Simulation of the continuous time random walk of the space-fractional diffusion equations

In this article, 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 α-stable Lévy distribution, 0 < α < 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.