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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 e− 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.

This paper is concerned with a p -energy functional with nonvanishing Dirichlet boundary condition. The authors prove the W 1,p loc convergence of the radial minimizer, and discuss the location of the zeros of this minimizer. In addition, an estimate of the convergence rate of the minimizer is given by means of the iterative approach. Mathematics subject classification (2000): 35J70, 49K20.

The projection algorithms for solving the constrained multiple-sets split feasibility problem are presented. The strong convergence results of the algorithms are given under some mild conditions. Especially, the minimum norm solution of the constrained multiple-sets split feasibility problem can be found.