Nemerical Stability Of Boundary Value Methods For Pantograph Differential Equations

Compared to ordinary differential equation, the delay differential equation isone of the more accurate mathematical models that can describe real world model.In traditional research algorithm, more scholars lay particular stress on the researchinto the existence and uniqueness of analytic solutions, but the progress is very slow.And in engineering calculation, with the gradually perfect of electronic computertechnology and the increasing demand, computational mathematics as a numericalmethod that can be solved to achieve a certain precision numerical solutiongradually develops. Especially in recent years, computational mathematics extendsout in many areas. On one hand the development of the results complement eachother with other subjects to form cross-disciplinary subjects and on the other handthey become some emerging sciences which are full of practical significance andpromote the mutual advancement. Scholars from various fields are increasinglyusing mathematical tool to deal with practical problems and produce dependence,which has produced more and more mathematical models demanding promptsolution. Pantograph differential equations is described as a class of infinite delaydifferential system as an important branch of delay differential equations. And itsstability analysis of analytical solution and numerical solution has gained widelyattention in physics, control science and engineering, precision instruments andother branches of science and engineering fields.Boundary Value Method(BVMs) and Block Boundary Value Method(B2VMs)are a relatively new class of numerical method to solve initial value problems of adifferential system, and can also be understood as a promotion of linear mult istepmethod. In addition, they overcome the barrier that the basic method of highcompatibility and stability can not coexist. What’s more, B2VMs takes two dividedin time, which ensures that the step size of B2VMs can be changed, effectivelyensures B2VMs of numerical precision and meanwhile allows B2VMs to use aparallel algorithm.The article is specific to the study of two types of widely used pantographdifferential equations. First, it discusses the stability problem when we use severalcommon BVMs to deal with nonautonomous pantograph differential equations andobtains corresponding conclusion. Second, it constructs a B2VMs for two-dimensional pantograph differential equation and the sufficient and necessarycondition of numerical stability is analyzed. Finally, numerical examples are givento verify the conclusion.