The fundamental solution of the space-time fractional diffusion equation

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.

46 pages, 14 figures (1+26=27 eps files)

References in corpus (2)

• Fractional Calculus: Integral and Differential Equations of Fractional Order
• Analytical properties and applications of the Wright function

Cited by in corpus (14)

• Matrix approach to discrete fractional calculus II: partial fractional differential equations
• Coupled continuous time random walks in finance
• Some aspects of fractional diffusion equations of single and distributed order
• Probability distributions generated by fractional diffusion equations
• Mellin transform and subordination laws in fractional diffusion processes
• Numerical simulations of anomalous diffusion
• Random walk models associated with distributed fractional order differential equations
• From Power Laws to Fractional Diffusion: the Direct Way
• Gibbs Partitions (EPPF's) Derived From a Stable Subordinator are Fox H and Meijer G Transforms
• Solutions of fractional reaction-diffusion equations in terms of the H-function
• Continuous time random walk, Mittag-Leffler waiting time and fractional diffusion: mathematical aspects
• Fractional Diffusion Processes: Probability Distributions and Continuous Time Random Walk
• Applications of integral transforms in fractional diffusion processes
• Some recent advances in theory and simulation of fractional diffusion processes