Meshless approach for solving nonlinear time-fractional fourth-order reaction–diffusion equation with convergence order analysis and stability analysis

This article presents a novel meshless approach to solve a nonlinear fourth-order reaction-diffusion equation with the time-fractional derivative of Caputo-type on arbitrary bounded domains by constructing a set of novel basis using the Sinc function as a shape function with higher accuracy and stability. The key contributions and innovations of this study are summarized as follows. Firstly, the time non-smooth problem is addressed by discretizing time term and approximating the time-fractional derivative with the piecewise fractional linear interpolation. What's more, a novel approach has been developed for the superconvergence estimation of the two-dimensional Sinc function approximation. Subsequently, the time-iterative stability analysis of the semi-analytical solution is presented, and it is demonstrated that the solution is absolutely stable on arbitrary bounded domains. We then present a detailed analysis of both local and global error estimates and prove that the space-time convergence order is O ( M 4 − m q + q ) with M being the number of basis and q being the length of time step, that is, the spatial convergence order is superconvergent. At last, a series of numerical examples validates the effectiveness of the proposed meshless method, and the low CPU time demonstrates its high computational efficiency.