Fractional Linear Fokker - Planck Equation with Applications

Abstract

Fractional calculus is a developing field of research in mathematics. It is used widely in other fields of science and finance with several applications. In this thesis, we solve the linear Fokker -Planck equation (FPE) and fractional Fokker Planck equation in three different ways: analytically, numerically and stochastically. We solved it for the normal diffusion case and the sub-diffusion case (0<  1 ). In the analytical method, the separation of variable technique are used, especially, we discuss the analytical solution in sections 2.2, and 2.4 in details, which was not discussed in details in the previous books and periodicals. In the numerical scheme, we used the explicit finite difference method. In Particular, we investigate the stability of the scheme in section 3.3, which was not discussed in the previous books and periodicals for the approach used in the proof. The stochastic method is used, with help of Euler-Maruyama method, to solve the motion of free diffusing ink molecules in a fluid. Using this method, we solved FPE for the unbounded and bounded case. Unexpectedly, in the bounded case, we noticed the appearance of transient fractional behavior.