Convergence And Stability Of Boundary Value Methods For Differential Equations With Piecewise Continuous Arguments

This dissertation deals with the convergence and asymptotic stability of block boundary value methods for a class differential equations of advanced type with piecewise constant arguments.Differential equations with piecewise continuous arguments represent hybrid of continuous and discrete dynamical systems,combine the properties of both differential and difference equations and have important applied value in information technology,electrodynamic,control science and many other scientific fields.Therefore,researching of differential equation with piecewise continuous arguments is of important theoretical values and practical significance.Some applications of differential equations with piecewise continuous arguments in different fields are presented.The development of numerical methods for differential equations with piecewise continuous arguments in the past years is introduced and introduce the application of block boundary value methods in the study of delay differential equation.There is a brief introduction to boundary value methods,and the numerical scheme of differential equations with piecewise continuous arguments is obtained by block boundary value methods.We analyzes the convergence of block boundary value methods for nonlinear differential equations with piecewise continuous arguments,and proves that the order of convergence is the same as that of the method.Further,the stability of block boundary value methods for linear test equation is discussed.The sufficient conditions for asymptotic stability of the numerical scheme are obtained.Meanwhile,we compare the asymptotic stable region of numerical solution and the asymptotic stability region of exact solutions,and it is proved that the asymptotic stability region of the numerical solution contains the asymptotic stability region of the exact solution under certain conditions.Finally,some numerical examples are given to illustrate the theoretical results.