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Solving singular nonlinear two-point boundary value problems in the reproducing kernel space. (English) Zbl 1154.34012

The boundary value problem

$H\left(u\left(x\right)\right){u}^{\text{'}\text{'}}\left(x\right)+\frac{1}{p\left(x\right)}{u}^{\text{'}}\left(x\right)+\frac{1}{q\left(x\right)}N\left(u\left(x\right)\right)=f\left(x\right),\phantom{\rule{0.277778em}{0ex}}t\in \left(0,1\right),$
$u\left(0\right)=0,\phantom{\rule{0.277778em}{0ex}}u\left(1\right)=0$

is considered for $H\left(u\right)$ and $N\left(u\right)$ continuous, $p,q\in C\left[0,1\right]$ vanish at ${\left\{{x}_{i}\right\}}_{i=1}^{m}\subset \left[0,1\right]$ and $f\in {W}_{2}^{1}\left[0,1\right]=\left\{u\left(x\right):u$ is absolutely continuous real valued function, $u,{u}^{\text{'}}\in {L}^{2}\left[0,1\right]\right\}$. The unique exact solution $u\in {W}_{2}^{3}\left[0,1\right]=\left\{u\left(x\right):{u}^{\left(i\right)},i=\overline{0,2},$ are absolutely continuous real valued functions, ${u}^{\left(i\right)}\in {L}^{2}\left[0,1\right],i=\overline{0,3},u\left(0\right)=u\left(1\right)=0\right\}$ is represented in the form of series in the reproducing kernel space ${W}_{2}^{3}\left[0,1\right]$. Some numerical examples demonstrate the present method.

##### MSC:
 34B16 Singular nonlinear boundary value problems for ODE 46E22 Hilbert spaces with reproducing kernels 47B32 Operators in reproducing-kernel Hilbert spaces