Runge-Kutta Methods For Two Classes Of Nonlinear Functional Boundary Value Problems

Functional differential equations are often used to describe actual phenomena in the fields of population dynamics,power control and automation,biomedicine,image processing,and ecology.Functional differential equations with variable delays are a complex and important model of functional differential equations,which can describe practical problems more accurately than functional differential equations without delay.So scholars begin to study this class of delay problems.The existence of time delay makes it difficult to obtain exact solutions of functional differential equations with variable delays.Therefore,scholars construct a series of numerical methods for solving this class of problems and study the convergence and stability of the methods.However,as far as we know,up to now no scholars use Runge-Kutta methods to solve boundary value problems of functional differential equations with variable delays.This thesis mainly focuses on the numerical calculation and analysis of two classes of boundary value problems of functional differential equations.We construct numerical methods for solving these two classes of boundary value problems based on Runge-Kutta methods and Lagrange interpolation methods,and study the convergence and stability of the methods.The thesis is divided into five chapters.In Chapter 1,we respectively introduce the research background and current research status of regular functional differential equations and singularly perturbed functional differential equations.In Chapter 2,we use Lagrange interpolation methods and Runge-Kutta methods to construct extended Runge-Kutta methods for solving a class of boundary value problems of regular functional differential equations.Under appropriate conditions,these methods are proven to be convergent and stable.Finally we verify the results through numerical experiments.In Chapter 3,we construct a class of high-order numerical methods for solving a class of boundary value problems of singularly perturbed functional differential equations.Besides,we study the convergence and stability of the methods,and prove that the methods are convergent and stable under appropriate conditions.Finally,the theoretical results are verified by numerical experiments.In Chapter 4,we summarize the research contents of the whole paper,and look forward to the future research contents.