The Contractivity Of Implict Euler Methods For The Nonlinear Systems Of Neutral Delay Integro-differential Equations

This paper is concerned with the contractivity of implict Euler methods for the nonlinear systems of neutral delay integro-differential equations (NDIDES).The con-dition number of this paper is a function of the variable t, not constant.For nonlinear neutral delay integro-differential equations,if the condition number meet certain con-ditions,then we use the implicit Euler method for numerical calculation with a variable step size,the final result can be contract. In particular,for the proof process, we use the classical interpolation scheme to calculate the points between two adjacent node val-ues.We use the Compound trapezoid quadrature formula to calculate the integral term of the equation.Finally, We prove the contractivity of Euler method for nonlinear neu-tral delay integro-differential equations(NDIDES) by sub-interval.At the end of the paper, numerical examples are given to verify the theoretical results obtained.This thesis consists of four chapters,and be organized as follows:The first is an introduction.We introduce the application of delay integro- differen-tial equations , backgrounds, and innovation of this paper.In the second chapter,we introduce nonlinear systems of neutral delay integro- dif-ferential equations (NDIDES).we also introduce the basic concepts and symbols used in this article.In the third chapter,we introduce the implicit Euler method, Compound trapezoid quadrature method, and the main results of the proof.And finally we obtained the con-tractivity of implicit Euler method for the nonlinear neutral delay intergro- differential equations.In the forth chapter,we use implicit Euler methods to solve several nonlinear neu-tral delay intergro-differential equations. These numerical results verify the conclu-sions obtained in Chapter III.