Time-fractional wave-diffusion equation in an inhomogeneous half-space

Abstract

We consider the fundamental solution of the time-fractional wave-diffusion equation in a three-dimensional half-space medium which contains an inhomogeneity in form of a plane parallel layer. The corresponding Green’s function which is derived by means of the Fourier and Laplace transforms can be accurately and efficiently evaluated without recourse to the Mittag-Leffler or the Fox H-function. Moreover, it is shown that in the one-dimensional case the fundamental solution in an inhomogeneous half-space is no longer a probability density function. In addition, we consider the advection equation for the fractional Laplacian and the Caputo time-fractional derivative of orders on a bounded domain. Simple algorithms for accurate evaluation of the M-Wright function and the Mittag-Leffler function are enclosed at the end of this article.