Fourier Convergence Analysis for Fokker-Planck Equation of Tempered Fractional Langevin-Brownian Motion and Nonlinear Time Fractional Diffusion Equation

Fourier analysis works well for the finite difference schemes of the linear partial differential equations. However, the presence of nonlinear terms leads to the fact that the method cannot be applied directly to deal with nonlinear problems. In the current work, we introduce an effective approach to enable Fourier methods to effectively deal with nonlinear problems and elaborate on it in detail by rigorously proving that the difference scheme for two-dimensional nonlinear problem considered in this paper is strictly unconditionally stable and convergent. Further, some numerical experiments are performed to confirm the rates of convergence and the robustness of the numerical scheme.