Research On Efficient Numerical Algorithms For Time Fractional Integro-Differential Equations

Time fractional integro-differential equation has always been one of the hot issues of research.Because of its non-locality,most of the analytical solutions of fractional integro-differential equations are extremely difficult to solve,and the numer-ical solution is more widely used than the analytical solution in practical engineering.Therefore,many scholars turn to study its numerical solution and try to find a more efficient numerical scheme.The main research of the article is on efficient numerical algorithms for time-fractional-order integral differential equations.The first part constructs a higher-order numerical scheme for the two-dimensional nonlinear fractional-order Volterra integral equation and its system of equations based on the segmented quadratic Lagrangian interpolation method.The local truncation error of the scheme is analyzed,and the convergence of the higher-order numerical format is rigorously proved with the optimal convergence order (h_x^4-α+h_y^4-β),0<α,β<1.Finally,the correctness of the theoretical analysis and the effectiveness of the algorithm are verified by numerical arithmetic examples.The second part is devoted to the study of the Volterra integral equation with Ca-puto fractional order derivatives and its extension to the form of a system of equations.The higher order numerical scheme of the equations and the system of equations is first constructed sequentially using the segmented quadratic Lagrangian interpolation method,followed by the analysis of the corresponding local truncation error of the scheme,whose order is (h_t^3-θ+h_x^4-α+h_y^4-β),0<θ,α,β<1.Finally,numer-ical simulations are performed by specific arithmetic examples to further verify the effectiveness of the numerical algorithm.The third part is the fast computation of the Volterra integral equation with Caputo fractional order derivatives with respect to time.First,the discrete and fast schemes are constructed based on linear interpolation and exponential and approximation methods,respectively,and the space is constructed in the same way as in the previous section.Then,the local truncation error analysis is given for the discrete scheme and the fas-tened scheme respectively,and finally,it is demonstrated by numerical examples that the fastened scheme is more efficient and requires less computation time than the dis-crete scheme.