Simulation of the continuous time random walk of the space-fractional diffusion equations. (English) Zbl 1153.65007

Abdel-Rehim, E. A.; Gorenflo, R.

J. Comput. Appl. Math. 222, No. 2, 274-283 (2008).

Summary: We discuss the solution of the space-fractional diffusion equation with and without central linear drift in the Fourier domain and show the strong connection between it and the \(\alpha \)-stable Lévy distribution, \(0<\alpha <2\). We use some relevant transformations of the independent variables \(x\) and \(t\), to find the solution of the space-fractional diffusion equation with central linear drift which is a special form of the space-fractional Fokker-Planck equation which is useful in studying the dynamic behaviour of stochastic differential equations driven by the non-Gaussian (Lévy) noises. We simulate the continuous time random walk of these models by using the Monte Carlo method.

MSC:

65C30	Numerical solutions to stochastic differential and integral equations
26A33	Fractional derivatives and integrals
45K05	Integro-partial differential equations
60J60	Diffusion processes
60G50	Sums of independent random variables; random walks
60G15	Gaussian processes
60H15	Stochastic partial differential equations (aspects of stochastic analysis)
60H35	Computational methods for stochastic equations (aspects of stochastic analysis)

Keywords:

space-fractional diffusion equation; space-fractional derivative; Fokker-Planck equation; stochastic processes; \(\alpha \)-stable distribution; continuous time random walk; Monte Carlo method; stochastic differential equations; numerical examples

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