具有双参数的弱非线性方程的奇摄动解

Singularly Perturbed Soluton for Weakly Nonlinear Equations With Two Parameters

讨论了一四阶具有双参数的弱非线性方程在有限区间上的奇摄动边值问题.在一定的假设下,首先,利用幂级数形式展开方法,构造了原问题的外部解.其次,利用伸长变量,在左端点附近构造问题解的第一边界层校正项.然后,利用更强的伸长变量,仍然在左端点附近构造问题解的第二边界层校正项.第二边界层的厚度比第一边界层的厚度更小,形成在左端点附近的边界层的套层.最后利用微分不等式理论,证明了边值问题解的存在性、和在整个区间内一致有效性和渐近性态,得到了满意的结果.

A class of singularly perturbed boundary value problem of weakly nonlinear equation for fourth order on the finite interval with two parameters is considered. Under suitable conditions, firstly, using the expansion method of power series, the reduced solution and formal outer solution are constructed. Secondly, using the transformation of stretched variable, the first boundary layer corrective term near the left endpoint is constructed which possesses exponential attenuation behavior. And then, using the stronger transformation of stretched variable, the second boundary layer corrective term near the left endpoint is constructed too, which also possesses exponential attenuation behavior. The thickness of second boundary layer smaller than first boundary layer and forms the cover layer near the left endpoint. Finally, using the theory of differential inequalities the existence, tmiform validity in the whole interval and asymptotic behavior of solution for the original boundary value problem are proved. The satisfying results are obtained.