Finite Difference and Spline Approximation for Solving Fractional Stochastic Advection-Diffusion Equation

Abstract

This paper is concerned with numerical solution of time fractional stochastic advection-diffusion type equation where the first order derivative is substituted by a Caputo fractional derivative of order $$\alpha $$ ( $$0 <\alpha \le 1$$ ). This type of equations due to randomness can rarely be solved, exactly. In this paper, a new approach based on finite difference method and spline approximation is employed to solve time fractional stochastic advection-diffusion type equation, numerically. After implementation of proposed method, the under consideration equation is transformed to a system of second order differential equations with appropriate boundary conditions. Then, using a suitable numerical method such as the backward differentiation formula, the resulting system can be solved. In addition, the error analysis is shown in some mild conditions by ignoring the error terms $$O(\Delta t^2)$$ in the system. In order to show the pertinent features of the suggested algorithm such as accuracy, efficiency and reliability, some test problems are included. Comparison achieved results via proposed scheme in the case of classical stochastic advection-diffusion equation ( $$\alpha =1$$ ) with obtained results via wavelets Galerkin method and obtained results for other values of $$\alpha $$ with the values of exact solution confirm the validity, efficiency and applicability of the proposed method.