×

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.

MSC:

34B16 Singular nonlinear boundary value problems for ordinary differential equations
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
PDF BibTeX XML Cite
Full Text: DOI