Numerical Stability Of Differential Equation With Piecewise Continuous Arguments

This paper is concerned with the analytical solution and numerical stability analysis of the Runge-Kutta methods for two kinds of delay differential equation of advanced type delay differential equations of both advanced type and retarded type.The numerical solutions of the three equations and the stability of the analytical solutions are also discussed.The first chapter of this paper mainly describes the development history and research status of delay differential equations.In the second chapter,we study delay differential equation of advanced type:u'?t?=au/?t?+a₁u?[t+3]?and obtain the asymptotically stable conditions of numerical solution.We make further use of the Order stars and?r,s?-Pade approximation theorems,the conditions that the analytic stability region is contained in the numerical stability region are obtained when the stability function is given by the?r,s?-Pade approximation to the exponential ex.At last some numerical experiments are given.In the third chapter,we study delay differential equations of advanced type:u'?t?=au?t?+a₀u?[t]?+a₁u?[t+1]?+a₂u?[t+2]?+a₃u?[t+3]?,the stability conditions of the analytical solution and the numerical solution of Runge-Kutta method are proved.We make further use of the Order stars and?r,s?-Pade approximation theorems,the conditions that the analytic stability region is contained in the numerical stability region are obtained when the stability function is given by the?r,s?-Pade approximation to the exponential ex.At last some numerical experiments are given.In the fourth chapter,we study delay differential equations of both advanced type and retarded type u'?t?=au?t?+a₀u?[t+1/2]?+a₁u?[t+1]?and obtain the asymptotically stable conditions of numerical solution.It is proved that the equation has a unique solution on(0,+?].The stability conditions of the analytical solution and the numerical solution of Runge-Kutta method are proved.We make further use of the Order stars and?r,s?-Pade approximation theorems,the conditions that the analytic stability region is contained in the numerical stability region are obtained when the stability function is given by the?r,s?-Padé approximation to the exponential e_x.At last some numerical experiments are given.