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

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)
Full Text: DOI
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