Reproducing kernel representation of the solution of second order linear three‐point boundary value problem

Abstract

This paper investigates the analytical solutions and numerical solutions of second order linear boundary value problems with three-point boundary conditions by means of the weak pre-orthogonal adaptive Fourier decomposition (W-POAFD) method introduced by Qian et al. Using the reproducing kernel functions of the Sobolev spaces on the unit interval [0, 1], we construct a dictionary to implement the W-POAFD. Based on orthonormalizataion, we can select the crucial parameters successively by adopting the weak maximal selection strategy of the W-POAFD method. Then the numerical solutions have "sparse" representations, which converge to the analytic solution with fast convergence rate. Some numerical experiments demonstrate the high accuracy and efficiency of the W-POAFD method in this three-point boundary value problem.