非线性向量边值问题的奇异摄动
Singular Perturbation of Nonlinear Vector Boundary Value Problem
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摘要: 本文研究摄动边值问题dx/dt=f(x,y,t;ε),εdy/dt=g(x,y,t;ε),a1(ε)x(0,ε)+a2(ε)y(0,ε)=a(ε)b1(ε)x(1,ε)+εb2(ε)y(1,ε)=β(ε)这里x,f,β∈Em,y,g,a∈En,0<ε《1,a1(ε),a2(ε),b1(ε),b2(ε)为适当阶数的矩阵.在gy(t)是非奇异矩阵及其它的适当限制下,证明了解的存在唯一性,作出了解的n阶渐近近似式,并得出余项估计.Abstract: In this paper we study the perturbed boundary value problem of the form in which x,f β∈Em and a1(ε),a2(ε),b1(ε) and b2(ε) are matrices of the appropriate size. Under the condition that gy(t) is nonsingular and other suitable restrictions, the existence of the solution is proved, the asymptotic expansion of solution of order n is constructed, and the remainder term is estimated.
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[1] O'Malley,R.E.,Singular perturbation of a boundary value problem for a system of nonlinear differential equations,J.Differential Equations,8(1970),431-447. [2] 江福汝,关于常微分方程非线性边值问题的奇异摄动,中国科学(A辑),3 (1985),232-242. [3] Hoppensteadt,F.,Properties of solutions of ordinary differential equations with small parameters,Comm.Pure.Appl.Math.,26(1971),807-840. [4] Chang,K.W.and W.A.Coppel,Singular perturbations of initial value problem over a finite interval,Arch.Rat.Mech.Anal.32(1969),268-280. -
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