Convergence And Stability Of BDF Method For Uniform Mesh Of Proportional Delay Differential Equations

This paper mainly deals with the convergence and stability of backward differ-ential formulas on uniform meshes for pantograph differential equations.This kind of equations,widely used in many areas of science and technology,such as electric mechanics,optics,nonlinear dynamic systems,etc,are of important significance.In the numerical analysis,the requirement of mesh for pantography differential equa-tions is very strict,so most of the numerical researchers apply one-step geometrical mesh to investigate the numerical solution of pantography differential equations,but the numerical analysis based on uniform mesh multi-step is of some difficulties.In this paper,we first review the research status of pantograph differential equation and present the main work.Second,the backward differential formula(BDF)combined with a linear interpolation on uniform meshes is introduced to solve pantograph differential equations and the corresponding difference formula is given.And the condition for the existence and uniqueness of the numerical solution is discussed.Third,it is proved that the numerical solution convergent to the exact solution with order 1.Next,the asymptotic stability of the numerical solution is investigated.Last,some numerical experiments are given to illustrate the theoretical results obtained in this paper.