Subordination Approach to Space-Time Fractional Diffusion

Emilia Bazhlekova,Ivan Bazhlekov

doi:10.3390/math7050415

Abstract

The fundamental solution to the multi-dimensional space-time fractional diffusion equation is studied by applying the subordination principle, which provides a relation to the classical Gaussian function. Integral representations in terms of Mittag-Leffler functions are derived for the fundamental solution and the subordination kernel. The obtained integral representations are used for numerical evaluation of the fundamental solution for different values of the parameters.

Highlights

This work is concerned with the n-dimensional space-time fractional diffusion equationDt u(x, t) = −(−∆)α u(x, t), β t > 0, x ∈ Rn ; u(x, 0) = v(x); (1)β where 0 < α, β ≤ 1, Dt is the Caputo time-fractional derivative [1,2] β Dt f ( t ) = d Γ(1 − β) dtZ t f ( t ) − f (0)(t − τ ) β dτ, t > 0, 0 < β < 1, (2)
Formula (10), we derive integral representations for the fundamental solution in terms of Mittag-Leffler functions, which are appropriate for numerical implementation
For the numerical implementation of Formula (34), the real and imaginary parts above can be numerically calculated by employing a method of computation of the Mittag-Leffler function of complex arguments

Summary

Introduction

This work is concerned with the n-dimensional space-time fractional diffusion equation. An alternative approach to dealing with the space-time fractional diffusion Equation (1) is based on the subordination formula, which relates the fundamental solution Gα,β,n (x, t) and the Gaussian function G1,1,n (x, t) as follows [14,15]. Subordination formulae for the one-dimensional space-time fractional diffusion equation have been studied in [6,9]. The subordination Formula (10) serves as a starting point for deriving integral representations for the fundamental solution Gα,β,n (x, t). Formula (10), we derive integral representations for the fundamental solution in terms of Mittag-Leffler functions, which are appropriate for numerical implementation.

Preliminaries

An Integral Representation for the Subordination Kernel

Integral Representations for the Fundamental Solution

Concluding Remarks