Sinc Methods For The Time Fractional Diffusion Equation

In recent years,the time fractional differential equations have attracted the attention of many scholars,and its application fields are becoming more and more extensive.In the previous research,the finite difference method and other methods were used to discrete time fractional derivatives,and the spatial derivative was processed by classical central difference scheme and spectral method.However,these methods have certain deficiencies in numerical precision and convergence,and the Sinc methods have attracted many scholars' attention because of its advantages of exponential convergence,high precision and small error.In this paper,the numerical schemes of one-dimensional and two-dimensional time fractional diffusion equations are studied by combining the L2-1?,difference formula and the Sinc methods,and the corresponding theoretical analysis is carried out.Specifically,there are two aspects of research content as follows:(1)The Sinc numerical schemes of the one-dimensional TFD equation are constructed.The difference formula of Caputo fractional derivative-L2-1? formula is used to discretize the time fractional derivative,and the semi-discrete scheme is obtained.The error analysis of the semi-discrete scheme is carried out and its convergence with O(?2).Then the Sinc-Collocation method,the Sinc-Galerkin method and the Quasi-wavelet method are used to discretize the spatial derivatives of the TFD equation,and the full-discrete schemes of the three methods for the one-dimensional TFD equation are obtained.It is proved that the Sinc-Collocation scheme is stable for all ?>0 in L2 norm.Finally,the validity of the schemes established by the three methods and the correctness of the theoretical analysis are verified by numerical examples.(2)The Sinc numerical schemes of the two-dimensional TFD equation are constructed.The discrete method of the time fractional derivative is the same as the one-dimensional problem described above.The L2-1? difference formula is used to discretize the time fractional derivative,and the time semi-discrete scheme is obtained.The stability and convergence of the semi-discrete scheme are proved.The Sinc-Collocation method and the Sinc-Galerkin method are used to discretize the spatial derivatives,and the full-discrete schemes for solving the two-dimensional TFD equation are constructed.The numerical examples are given to verify the effectiveness of the numerical schemes constructed by the two methods.The results show that the errors obtained by the Sinc-Collocation method and the Sinc-Galerkin method are exponentially convergent.