The Analysis Of Stability For Rosenbrock Methods Of Two Types Of Delay Differential Equaions

Differential equations are widely applied in various areas of science. Recently, delay differential equations play an important role in mathematical models. Furthermore, the study about the stability for delay differential equations has rapidly developed. As two special cases, the constant delay differential equations and the pantograph delay differential equations are also paid special attention by the academic experts. This paper mainly studies on the problem of the stability for Rosenbrock methods of two types of delay differential equations, i.e., the constant delay differential equations and the pantograph delay differential equations.In the chapter one of this paper, we mainly introduce the background and the significance of the delay differential equations and the corresponding numerical analysis. And we review the research history about the stability of delay differential equations and the numerical methods. In particular, the studies of Rosenbrock methods for these kinds of equations are summarized.In the chapter two, we introduce the creation process of Rosenbrock methods in detail. And the formula about the Rosenbrock methods for solving two-dimensional constant delay differential equations is listed. Furthermore, we study its GP ?stability. And, some numerical examples are given out to demonstrate the conclusions.In the chapter three, we consider the two-dimensional pantograph equations. We get the formula about Rosenbrock methods for solving two-dimensional pantograph equations with variable step size, and we discuss the H ? stability of numerical solutions. Moreover, we confirm the conclusions with numerical examples.Finally, we summarize this paper.