Numerical Stability Of Linear Multistep Multiderivative Methods For Delay Differential Equations

The pantograph differential equations arise widely in scientific fields such asbiology, ecology, medicine and physics. These classes of equations play animportant role in modeling diverse problems of engineering and natural science. Asthe deep understanding of delay differential equations and the complexity ofproblems, researchers focus their attention on pantograph equations and delayintegro-differential equations, which are widely used. Now, many problems can bemodeled as pantograph equations and delay integro-differential equation. Because itis difficult to gain the analytic solutions, the researchers deal with numericalanalysis and numerical computation of these two kinds of equations. It is importantthat whether the numerical methods can inherit the stability of the analytic solutionsof the equations.Higher derivative method is a kind of traditional numerical method, which iswidely used in ordinary differential equations. The SDAM second derivative methodis a method of higher order derivatives. This article applies it in the solution ofpantograph differential equation and the delay integro-differential equation,provides a new method for solving the pantograph differential equation and thedelay integro-differential equation.This article first introduces the backgrounds of the pantograph differentialequations and the delay integro-differential equations and their status of research.Then, it studies the numerical stability of the SDAM second derivative methodwhich is used for solving them. On the one hand, it studies the numerical stabilitywhen the method is used for pantograph differential equation，that is use the SDAMsecond derivative method D to the pantograph differential equation witch is aftertransformation and draws numeric format, carries on an analysis to the coefficientmatrix of the numeric format and draws the corresponding conclusion. On the otherhand, use the SDAM second derivative method to solve delay integro-differentialequation and draws numeric format, through the discussion that the distribution ofroots of the characteristic equation of the numeric format, then draws the necessaryand sufficient conditions for asymptotic stability. Finally, some numerical examplesare presented to verify the validity of the conclusions.