A Hp - Legendre - Gauss Spectrum Configuration Method For Independent Variable Stochastic Delay Differential Equations

This paper mainly discusses Legendre-Gauss collocation methods and hp-LegendreGauss collocation methods for delay di?erential equations with piecewise continuous arguments, and analyzes the errors of the two di?erent collocation methods. Because the mathematical model has been widely applied in many scienti?c ?elds, so researching of delay di?erential equations with piecewise continuous arguments has important theoretical signi?cance and practical value.First, the research purpose and meaning,the research status of delay di?erential equations with piecewise continuous arguments at home and abroad are introduced.Second, the Legendre-Gauss collocation method is applied to solve delay di?erential equations with piecewise continuous arguments, and the corresponding error analysis is given. At last, a new hp-Legendre-Gauss collocation method for delay di?erential equations with piecewise continuous arguments is presented, and the corresponding error analysis is also derived. The results obtained in this paper show that the convergence condition of the Legendre-Gauss collocation method only depends on delay di?erential equations with piecewise continuous arguments, and it can not be improved. However, the convergence condition of the hp-Legendre-Gauss collocation method depends not only on delay di?erential equations with piecewise continuous arguments but also on stepsize. Therefore, we can always change stepsize in order to meet the convergence condition. This showes that the hp-Legendre-Gauss collocation method is superior to the Legendre-Gauss collocation method.