A Hp - Legendre - Gauss Spectrum Allocation Method For Piecewise Continuous Delay Differential Equations With Independent Variables

This paper mainly discusses Legendre-Gauss collocation methods and hp-LegendreGauss collocation methods for differential equation with piecewise continuous arguments, and analyzes the errors of these two kinds of different collocation methods.Because differential equation with piecewise continuous arguments have important applied value in information technology, life science, electronic physical and many other scienti?c ?elds. Therefore, researching of differential equation with piecewise continuous arguments is of important applied value.Firstly, this paper introduces delay differential equations and differential equation with piecewise continuous arguments’ s numerical method and present situation.Secondly, this parer research numerical implementation process of Legendre-Gauss collocation method, and the corresponding error is analyzed. Thirdly, this paper introduces a new hp-Legendre-Gauss collocation method numerical implementation,and the corresponding error is analyzed. The results obtained in this paper show that the convergence condition for the usual Legendre-Gauss collocation method depends on the differential equation with piecewise continuous arguments, and it cannot be improved, however, the convergence condition for the hp-Legendre-Gauss collocation method depends both on the differential equation with piecewise continuous arguments and the stepsize, and we always can choose the stepsize to satisfy it. Therefore, the hp-Legendre-Gauss collocation method is superior to the usual Legendre-Gauss collocation method.