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

This paper mainly discusses two kinds of different collocation methods of the pantograph differential equation with piecewise continuous arguments,and their convergence is analyzed respectively. As an important mathematical models, this kind of equations often arises in control theory, physics, biology systems etc. There-fore, researching this kind of equations is of great significance and practice.The history of pantograph differential equations and differential equations with piecewise continuous arguments is introduced, and development status is reviewed. The Legendre-Gauss-Radau method is applied to solve pantograph differential equa-tions with piecewise continuous arguments, and the corresponding convergence is an-alyzed. A new hp-Lengendre-Gauss-Radau is used to solve pantograph differential equations with piecewise continuous arguments, and the corresponding convergence is also analyzed.By comparison, we know that the convergence condition of the hp-Legendre-Gauss-Radau method not only depends on the pantograph differential equation with piecewise continuous arguments, but also on stepsize. Therefore, we can always choose the stepsize to satisfy the convergence condition. However, the convergence condition of the Legendre-Gauss-Radau method only depends on the differential equation itself. Hence, the hp-Legendre-Gauss-Rada method is better than the Legendre-Gauss-Radau method.