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Numerical Analysis of Impulsive Differential Equations and Differential Equations with Piecewise Continuous Arguments

Author: LiuXing
Tutor: LiuMingZhu
School: Harbin Institute of Technology
Course: Basic mathematics
Keywords: Runge-Kutta methods linear multistep methods stability delay differentialequations impulsive differential equations
CLC: O241.8
Type: PhD thesis
Year: 2012
Downloads: 56
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Abstract


In this thesis, the properties of exact and numerical solutions to delay differentialequations and impulsive differential equations are investigated. Differential equations arean important tool in describing system evolution, which play an important role in manyapplications, such as control theory, biology and physics and so on. However, it is alwaysimpossible to obtain the explicit exact solutions of differential equations. Therefore, ingeneral, a numerical method is useful for some approximations. The numerical con-vergence indicates the accuracy to the exact solution at each grid point. The numericalstability shows the insensitivity to perturbations. Hence it is extremely worth to studythese two properties of numerical methods.In this thesis, we present the application background of delay differential equationsand impulsive differential equations, besides the history and development of the corre-sponding numerical solutions.For the classical linear multistep methods may not preserve their original order oreven not converge when applied to differential equations with piecewise continuous ar-guments, an improved linear multistep method for ordinary differential equations is con-structed and the convergence order for differential equations with piecewise continuousarguments is studied.For semi-linear differential equations with piecewise continuous arguments, theasymptotic stability of the exact solution is investigated and the exponential-methodsare introduced and the numerical asymptotic stability is investigated. As a sequence re-sult, the numerical scheme of exponential Runge-Kutta methods for semi-linear differen-tial equations with piecewise continuous arguments is constructed and the correspondingnumerical stability is studied.Based on the stability of the exact solutions to nonlinear differential equations withpiecewise continuous arguments, a sufficient and necessary condition for Runge-Kuttamethods is given for the numerical asymptotic stability of scalar nonlinear differentialequations with piecewise continuous arguments. For general nonlinear differential equa-tions with piecewise continuous arguments, a sufficient condition is proposed by a newstrategy. For linear impulsive differential equations, numerical experiments demonstrate thatclassical linear multistep methods may not preserve their original convergence order. An-other improved strategy is proposed and the corresponding convergence and stability arestudied.For nonlinear impulsive differential equations, the exponentially asymptotic stabil-ity of the exact solution is studied through transforming into differential equations withoutimpulsive. Numerical schemes of Runge-Kutta methods for impulsive differential equa-tions are built based on this transformation and the convergence together with the stabilityis investigated.

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CLC: > Mathematical sciences and chemical > Mathematics > Computational Mathematics > Numerical Analysis > The numerical solution of differential equations, integral equations
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