Continuous Block Implicit Hybrid One-sdtep Methods For Ordinary And Delary Differential Equations

Ordinary differential equations (ODEs) are used frequently for describing and modeling mathematically many real-life phenomena in physics, engineering, biology, medicine and economics, etc. There are many types of methods for solving numerically the initial value problem for ordinary differential equations, such as Runge-Kutta methods, linear multistep methods and block methods. The block methods can be regarded as a set of linear multistep methods simultaneously applied to ODEs and then combined to yield an approximation with better accuracy, better stability, and better e±ciency in parallel computing. Continuous numerical methods have many applications in the numerical solution of dis-continuous ordinary differential equations, delay differential equations, neutral differential equations and integro-differential equations. In the past decades, some kinds of continuous extensions of Runge-Kutta methods, linear multistep methods and one-blockθ-methods have been constructed.In this thesis, we present a continuous extension for the discrete approximate solution of ODEs generated by the block implicit hybrid methods. We construct a class of continuous block implicit hybrid methods to obtain the numerical dense output of ODEs. Existence and uniqueness for the continuous methods are studied. It is shown that the continuous block implicit hybrid method of order 2k + 2 exists, and these methods are Aω-stable for k = 1,2,L ,5. We then reconstruct the continuous extension of the block implicit hybrid methods to solve delay differential equations and show that these methods are convergent of order 2k + 2. Several numerical experiments are conducted to verify our theoretic results.