摘要
本文考虑如下积分微分方程边值问题: εx″=f(t,x,T_εx,ε)x′+g(t,x,T_εx,ε), x(0,ε)=A(ε),x(1,ε)=B(ε),其中ε>0是小参数,〔T_εx〕(t,ε)=φ(t,ε)+integral from n=0 to t (K(t, s)x(s,ε)ds),K(t,s)≥0是〔0,1〕×〔0,1〕上的连续函数,φ(t,ε)是〔0,1〕×〔0,ε_0〕上关于ε的无穷次连续可微函数。在适当的假设下,利用复合展开法和微分不等式技巧,我们获得所述问题的解的存在性和高阶渐近估计。
Abstract This paper concerns with the following boundary value problems of the integral-differential equations: εx″. = f(t, x, T_εx, ε)x′ + g (t, x, T_εx, ε), x(0, ε) = A(ε), x(1, ε)=B(ε), where ε>0 is a small parameter, and [T_εx](t, ε)=ψ(t, ε) + integral from n=0 to t K(t, s)x(s, ε)ds, K(t, s)≥0 is continuous on, [0, 1] x ψ(t, ε) is arbitrary order continuous differentiable with respect to ε on [0, 1]×[0, ε_0]. Under the pro- per assumptions, using the method of composite expand and the differential inequalities theory, we obtain the existence and a higher order approximation of the solution.
基金
国家自然科学基金
关键词
积分微分方程
奇摄动
高阶近似
Singular perturbation
Integral differential equations
Differential inequalities
Higher order approximation
