2017/09/22 by Alessandro De Gregorio, Enzo Orsingher, De Gregorio, Alessandro +1 · 1 citation
Biochemistry, Genetics and Molecular Biology · #Diffusion and Search Dynamics
paper · pdf · doi:10.48550/arxiv.1709.07663
In this paper we consider the Porous Medium Equation and establish a\nrelationship between its Kompanets-Zel'dovich-Barenblatt solution u( xd,t),\n xd\∈ mathbb Rd,t>0 and random flights. The time-rescaled version of\nu( xd,t) is the fundamental solution of the Euler-Poisson-Darboux equation\nwhich governs the distribution of random flights performed by a particle whose\ndisplacements have a Dirichlet probability distribution and choosing directions\nuniformly on a d-dimensional sphere (see, e.g., citedgo).\n We consider the space-fractional version of the Euler-Poisson-Darboux\nequation and present the solution of the related Cauchy problem in terms of the\nprobability distributions of random flights governed by the classical\nEuler-Poisson-Darboux equation. Furthermore, this research is also aimed at\nstudying the relationship between the solutions of a fractional Porous Medium\nEquation and the fractional Euler-Poisson-Darboux equation.\n A considerable part of the paper is devoted to the analysis of the\nprobabilistic tools of the solutions of the fractional equations. Also the\nextension to higher-order Euler-Poisson-Darboux equation is considered and the\nsolutions interpreted as compositions of laws of pseudoprocesses.\n