``Shooting method'' for the solution of singularly perturbed two-point boundary-value problems having less severe boundary layer.
This paper presents a numerical method to compute the solution of the problem \[ \begin{matrix}\epsilon u''+ a(x)u'-b(x)u= f(x), \\ a(0)u'(0)-b(0)u(0)=f(0),\,\, u(1)=B. \end{matrix}\tag{1} \] Boundary conditions are said to be ``less severe'' as there is a solution \(u_0\) of the reduced problem \[ a(x)u'-b(x)u= f(x) \] that verifies both boundary conditions in (1). Basic assumptions are: \(\epsilon >0\) is a small parameter, \(a(x)\geq \alpha>0\) and \(b(x)\geq 0\). The proposed method is to solve an inner region problem \[ \begin{matrix} \epsilon u''+ a(x)u'-b(x)u= f(x), \\ u(0)=u_0(0),\,\, u'(0)=[f(0)+b(0)u_0(0)]/a(0), \end{matrix} \] defined for \(x\in [0,k_1]\), together with an outer region problem \[ \begin{matrix}\epsilon u''+ a(x)u'-b(x)u= f(x), \\ u(k_2)=u_0(k_2),\,\, u(1)=B, \end{matrix} \] where \(k_1\leq k_2\). The approximate solution of (1) is then obtained as the juxtaposition of these inner and outer solutions. Error estimates are derived. Numerical examples are given.
scientific article; zbMATH DE number 1861279
scientific article