含有δ函数的弱非线性微分方程的摄动解
Perturbation Solution of the Weak-Nonlinear Differential Equation with δ-Function
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摘要: 本文从Heaviside函数和δ函数的基本性质出发,利用奇异摄动法,提出了一个方法来求方程M(u)=ef(u)+λδ(t-a)的渐近解析解,这里M是n阶线性微分算子,f(u)是多项式,利用这个方法讨论了一些具体例子,得到的结果有很满意的物理解释.Abstract: In this paper,starting from some fundamental properties of Heaviside function and δ-function,making use of singular perturbation methods we provide a method of finding the asymptotic analytic solution of equationwhere M(u)=eƒ+λ#948;(t-a) where M is an n-order linear differential operator,ƒ(u) is a polynomial. By means of this method,we discuss some examples concretely. The results can be explained satisfactorily in physics. If we deal with linear problem by this method,the result will agree with that drawn from theorem of impulse.
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[1] Chia-shun Hsu,Advances in Appl.Mech.,17,(1977),245-301. [2] Chia-shun Hsu,J.Appl.Mech.,39,(1972),551-559. [3] Nayfeh,A.H.and Deam T.Mook,Nonlinear Oscillation,Wiley-Interscience,(1979). [4] Nayfeh,A.H.,Perturbation Methods,Wiley-Interscience,(1973). [5] Pan,H.H.and R.M.Hohenstein,Quart.Appl.Math.,39,(1981),131-137. [6] Kevorkian,J.and J.D.Cole.Perturbation Methods in Applied Mathematics,Springer-Verlag,New York,Inc.Published,(1981).
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