Research On Efficient Numerical Algorithms For High Dimensional Nonlocal Integro-differential Equations

Fractional integro-differential equations are a class of common nonlocal differential equations,which have wider application scope and stronger analytical ability.They are often applied to viscoelastic mechanics,fractional physics,molecular biology,medical image processing,rheology,stochastic processes and control robots.Fractional integro-differential equations are a new type of differential equations,which have important research value.It not only challenges the thinking mode of traditional integer-order differential equations,promotes the development of differential equations,but also opens up the application prospects of various scientific fields.Therefore,the study of fractional integro-differential equations is an important and challenging topic in the field of scientific research.This paper mainly studies the efficient numerical algorithm of high-dimensional nonlocal integro-differential equations.In the first part of this paper,two different methods are used to construct the high-order numerical scheme of three-dimensional nonlinear Volterra integral equations.Firstly,the Simpson formula and the quadratic Lagrange interpolation method are used to construct the high-order numerical scheme of three-dimensional nonlinear Volterra integral equations,and the local truncation error and convergence are strictly analyzed.Secondly,another high-order numerical scheme for the three-dimensional nonlinear Volterra integral equation is constructed by using the piecewise cubic Lagrange interpolation method.Based on the local truncation error and convergence analysis,the convergence order of the scheme is obtained to be fourth order,and the convergence order of the different numerical schemes constructed by the two different methods is fourth order,and the validity of the two numerical algorithms and the correctness of the theoretical analysis are verified by the calculation of typical problems.In the second part of this paper,we study the fast scheme of Volterra integral equation with Caputo fractional derivative.Firstly,an ordinary numerical algorithm(L2scheme)for Volterra integral equation with Caputo fractional derivative is constructed by using the quadratic Lagrange interpolation method,and the corresponding error analysis and numerical examples are given.Secondly,the fast scheme and error analysis of the Volterra integral equation with Caputo fractional derivative are given.At the same time,three numerical examples are given,and these three numerical examples are based on the L2 scheme and the fast scheme.The error and CPU time obtained by the operation further verify the applicability of the efficient numerical algorithm.