Piecewise Continuous Delay Differential Equations, Numerical Stability

This paper discusses the convergence and the numerical stability of the de-lay differential equations with piecewise continuous arguments. This kind of equa-tions has extremely extensive application background. In some fields such as con-trol theory,biomedical sciences and physics,many problems can be described byit.Therefore,It is important to study this kind of equations.A typical delay differential equations with piecewise continuous arguments con-tains arguments that are constants on certain intervals. A solution is defined as acontinuous,sectionally smooth function that satisfies the equation within these in-tervals . Continuity of a solution at a point joining any two consecutive intervalsleads to some recursion relation for the solutions at such points.Hence,the solutionsof this kind of equations are determined by a finite set of initial data rather than byan initial function as in the case of general functional differential equations.The thesis investigates the convergence and stability of the numerical solutionof the Euler-Maclaurin method for the advanced delay differential equations withpiecewise continuous arguments.It is proved that the convergence order of the ad-vanced delay differential equations with piecewise continuous arguments is of 2n+2for the nth-Euler-Maclaurin method , the conditions that the stability region ofthe numerical solutions contains the stability region of the analytic solutions areobtained.The convergence of the numerical solutions of the retarded delay differentialequations with piecewise continuous arguments is discussed in this paper.It is provedthat the stability region of the numerical method contains the stability region of theanalytic solutions if and only if n is odd.The thesis analyses the numerical stability of the alternatively advanced andretarded equations with piecewise continuous arguments with the term [t + 21] .Thesuffcient and necessary conditions that the numerical solutions preserve the asymp-totic stability of the analytic solutions are given,which is completely different to theadvanced type.The correctness of the conclusions given in this thesis are demonstrated bynumerical examples.