A new algorithm for a class of linear nonlocal boundary value problems based on the reproducing kernel method (Q609977)

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.