Pouzet-Runge-Kutta Methods For Singularly Perturbed Delay-Integro-Differential Equadtions

Delay differential equations(DDEs) often appear in auto-control, biology, physics,aerospace and economics.Singular perturbation delay problem(DSPPs) and delay dif-ferential algebraic problem are a special subproblem of delay differential equation.Fordiscrete type singularly perturbed delay problem, Gun SiQing had studied error behaviorabout Runge-Kutta methods and linear multi-step methods.Tian HongJiong had proved thesingularly perturbed delay problem was exponential stability,and gave out the expansionof true solution.Up to now,both abroad and home have no results about numerical solu-tion of singular perturbation delay problem.Moreover for singularly perturbed problem,because the solution near the origin is decay very fast, so studying the convergence ofnumerical solution is very important.Therefor,In my paper we study the convergence ofPouzet-Runge-Kutta methods for singular perturbation delay integro-differential problem.At the first chapter,we review the history of numerical solution for delay differentialequation.At the second chapter,we discuss theε-expansion of single variable and singlestiffness singular perturbation delay differential-integral problem.At the third chapter,westudy the multi-stiffness singular perturbation delay differential-integral problem,and getthe global error of Pouzet-Runge-Kutta method for that problem.In the end,numericalexperiments can confirm the results.At the forth chapter,we study the two variables andsingle stiffness singular perturbation delay differential-integral problem,and get the globalerror of Pouzet-Runge-Kutta method for that problem.