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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 $$H(u(x))u''(x)+{1\over p(x)}u'(x)+{1 \over q(x)}N(u(x))=f(x),\;t\in(0,1),$$ $$u(0)=0,\;u(1)=0$$ is considered for $H(u)$ and $N(u)$ continuous, $p,q\in C[0,1]$ vanish at $\{x_i\}_{i=1}^m\subset[0,1]$ and $f\in W_2^1[0,1]=\{u(x): u$ is absolutely continuous real valued function, $u,u'\in L^2[0,1]\}$. The unique exact solution $u\in W_2^3[0,1]=\{u(x):u^{(i)}, i=\overline{0,2},$ are absolutely continuous real valued functions, $u^{(i)}\in L^2[0,1], i=\overline{0,3}, u(0)=u(1)=0\}$ is represented in the form of series in the reproducing kernel space $W_2^3[0,1]$. Some numerical examples demonstrate the present method.

34B16Singular nonlinear boundary value problems for ODE
46E22Hilbert spaces with reproducing kernels
47B32Operators in reproducing-kernel Hilbert spaces
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