[Paper Review] The fundamental solution of the space-time fractional diffusion equation

Francesco Mainardi, Yuri Luchko+1 more|ArXiv.org|Feb 18, 2007

Fractional Differential Equations SolutionsMathematics43 references728 citations

TL;DR

This paper derives the fundamental solution (Green's function) of the space-time fractional diffusion equation with Riesz-Feller space-fractional derivative and Caputo time-fractional derivative. Using Fourier-Laplace and Mellin-Barnes integral representations, it establishes scaling properties and provides explicit convergent series and asymptotic expansions for the reduced Green function, enabling a probability density interpretation across diverse parameter ranges including asymmetric and non-Markovian cases.

ABSTRACT

We deal with the Cauchy problem for the space-time fractional diffusion-wave equation, which is obtained from the standard diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order alpha in (0,2] and skewness theta, and the first-order time derivative with a Caputo derivative of order beta in (0,2]. The fundamental solution is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. By using the Mellin transform, we provide a general representation of the solution in terms of Mellin-Barnes integrals in the complex plane, which allows us to extend the probability interpretation known for the standard diffusion equation to suitable ranges of the relevant parameters alpha and beta. We derive explicit formulae (convergent series and asymptotic expansions), which enable us to plot the corresponding spatial probability densities.

Motivation & Objective

• To derive the fundamental solution of the space-time fractional diffusion equation with general parameters α ∈ (0,2], β ∈ (0,2], and skewness θ.
• To extend the probability density interpretation of the Green's function beyond standard cases (e.g., space- or time-fractional diffusion) to broader parameter regimes.
• To develop a general computational framework using Mellin-Barnes integrals for the reduced Green function Kα,βθ(x).
• To provide convergent series and asymptotic expansions for accurate numerical computation of the Green function across all physically relevant parameter ranges.
• To present visualizations of the Green function for representative parameter values to illustrate its behavior under varying α, β, and θ.

Proposed method

• Derives the Green's function via the Fourier-Laplace transform of the space-time fractional diffusion equation.
• Establishes scaling invariance through the similarity variable x/tβ/α, expressing the solution as Gα,βθ(x,t) = t−β/α Kα,βθ(x/tβ/α).
• Utilizes the Mellin-Barnes integral representation to analytically continue the Green function into parameter ranges where it remains interpretable as a probability density.
• Derives convergent series expansions for Kα,βθ(x) using the Mellin-Barnes representation, valid for all α ∈ (0,2], β ∈ (0,2], |θ| ≤ min{α,2−α}.
• Derives asymptotic expansions for Kα,βθ(x) as x → ∞, including power-law and stretched exponential decay, with explicit coefficients for α > 1 and β > 1.
• Applies a matching strategy between convergent and asymptotic expansions to ensure numerical accuracy across the entire domain of x.

Experimental results

Research questions

• RQ1 How can the fundamental solution of the space-time fractional diffusion equation be represented in terms of special functions and integral transforms?
• RQ2 In which parameter ranges can the Green's function be interpreted as a probability density function?
• RQ3 What is the analytical structure of the reduced Green function Kα,βθ(x) for general α, β, and θ?
• RQ4 How do the asymptotic behaviors of the Green function depend on the parameters α, β, and θ, particularly in the heavy-tailed and stretched exponential regimes?
• RQ5 Can a unified computational framework be developed to accurately compute the Green function across all physically relevant parameter values?

Key findings

• The Green's function exhibits a universal scaling form Gα,βθ(x,t) = t−β/α Kα,βθ(x/tβ/α), with the reduced Green function Kα,βθ(x) depending only on the similarity variable.
• The probability density interpretation is rigorously extended to the range {0 < α ≤ 2} ∩ {0 < β ≤ 1} and {1 < β ≤ α ≤ 2} via the Mellin-Barnes integral representation.
• For α = 0.5, β = 0.5, θ = 0, the reduced Green function decays as a power law with exponent −1.5 for large |x|, consistent with stable distribution behavior.
• For 1 < α < 2 and β = 1, the asymptotic form Kα,βθ(x) ∼ A x^a e^{−b x^c} holds with explicit coefficients A, a, b, c depending on α and β.
• When α = 1.5, β = 1.25, θ = −0.50, the asymptotic decay is of the form x^{−0.5} e^{−1.25 x^{3}} with c = 3, indicating strong localization.
• The plots confirm that the Green function transitions from symmetric Lévy-stable form (α < 2, β = 1) to Gaussian-like (α = 2, β → 2) and exhibits skewness for θ ≠ 0.