Exponentially Fitted Runge-Kutta Methods For Singularly Perturbed Delay Differential Equations

A singularly perturbed delay di?erential equation is an ordinary di?erential equation in whichthe highest derivative is multiplied by a small parameter ε and involving at least one delay term.Singularly perturbed delay di?erential equations arise in the study of an “optically bistable device”and in a variety of models for physiological processes or diseases. In general, equation of thistype is too complex for us to seek an analytic solution, and for quantitative results one resorts toreliable numerical techniques. Typical for these problems is the presence of fast and slow phasesand, numerically speaking, such problems can be regarded as a subclass of “sti?” systems and cancause a number of numerical di?culties.The main purpose of this thesis is to construct an e?cient numerical scheme of the initialvalue problem for the delayed recruitment equation. In Chapter 1, we briefly recall some basicproperties of the delayed recruitment equation and provide a gentle introduction to its existednumerical methods.In Chapter 2, we will propose a new family of numerical schemes of exponential fittingtype, which is known as exponentially fitted Runge-Kutta methods, for the delayed recruitmentequation. Convergence of the numerical methods are studied. Some numerical examples arepresented to illustrate the performance of the proposed methods.In Chapter 3, we study the absolute stability of exponentially fitted Runge-Kutta methods forthe linear delayed recruitment equation and obtain a su?cient condition for the numerical solutionto be asymptotically stable. Several numerical experiments are presented to confirm our theoreticresult.