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Constructive approximation of solution for fourth-order nonlinear boundary value problems. (English) Zbl 1170.34015

The paper deals with the fourth-order nonlinear boundary value problem
\[ u^{(4)}(x)=f(x,u(x),u'(x),u''(x), u'''(x)),\quad 0<x<1, \tag{1} \]
\[ u(0)=0,\;u'(1)=0,\;u''(0)=0,\;u'''(1)=0. \tag{2} \]
The authors obtain sequences of approximate solutions uniformly converging to an exact solution \(u\) of problem (1), (2). The approximate solutions \(u_n\) are found in the reproducing kernel space \(W_2^5[0,1]\), which is defined as \[ W_2^5[0,1]=\{u\in AC^4[0,1],\;u^{(5)}\in L^2[0,1],\;u \;\text{fulfils}\;(2)\}, \] with the inner product \[ <u(x),v(x)>_{W_2^5}\, =\, \sum_{i=0}^4u^{(i)}(0)v^{(i)}(0)+\int_0^1u^{(5)}(x)v^{(5)}(x) dx. \] Two equivalent approximate sequences \(\{u_n\}\) are developed and the convergence \(\lim_{n\to \infty} \|u_n-u\|_{C^4[0,1]}\to 0\) is proved. The error estimate is given.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
34A45 Theoretical approximation of solutions to ordinary differential equations
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References:

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