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Theory and computation in singular boundary value problems. (English) Zbl 1134.65045

The problem under consideration is the singular boundary value problem

${\left(p\left(x\right){y}^{\text{'}}\right)}^{\text{'}}/p\left(x\right)-q\left(x\right)y=f\left(x\right),\phantom{\rule{1.em}{0ex}}x\in \left(0,1\right),\phantom{\rule{1.em}{0ex}}\underset{x\to {0}^{+}}{lim}p\left(x\right){y}^{\text{'}}\left(x\right)=0,\phantom{\rule{1.em}{0ex}}y\left(1\right)=0·$

The author applies and investigates to this problem two numerical methods. The first is the Galerkin method with the base system generated by the sinc function $sinc\left(x\right)=sin\left(\pi x\right)/\left(\pi x\right)$, and the second method is some variant of the well-known parametric continuation method which is developed for boundary value problems in recent works under the denotation “homotopy perturbation method”. A numerical example is given to demonstrate the computational efficiency of the two methods.

##### MSC:
 65L10 Boundary value problems for ODE (numerical methods) 65L60 Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE 34B05 Linear boundary value problems for ODE