Piecewise Continuous Configuration Pantograph Differential Equations

This paper mainly discusses collocation methods of the pantograph delay dif-ferential equation with piecewise continuous arguments. As an important mathe-matical models, this kind of equations often arises in physics, biology systems andcontrol theory. Therefore, researching this kind of equations is of great significanceand practice.Firstly, the domestic and overseas development states of pantograph delay dif-ferential equations and delay diferential equations with piecewise continuous argu-ments are introduced, and the diferences between pantograph diferential equationsand pantograph diferential equations with piecewise continuous arguments are com-pared.Secondly, some basic definitions of collocation methods are given. According toLagrange interpolation formula, the general form of the pantograph delay diferentialequations with piecewise continuous arguments is obtained and the existence anduniqueness of the collocation solution are proved.Then, the global convergence of the collocation solution for the pantographdelay diferential equations with piecewise continuous arguments is studied for marbitrary collocation parameters.In addition, when m collocation parameters are subject to some orthogonalityconditions, the global supconvergence of the collocation solution for the pantographdelay diferential equations with piecewise continuous arguments is analyzed.At last, some corresponding numerical experiments are given to verify the cor-rectness of the conclusions obtained in this paper.