Robust Implicit Difference Approach for the Time-Fractional Kuramoto–sivashinsky Equation With the Non-Smooth Solution

Abstract

This paper formulates the L1 implicit difference scheme (L1IDS) for the time-fractional Kuramoto–Sivashinsky equation (TFKSE) with non-smooth solution. The TFKSE is one of useful descriptions for modeling flame-propagation, viscous flow problems, and reaction–diffusion systems. The proposed method approximates the unknown solution by using two main stages. At the first stage, the L1 method with nonuniform meshes and the general centered difference method is adopted to discretize the Caputo fractional derivative and the spatial derivative, respectively. In the second stage, the fully-discrete L1IDS is established with the help of the Galerkin scheme based on piecewise linear test functions. Meanwhile, an iterative algorithm is adopted to solve the nonlinear systems. Furthermore, the convergence and stability of the proposed method are both demonstrated and confirmed numerically. Finally, three numerical examples highlight the accuracy and efficiency of the proposed strategy.