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A new algorithm for a class of linear nonlocal boundary value problems based on the reproducing kernel method. (English) Zbl 1215.34014

The authors study a numerical algorithm for solving fourth-order multi-point boundary value problems
\[ \begin{aligned} &u^{(4)}+\sum_{i=0}^{3}a_i(x)u^{(i)}(x)=f(x),\quad 0\leq x\leq 1,\\ &u(\xi_{1})=b_1,\quad u'(\xi_{1})=b_2,\quad u''(\xi_{1})=b_3,\\ &u(\xi_{2})-u(\xi_{3})=b_4, \end{aligned} \]
where \(a_i\in C[a,b]\), \(0<\xi_{1}<\xi_{2}<\xi_{3}<1\) and \(b_i(i=1,2,3,4)\) are real numbers. They present an algorithm for solving the above problems based on the reproducing kernel method. The characteristic feature of this method is that a global approximation can be established on the whole solution domain and the convergence is uniform.

MSC:

34A45 Theoretical approximation of solutions to ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint 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)
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References:

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