The Numerical Solution Of BVPs For A Class Of Functional Differential Equations

Boundary value problems (BVPs) of functional differential equations(FDEs) arise frequently in application of optimal control theory and circuit theory. Compared with initial value problems (IVPs) of FDEs, it is very difficult to solute this kind of problems in theory. Accordingly, to construct its numerical algorithm and the corresponding theoretical analysis is particularly important. Collocation methods and some difference methods have been used to solve BVPs of FDEs. However, theoretical analysis is not quite consistent. This article aims to construct several types of high order difference method and its uniform convergence analysis.It is well known that the theory about BVPs of ordinary differential equations(ODEs) and numerical solutions are closely related to the corresponding theory and numerical solutions of initial value problems (IVPs). Keller has proved the a difference scheme are stable and consistent for the BVPs of first order ODEs with unique solutions if and only if the corresponding scheme are stable and consistent ,which is equivalent to the convergence for the corresponding IVPs. So, is there any similar inherent relationships between IVPs and BVPs for FDEs? This paper will give a positive answer for linear problems.In this paper, we deal with the adaptation of Runge-Kutta methods and block BVMs to numerical solution for BVPs for a class of linear FDEs and give analysis of convergence. For BVPs of FDEs with unique solution, we prove that a difference scheme is p-order convergent for the BVP of FDEs if and only if it is p-order convergent for the corresponding IVP of FDEs. Then the convergence of Runge-Kutta methods and block BVMs for BVPs of FDEs is easily obtained due to the exiting theory on those numerical methods for IVPs of FDEs. Finally, some numerical examples are presented to illustrate our theoretical analysis.