From stochastic processes to numerical schemes for fractional DEs, and PDEs

Abstract: 

In this talk I will outline a method of obtaining numerical schemes for certain classes of DEs and PDEs as well as their generalisations to fractional DEs and PDEs. The fundamental idea is to use a discrete time and space random walk and show that in the diffusive limit, that in the limit as the time and space grid spacings go to zero, the governing equations of the random walk become the equation of interest. In such a manner the governing equations of the discrete random walk can then be taken as approximating the diffusion limit equations. As a the numerical scheme corresponds to the governing equation of a stochastic process the resulting solution must obey certain regularity conditions, for example the solution is positive definite .
For simple PDEs, such as the diffusion equation, this random walk scheme corresponds to well known explicit numerical schemes. But in slightly more complicated cases, such as Fokker-Planck equations, the discrete random walk gives different schemes. In the case of fractional DEs and PDEs a discrete time random walk is chosen such that the waiting time probability is dependent on the time since arrival at the site. With the appropriate choice of probabilities, fractional derivatives will appear in the limit of the governing equations.
Examples of where we have used this include solving fractional Fokker-Planck equations, fractional reaction-diffusion equations, as well as a fractional SIR compartment model.