-II , 1967 .

[17]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[18]  Socio Michele Caputo.  The Green function of the diffusion of fluids in porous media with memory , 1996 .

[19]  R. Gorenflo,et al.  Fractional Calculus: Integral and Differential Equations of Fractional Order , 2008, 0805.3823.

[20]  W. Ames.  Mathematics in Science and Engineering , 1999 .

[21]  M. Caputo,et al.  A new dissipation model based on memory mechanism , 1971 .

[22]  Francesco Mainardi,et al.  Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics , 2012, 1201.0863.

[23]  Andreas Kreis,et al.  Viscoelastic pulse propagation and stable probability distributions , 1986 .

[24]  Anders E Boström,et al.  Acoustic Interactions with Submerged Elastic Structures: Part IV: Nondestructive Testing, Acoustic Wave Propagation and Scattering , 2002 .

[25]  F. Mainardi.  The fundamental solutions for the fractional diffusion-wave equation , 1996 .

[26]  F. Mainardi.  Fractional Relaxation-Oscillation and Fractional Diffusion-Wave Phenomena , 1996 .

[27]  R. Nigmatullin.  The Realization of the Generalized Transfer Equation in a Medium with Fractal Geometry , 1986, January 1.

[28]  Francesco Mainardi,et al.  Fractional Diffusive Waves , 2001 .

[29]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[30]  J. Prüss.  Evolutionary Integral Equations And Applications , 1993 .

[31]  Francesco Mainardi.  FRACTIONAL DIFFUSIVE WAVES , 2001 .

[32]  F. Mainardi,et al.  Wave front expansions for transient viscoelastic waves , 1975 .

[33]  Michele Caputo,et al.  Vibrations of an infinite plate with a frequency independent Q , 1976 .

[34]  M. Caputo.  Linear models of dissipation whose Q is almost frequency independent , 1966 .

[35]  I. Podlubny.  Fractional differential equations , 1998 .

[36]  M. Shitikova,et al.  Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids , 1997 .

[37]  W. Schneider,et al.  Fractional diffusion and wave equations , 1989 .

[38]  R. Gorenflo,et al.  Time-fractional derivatives in relaxation processes: a tutorial survey , 2008, 0801.4914.

[39]  Francesco Mainardi.  Transient Waves in Linear Viscoelastic Media , 2002 .

[40]  E. Strick,et al.  A PREDICTED PEDESTAL EFFECT FOR PULSE PROPAGATION IN CONSTANT‐Q SOLIDS , 1970 .

[41]  V. Zolotarev,et al.  Chance and Stability, Stable Distributions and Their Applications , 1999 .

[42]  M. T. Cicero.  FRACTIONAL CALCULUS AND WAVES IN LINEAR VISCOELASTICITY , 2012 .

[43]  Francesco Mainardi,et al.  The Wright functions as solutions of the time-fractional diffusion equation , 2003, Appl. Math. Comput..

[44]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[45]  Yasuhiro Fujita,et al.  INTEGRODIFFERENTIAL EQUATION WHICH INTERPOLATES THE HEAT EQUATION AND THE WAVE EQUATION I(Martingales and Related Topics) , 1989 .

[46]  J. Davis.  On wave propagation in viscoelastic media , 1961 .