Numerical Methods Of A Class Of Nonlinear Partial Differential Equations With Many Delays

The study on delay differential equations has a rapid development during the pasttwo decades. Delay partial differential equations, as an active branch, are widely appliedin many scientific fields, such as population dynamics, epidemiology, ecology, nuclearengineering, traffic scheduling, engineering Controls and so on, which play a crucial rolein describing various phenomena in natural and social science.Scholars have got more and more theoretical researchs on delay partial differentialequations, such as stability, convergence, periodicity, numerical oscillations and othersolutions. Due to the existence of delay term, it is very difficult to get analytical solutionsfor delay differential equations. Moreover, the delay term makes theoretical analysiscomplex to some extent. So it is necessary to study the numerical solution of delaydifferential equations.For a class of nonlinear partial differential equations with single delay, Ferreira J A.gives the Euler backward difference scheme. Then Sun Zhi-zhong and Zhang Zai-binestablish Crank-Nicolson scheme and the compact difference scheme. What’s more,proofs of the numerical stability and convergence are also given. And it is applicable toapply those numerical methods to a class of multi-delay delay partial differentialequations.This paper is concerned with three numerical methods of a class of multi-delaypartial differential equations and the numerical stability and convergence. For themulti-delay differential equations, the appropriate backward Euler difference scheme,Crank-Nicolson scheme and compact difference scheme. Applying energy analysismethods, proofs of the numerical stability and convergence are easy to be obtained. Atlast the validity of the theoretical analysis is verified by several numerical experiments.