Predictor-corrector schemes for nonlinear space-fractional parabolic PDEs with time-dependent boundary conditions

Abstract

We propose novel second-order accurate methods for nonlinear space-fractional PDEs with time-dependent boundary conditions. The matrix transfer technique is used for spatial discretization of the parabolic PDEs to obtain systems of ordinary differential equations. The treatment of the boundary conditions gave rise to a source term which depends on the temporal variable only. Methods based on rational approximations to the exponential function and which uses Gaussian quadrature points are developed for integrating the systems of differential equations. We develop new schemes based on the (1,1)-, (0,2)-Padé approximations and the real distinct pole approximation to the exponential function with Gaussian quadrature points. Error estimates are provided for the convergence analysis. Stability and reliability of the resulting schemes are discussed. A partial fraction decomposition technique is used for the efficient implementation of the schemes. Finally, numerical results in one and two spatial dimensions are presented to corroborate our theoretical observations.