Wavelet optimized finite difference method using interpolating wavelets for self-adjoint singularly perturbed problems

Abstract

We design a wavelet optimized finite difference (WOFD) scheme for solving self-adjoint singularly perturbed boundary value problems. The method is based on an interpolating wavelet transform using polynomial interpolation on dyadic grids. Small dissipation of the solution is captured significantly using an adaptive grid. The adaptive feature is performed automatically by thresholding the wavelet coefficients. Numerical examples have been solved and compared with non-standard finite difference schemes in [J.M.S. Lubuma, K.C. Patidar, Uniformly convergent non-standard finite difference methods for self-adjoint singular perturbation problems, J. Comput. Appl. Math. 191 (2006) 228–238]. The proposed method outperforms the non-standard finite difference for studying singular perturbation problems for small dissipations (very small ϵ ) and effective grid generation. Therefore, the proposed method is better for studying the more challenging cases of singularly perturbed problems.