Research On Numerical Algorithm Of Fractional Differential Equation Based On Spectral Deferred Correction

In recent years,fractional differential equations and their applications have received widespread attention,mainly due to the application of fractional differential equations in various disciplines such as fluid mechanics,nonlinear acoustics,rheology,materials science and medicine,and the rapid development of their own theories.Fractional differential equations are the generalization of integral order differential equations.The difference is that fractional operators are nonlocal,which makes many numerical algorithms that can effectively calculate integral differential equations unable to be used to calculate fractional differential equations.Moreover,because of the“memory”property of the fractional differential operator and the singularity of the kernel,the solution of the fractional differential equation usually has low regularity,which leads to the slow convergence speed of the numerical algorithm and can not reach the expected accuracy.Therefore,it is particularly important to study the high precision and fast numerical algorithms for fractional differential equations.In this paper,we study the numerical solution of fractional differential equations,and propose an effective high-accurate numerical method based on spectral deferred correction.The spectral deferred correction method was first proposed by Dutt et al as an efficient,high-accuracy and stable numerical method for solving ordinary differential equations.In this paper,the space is discretized by the Fourier-Galerkin spectral method,and the original equation is transformed into a system of fractional order ordinary differential equations.At the same time,the convergence and stability of the space semi-discrete scheme are analyzed.Then,the spectral deferred correction method is used to solve the fractional ordinary differential equations,and the fully discrete scheme of the original equation is obtained.Through the combination of Fourier-Galerkin spectral method and spectral deferred correction method,the numerical solution can achieve high numerical accuracy in space and time.