A class of boundary value problems for third-order differential equation with a turning point
Consider a boundary value problem for a class of singularly perturbed third-order differential equations with a turning point, of the form \[ \begin{aligned} &L_{\varepsilon}y=\varepsilon \frac{d^3y}{dx^3}+f(x)\frac{d^2y}{dx^2}+g(x)\frac{dy}{dx}=h(x), \qquad x\in(-1,1),\\ &y(-1)=A_1, \quad y'(-1)=A_2, \quad y(1)=B,\end{aligned}\tag{1} \] where \(\varepsilon\) is a small positive parameter, \(A_1, A_2, B\) are constants and \(f(0)=0\). \(x=0\) is a turning point for equation (1). The following assumptions are made: (I) \(f, g, h\) are sufficiently smooth functions on \((-1,1)\); (II) \(f'(x)>0\) and \(\int_{-\delta}^{\delta}\frac{1}{|f(x)|}\,dx \leq M\), where \(\delta\) and \(M\) are positive constants. Using the method of multiple scales and others, the uniformly valid asymptotic expansion of a solution for the boundary value problem is constructed.
scientific article; zbMATH DE number 5818473
scientific article