A new multiscale algorithm for solving second order boundary value problems

Abstract

In this study, a new multiscale algorithm was proposed to solve the boundary value problems of second order differential equations. A multiscale basis consisting of two sets of multiscale functions was constructed in the reproducing kernel space, and the proposed multiscale basis was proved to be orthonormal. The ε− approximate solution was defined, and then it was proved to be the optimal solution. In addition, the stability, convergence and complexity of this algorithm were discussed and illustrated in this study. Numerical examples verify the effectiveness and feasibility of the algorithm, and the results show that the proposed intelligent multiscale algorithm has advantages in accuracy and stability compared with other methods.