Blended Block Boundary Value Methods Of Initial Value Problems For Delay Differential Equations

Initial value problems for delay differential equations are very useful in our life. With more and more study on the initial value problems, many new methods have been proposed. Most of the methods are derived from the methods of initial value problems for ordinary differential equations, including linear multistep method, boundary value methods, block boundary value methods and the blended block boundary value methods that Brugnano and Magherini have proposed. But the study of blended block boundary value methods that people have done was only applied on the ordinary differential equations. In this paper, I will apply the method on the delay differential equations and delay differential algebraic equations.This article first introduces the boundary value methods, block boundary value methods and the blended block boundary value methods of initial value problems for ordinary differential equations at first , from which we can see the advantages of the blended block boundary value methods. Then I put blended block boundary value methods applied to the initial value problems of delay differential equations and delay differential algebraic equations. At the same time, I do some experiments. From the experiments, we can understand the thought and advantages of the blended block boundary value methods more deeply.This paper mainly introduces two kinds of blended block boundary value methods: B₂ GAMs and HB₂ GAMs . Through the CPU running time comparison of three kinds of concrete methods solving the delay differential equation and delay differential algebraic equations, we get the results:blended block boundary value methods are mainly suitable for the large-dimension problems, and when problems dimension is bigger, the time efficiency of the two blended block boundary value methods are much higher than boundary value methods, also the time efficiency of HB₂ GAMs is higher than B₂ GAMs .