A comparative study on a singular perturbation problem with two singular boundary points (Q1294288)

A singularly perturbed second-order differential equation with two singular boundary points \[ x(b-x)[ \varepsilon f(x)u_{xx}+g(x)u_x ] =su, \quad u=u(x,\varepsilon) \quad 0< x < b, \tag{1} \] is investigated. Here, \(f\) and \(g\) are assumed to be real, positive, and several times continuously differentiable in \(x\). The author compares two distinct methods, usually used in singular perturbation problems. First, a leading-order asymptotic solution to (1) is obtained by using the WKB method in the interior region between the boundary points and using a boundary layer analysis near the singular boundary points \(x=0\) and \(x=b\). In the second approach, the uniform reduction method by \textit{A.-M. Wazwaz} [IMA J. Appl. Math. 49, No. 3, 231-244 (1992; Zbl 0780.34043)] is used. In this way, a two-part uniform approximation is obtained. The matching concept is applied to each type of analysis where dominant and recessive terms are matched independently.