An enhanced wavelet-based numerical scheme with higher convergence order for a class of high-order boundary value problems

Abstract

In this paper, a new numerical scheme is proposed to effectively solve a class of high-order boundary value problems (BVPs) with arbitrary orders and three types of boundary conditions. The method can reduce computational cost. By constructing an enhanced wavelet basis based on Legendre polynomials within a defined reproducing kernel space $W_m[0,1]$, the ε-approximate solution of BVPs can be obtained applying the least squares method. These enhanced wavelets maintain compact support, ensuring superior approximation performance compared to classical wavelets. The convergence of the proposed scheme is characterized by its analytical order, while stability and complexity are also investigated. Specifically, for high-order nonlinear BVPs, the Quasi-Newton method is utilized effectively. We present numerical examples of several high-order linear and nonlinear BVPs with various boundary conditions, including Dirichlet, Neumann, and Robin conditions, to assess the stability and efficiency of the method. The numerical results show that our approach provides more accurate approximations, which are consistent with analytical solutions and show a high order of accuracy, outperforming several existing methods in the literature.