A Lie Group Approach To Solve the Fractional Poisson Equation

Abstract

In the present paper, approximate solutions of fractional Poisson equation (FPE) have been considered using an integrator in the class of Lie groups, namely, the fictitious time integration method (FTIM). Based on the FTIM, the unknown dependent variable u(x, t) is transformed into a new variable with one more dimension. We use a fictitious time tau as the additional dimension (fictitious dimension), by transformation: v(x, t, tau) := (1 + tau)(k) u(x, t), where 0 < k <= 1 is a parameter to control the rate of convergency in the FTIM. Then the group preserving scheme (GPS) is used to integrate the new fractional partial differential equations in the augmented space R2+1. The power and the validity of the method are demonstrated using two examples.