Waveform Relaxation Methods For Singular Perturbed Differential Equations With Delay

The singular-perturbed delay diferential equations have been widely appliedin many areas, such as fuid dynamics, optimal control, chemical reactions, popu-lation dynamics, environmental, medical, etc. Because the classical Lipschitz con-stants are of size O(1/)(is a small parameter), the classical convergence theorycannot be directly applied to the singular-perturbed delay diferential equations.We are devoted to solving singular-perturbation delay diferential equations bythe waveform relaxation method in this paper. In Chapter1, we mainly introducethe background and the current research situation of numerical methods for thesingular-perturbed delay diferential equations and the waveform relaxation meth-ods. In Chapter2, we give the iterative formulas of the continuous-time waveformrelaxation for the linear singular-perturbed delay diferential equations, and provetheir convergence. Then, we apply the s-step linear multi-step method to solvethe continuous-time iterative formulas and obtain the discrete waveform relaxationmethods for the linear singular-perturbed delay diferential equations. The conver-gence of the discrete waveform relaxation methods are proved. In Chapter3, weobtain the convergence of the continuous-time waveform relaxation methods forthe nonlinear singular-perturbed delay diferential equations. Then, we obtain thecorresponding discrete waveform relaxation methods by using the Runge-Kuttamethods, and the corresponding convergence is also proved. In Chapter4, thecorrectness of the theoretical results is verifed by numerical experiments.