Solution of stochastic nonlinear time fractional PDEs using polynomial chaos expansion combined with an exponential integrator

Abstract

The aim of this paper is to introduce an efficient numerical algorithm for the solution of stochastic time fractional stiff partial differential equations (PDEs). The time fractional derivative is described in the Caputo sense. By applying polynomial chaos (PC) expansion to discretize the random variable, a coupled time fractional deterministic system of PDEs is obtained. For the resulting system of time fractional PDEs we apply the Fourier spectral method to discretize the spatial variable, and use the fast Fourier transform (FFT) during the computation. Then, we use an exponential integrator method to overcome the stability issues due to the stiffness in the resulting time fractional semi-discrete system. The considered models have two challenging parameters, fractional order of equations and noise amplitude. In addition to exponential integrator, we also implemented a predictor–corrector method of Adams–Bashforth–Moulton type. We include numerical results of applying the developed method to stochastic time fractional Burgers and Kuramoto–Sivashinsky (KS) equations, for various values of noise intensity. A comparison of performance of the proposed scheme with fractional Adams method is also reported which confirms the efficiency and applicability of the proposed method based on the exponential integrator scheme.