From the space–time fractional integral of the continuous time random walk to the space–time fractional diffusion equations, a short proof and simulation

Abstract

In this paper, I prove the relation between the integral equation of the continuous time random walk (CTRW) and the space–time fractional diffusion equations by using the definitions of the Weyl fractional derivatives and integrals. I generalize a transformation theorem between the independent variables of the solution of the space–time fractional diffusion equation (stfde) and the solution of the space–time fractional Fokker–Planck equation(stffpe). I simulate the symmetric and the non symmetric random walks for the two mentioned models. I use the asymptotic behavior of the Mittag-Leffler function for generating the waiting time random variable and use the Lévy flight distribution function to generate the jump width random variable for the both models. The simulation results are investigated for different values of the space and time fractional orders.