Numerical solution of a fourth‐order singularly perturbed boundary value problem with discontinuities via Haar wavelets

Abstract

In this paper, an efficient wavelet-based numerical scheme is presented for the solution of fourth-order singularly perturbed boundary value problems with discontinuous data. The fourth-order derivative of the solution function is expanded in Haar series and then integrated to get the approximations for the lower order derivatives of the solution function. The convergence of the proposed method is analyzed. The analysis shows that the error bound is inversely proportional to the wavelet resolution. The method is applied on three test problems to check its performance. The proposed method is computationally fast, cheap, and reliable, even considering low-order wavelet resolutions. Another important advantage of the proposed method is that it can be easily extended to similar classes of problems with a variety of boundary conditions.