The Application Of Contour Integral Method In Non-local Differential Equations

In recent years,the research on non-local differential equations has received extensive attention.Among them,the research on high-precision numerical methods of non-local differential equations has been a hot topic in the field of computational mathematics and applied mathematics.Based on this research background,this paper discusses a class of efficient numerical methods,that is,Contour Integral Method(CIM).Among them,the birth and current research status of the contour integral method were studied,the mathematical mechanism of the contour integral method was explained,the existing integral contour was classified,and the selection of the parameters of each integral contour is summarized.Based on the above analysis,this paper uses the contour integral method to solve a representative scalar problem and the time-fractional diffusion equation,the numerical results show the high numerical performance of the CIM and take this as an prelude of this paper.In the process of solving,the six types of integral contours mentioned in the article are selected.In addition,under given conditions,by comparing the corresponding numerical performances of the contour integral method under different integral contour,the integral contour used in this paper is qualitatively determined.Next,this paper mainly discusses the use of the contour integration method to solve the Feynman-Kac equation with multiple internal states.Since the FeynmanKac equations for multiple internal states are newly established,which contain fractional-order matter derivatives,and they are spatio-temporal coupling operators,which pose challenges to analysis and solution,so it becomes obvious to study the efficient numerical method of this equation.Therefore,this paper first performs a regularity analysis on the Feynman-Kac equations for the two internal states to obtain the singularity of the equation at the origin.The CIM scheme of the Feynman-Kac equations with two internal states is constructed,and the strict error analysis is given.The predictor-corrector scheme of the Feynman-Kac equations with two internal states is also constructed.Taking the step size is sufficiently small,and letting this solution as an exact solution and compared with the numerical solution obtained from the CIM scheme,thus the high numerical performance of the CIM format is verified.Finally,the research results of this article are prospected and summarized.