[Paper Review] The explicit formula for solution of anomalous diffusion equation in the multi-dimensional space

D. K. Durdiev, Elina Shishkina+1 more|arXiv (Cornell University)|Sep 20, 2020

Differential Equations and Numerical Methods4 citations

TL;DR

This paper derives an explicit analytical solution for the n-dimensional anomalous diffusion equation with a Gerasimov–Caputo fractional derivative using Laplace and Fourier transforms. By showing equivalence between the fractional diffusion equation and a parabolic integro-differential equation with a Mittag-Leffler kernel, the solution is expressed in terms of Prabhakar fractional integrals and Fox H-functions, providing a closed-form representation for initial-boundary value problems in infinite domains.

ABSTRACT

This paper intends on obtaining the explicit solution of $n$-dimensional anomalous diffusion equation in the infinite domain with non-zero initial condition and vanishing condition at infinity. It is shown that this equation can be derived from the parabolic integro-differential equation with memory in which the kernel is $t^{-\\alpha}E_{1-\\alpha, 1-\\alpha}(-t^{1-\\alpha}),\\alpha\\in(0, 1),$ where $E_{\\alpha, \\beta}$ is the Mittag-Liffler function. Based on Laplace and Fourier transforms the properties of the Fox H-function and convolution theorem, explicit solution for anomalous diffusion equation is obtained.

Motivation & Objective

• To derive an explicit analytical solution for the n-dimensional anomalous diffusion equation with non-zero initial condition and vanishing boundary conditions at infinity.
• To establish the equivalence between the time-fractional diffusion equation and a parabolic integro-differential equation with a memory kernel involving the Mittag-Leffler function.
• To apply Laplace and Fourier transforms combined with the convolution theorem to obtain a closed-form solution in terms of special functions.
• To express the solution using the Prabhakar fractional integral and Fox H-function, enabling analytical treatment of anomalous diffusion in complex media.

Proposed method

• Use of the Gerasimov–Caputo fractional derivative to model time non-locality in anomalous diffusion.
• Transformation of the problem via Laplace transform in time and Fourier transform in space to decouple the system.
• Application of the convolution theorem to handle the integral terms arising from the fractional derivative and memory kernel.
• Derivation of the resolvent kernel for the memory integral equation using Laplace transform techniques.
• Identification of the memory kernel as $ t^{-eta} E_{1-eta,1-eta}(-t^{1-eta}) $, where $ \beta = \alpha $, linking it to the Mittag-Leffler function.
• Expression of the final solution in terms of the Fox H-function and Prabhakar integral operators through inverse transform inversion.

Experimental results

Research questions

• RQ1 Can an explicit analytical solution be derived for the n-dimensional anomalous diffusion equation with a Gerasimov–Caputo fractional derivative?
• RQ2 How is the anomalous diffusion equation related to a parabolic integro-differential equation with a Mittag-Leffler kernel?
• RQ3 What role do the Laplace and Fourier transforms play in deriving a closed-form solution for such fractional PDEs?
• RQ4 How can the Fox H-function and Prabhakar integral be used to represent the solution in a compact, analytically tractable form?

Key findings

• The solution to the n-dimensional anomalous diffusion equation with $ f(x,t) = 0 $ is given explicitly via the Fox H-function and Prabhakar fractional integral operators.
• The integro-differential equation with kernel $ t^{-eta} E_{1-eta,1-eta}(-t^{1-eta}) $ is mathematically equivalent to the time-fractional diffusion equation with the Gerasimov–Caputo derivative.
• The inverse Laplace transform of the resolvent kernel yields $ r(t) = t^{-eta}/\Gamma(1-\beta) $, confirming the connection to fractional calculus.
• The solution satisfies the initial condition $ u(x,0) = g(x) $ and vanishes at infinity in space, as required by the boundary conditions.
• The final solution is expressed as a series involving Fox H-functions: $ G(x,t) = \frac{1}{(2\pi)^{n/2}|x|^n} \sum_{j=0}^\infty \frac{(-t^{1-\alpha})^j}{j!} \left[ H_{2,0}^{1,2}(...) + H_{2,0}^{1,2}(...) \right] $.