Stability Analysis Of Exponential Runge-Kutta Methods For Pantograph Equations

Delay differential equations (DDEs) as an important mathematical model have been widely used in many research fields. With the deepening of the study, pantograph delay differential equations have gradually become the focus of research in the field,This paper studies numerical stability for exponential R-K methods for the two forms of pantograph delay differential equations, its content is organized as follows: it is usually difficult to obtain the analytic expression of solutions of the systems, and therefore, to study the behavior of numerical solutions becomes particularly important.Firstly, we review the development history and meaning of delay differential equations. Further we present applications and relative theories on pantograph delay differential equations and exponential R-K methods.Secondly, we investigate the linear pantograph delay differential equations based on exponential R-K methods. The paper gives the definition of asymptotic stability and stability region. And the exponential R-K methods which use the variable step are given out, and the asymptotic stability of the numerical methods for linear pantograph delay differential equations is proved.Thirdly, we are concerned with the semi-linear pantograph delay differential equations based on exponential R-K methods. We define the concept of EB stability, exponential algebraic stability and global stability. Finally, we prove that if an exponential R-K method is exponential algebraic stable, it is EB stable. And when an exponential R-K method satisfies certain conditions, if it is exponential algebraic stable, it is global stable.In addition, the corresponding numerical examples are made to verify the correctness of the theoretical conclusions.