Research On Non-standard Finite Difference Methods For Spatial Fractional Partial Differential Equations

In this thesis,the non-standard finite difference method for three kinds of spatial fractional partial differential equations is studied,and the stability and convergence of the nonstandard finite difference scheme are discussed.Finally,the accuracy of the results is proved by numerical examples.There are four chapters in this thesis.The structure is as follows:In chapter one,the history,background and significance of the research on spatial fractional partial differential equation and non-standard finite difference method are introduced,and some research status at domestic and foreign are analyzed.In chapter two,we constructed a non-standard finite difference scheme for spatial fractional diffusion equation,and then proved that the scheme is stable by using Fourier transformation method.The numerical experiments show that the denominator functions are constructed in various forms,and the maximum error can be reduced by using different denominator functions,which further verifies the effectiveness of the non-standard finite difference method.In chapter three,the non-standard finite difference method for space fractional convection-diffusion equation is studied.The fractional derivative of space is discretized by the displaced Grünwald-Letnikov formula,and the non-standard finite difference scheme is constructed by substituting the denominator of the discrete scheme with the denominator function with step size.Then the stability of the difference scheme is discussed by using Fourier transform method.In numerical examples,the comparison of experimental data shows that the maximum error can be reduced by choosing the appropriate denominator function.In chapter four,the space fractional heat equation is numerically studied by combining Crank-Nicolson difference method and non-standard finite difference method.The non-standard finite difference scheme is constructed by discretizing two spatial fractional derivatives with Grünwald-Letnikov formula and displacement Grünwald-Letnikov formula.The stability of the scheme is analyzed by Fourier transformation method.The numerical example not only verifies the correctness of the conclusion,but also shows that the maximum error can be reduced and the accuracy can be improved by constructing a proper denominator function.Finally,the research content of this thesis is summarized and the future work is prospected.