Subordination principles for the multi-dimensional space-time-fractional diffusion-wave equation

Abstract

This paper is devoted to an in deep investigation of the first fundamental solution to the linear multi-dimensional space-time-fractional diffusionwave equation. This equation is obtained from the diffusion equation by replacing the first order time-derivative by the Caputo fractional derivative of order β, 0 < β ≤ 2 and the Laplace operator by the fractional Laplacian (−∆) α 2 with 0 < α ≤ 2. First, a representation of the fundamental solution in form of a Mellin-Barnes integral is deduced by employing the technique of the Mellin integral transform. This representation is then used for establishing of several subordination formulas that connect the fundamental solutions for different values of the fractional derivatives α and β. We also discuss some new cases of completely monotone functions and probability density functions that are expressed in terms of the Mittag-Leffler function, the Wright function, and the generalized Wright function.

Subordination Approach to Space-Time Fractional Diffusion

“…In the one-dimensional case −(−∆) α is the Riesz space-fractional derivative of order 2α. The space-time fractional diffusion Equation (1) has been extensively studied [4][5][6][7][8][9][10][11][12][13][14][15]. The solution u(x, t) of Problem (1) is given in terms of the fundamental solution G α,β,n (x, t) and the initial function v(x), as follows:…”

“…Extensive research has been devoted to representations of the fundamental solution in the form of the Mellin-Barnes integral or the Fox H-function, such as in [5,6,11] for the one-dimensional and [12][13][14] for the multi-dimensional space-time fractional diffusion-wave equation. One of the advantages of such representations is that the asymptotic behavior of the fundamental solution can be derived from them, because the asymptotic behavior of the Fox H-function has been well-studied (see e.g., [18] or [19]).…”

“…An alternative approach to dealing with the space-time fractional diffusion Equation (1) is based on the subordination formula, which relates the fundamental solution G α,β,n (x, t) and the Gaussian function G 1,1,n (x, t) as follows [14,15]…”

“…The subordination kernel ψ α,β (t, τ) depends on the similarity variable τt −β/α and admits the representation [14]…”

“…The Laplace transform pair (13) was first derived in [14] (see Remark 4.4). It is worth noting that some known basic properties of G α,β,n (x, t) follow in a straightforward way from the subordination relation (10), taking into account that the subordination kernel is a pdf.…”