Multi-dimensional α-fractional diffusion-wave equation and some properties of its fundamental solution

In this paper, a multi-dimensional -fractional diffusionwave equation is introduced and the properties of its fundamental solution are studied. This equation can be deduced from the basic continuous time random walk equations and contains the Caputo time-fractional derivative of the order /2 and the Riesz space-fractional derivative of the order so that the ratio of the derivatives orders is equal to one half as in the case of the conventional diffusion equation. It turns out that the -fractional diffusionwave equation inherits some properties of both the conventional diffusion equation and of the wave equation. In particular, in the one- and two-dimensional cases, the fundamental solution to the -fractional diffusionwave equation can be interpreted as a probability density function and the entropy production rate of the stochastic process governed by this equation is exactly the same as the case of the conventional diffusion equation. On the other hand, in the three-dimensional case this equation describes a kind of anomalous wave propagation with a time-dependent propagation phase velocity.

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

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

[3]  L. Boyadjiev,et al.  Comments on employing the Riesz-Feller derivative in the Schrödinger equation , 2013 .

[4]  Alan D. Freed,et al.  Fractional-order Viscoelasticity (FOV): Constitutive Development Using the Fractional Calculus: First Annual Report , 2002 .

[5]  William Rundell,et al.  Uniqueness and reconstruction of an unknown semilinear term in a time-fractional reaction–diffusion equation , 2013 .

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

[7]  J. Klafter,et al.  The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics , 2004 .

[8]  Andrey G. Cherstvy,et al.  Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. , 2014, Physical chemistry chemical physics : PCCP.

[9]  V. Kiryakova,et al.  The mellin integral transform in fractional calculus , 2013 .

[10]  Karl Heinz Hoffmann,et al.  Fractional Diffusion and Entropy Production , 1998 .

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

[12]  R. Askey,et al.  HANDBOOK OF INTEGRAL TRANSFORMS OF HIGHER TRANSCENDENTAL FUNCTIONS: Theory and Algorithmic Tables (Ellis Horwood Series: Mathematics and Its Applications) , 1983 .

[13]  Andrzej Hanygad,et al.  Multidimensional solutions of time-fractional diffusion-wave equations , 2002, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[14]  Yury Luchko,et al.  Modeling anomalous heat transport in geothermal reservoirs via fractional diffusion equations , 2011 .

[15]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[16]  Yu. A. Brychkov,et al.  Integrals and series , 1992 .

[17]  Yuri Luchko,et al.  Wave-diffusion dualism of the neutral-fractional processes , 2015, J. Comput. Phys..

[18]  V. Uchaikin.  Fractional Derivatives for Physicists and Engineers , 2013 .

[19]  Francesco Mainardi,et al.  Propagation speed of the maximum of the fundamental solution to the fractional diffusion-wave equation , 2012, Comput. Math. Appl..

[20]  Y. Povstenko.  Linear Fractional Diffusion-Wave Equation for Scientists and Engineers , 2015 .

[21]  Yuri Luchko.  Models of the neutral-fractional anomalous diffusion and their analysis , 2012 .

[22]  R. Herrmann.  Fractional Calculus: An Introduction for Physicists , 2011 .

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

[24]  A. Hanyga,et al.  Multidimensional solutions of space–fractional diffusion equations , 2001, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[25]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

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

[27]  I. Sokolov,et al.  Anomalous transport : foundations and applications , 2008 .

[28]  Christopher Essex,et al.  Tsallis Relative Entropy and Anomalous Diffusion , 2012, Entropy.

[29]  Karl Heinz Hoffmann,et al.  Fractional Diffusion, Irreversibility and Entropy , 2003 .

[30]  F. Mainardi,et al.  Some properties of the fundamental solution to the signalling problem for the fractional diffusion-wave equation , 2013 .

[31]  Patricia Melin,et al.  Background and Theory , 2021, New Medical Diagnosis Models Based on Generalized Type-2 Fuzzy Logic.

[32]  Yuri Luchko,et al.  Entropy Production Rate of a One-Dimensional Alpha-Fractional Diffusion Process , 2016, Axioms.

[33]  Yuri Luchko,et al.  A New Fractional Calculus Model for the Two-dimensional Anomalous Diffusion and its Analysis , 2016 .

[34]  R. Metzler,et al.  Space- and time-fractional diffusion and wave equations, fractional Fokker-Planck equations, and physical motivation , 2002 .

[35]  Yuri Luchko.  Multi-dimensional fractional wave equation and some properties of its fundamental solution , 2013, 1311.5920.

[36]  Yuri Luchko.  Fractional wave equation and damped waves , 2012, 1205.1199.

[37]  O. I. Marichev,et al.  Handbook of Integral Transforms of Higher Transcendental Functions , 1983 .

[38]  Karl Heinz Hoffmann,et al.  The superdiffusion entropy production paradox in the space-fractional case for extended entropies , 2010 .

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

[40]  Alexander I. Saichev,et al.  Fractional kinetic equations: solutions and applications. , 1997, Chaos.

[41]  Sverre Holm,et al.  On a fractional Zener elastic wave equation , 2012 .

[42]  A. Hanyga,et al.  Multi–dimensional solutions of space–time–fractional diffusion equations , 2002, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[43]  Yuri Luchko.  Fractional Schrödinger equation for a particle moving in a potential well , 2013 .