Fractional Differential Spectra Delay Correction Method And Spectral Galerkin Method

Fractional calculus has recently gained considerable popularity and importance due to its attractive applications in widespread fields of science and engineering. These applications greatly highlight distinct superiorities and unsubstitutability of fractional calculus. However, as is well known that the fractional-calculus operators are nonlocal, being quite different from the classical-calculus ones, and many most effective numerical methods for classical differential equations would be totally ineffective in solving fractional differential equations. Thus, the study of numerical methods for fractional differential equations is getting to be more and more important.In this paper, a new high order numerical method is given to calculate a class of fractional differential equations, the main works are as follows:(1) Based on the residual function and the error equation deduced from Volterra integral equations equivalent to the fractional differential equations, a new high order numerical method for fractional differential equations(FDEs) and a system of fractional differential equations(FDES) is constructed according to the idea of spectral deferred correction. The proposed method allows that one can use a relatively few nodes to obtain the high accuracy numerical solutions of FDES without the penalty of a huge computational cost due to the non-locality of Caputo derivative. Finally, numerical experiments are given to verify the high accuracy and efficiency of this method.(2) The variational form of time-space fractional advection-diffusion equation is given. And a spectral Galerkin method is designed for time-space fractional advection-diffusion equation with initial boundary conditions. The existence and uniqueness of the variational form are proved, and it is also proved that the proposed method has spectral accuracy both in temporal and spatial direction. Numerical experiments are provided to check the theoretical results.