Haar wavelets for the numerical study of a singularly perturbed reaction-diffusion problem with discontinuities

Abstract

In this article, we presented a numerical approach based on non-uniform Haar wavelets to approximate the solution of a second-order singularly perturbed problems with discontinuous data. The solutions to such problems have strong interior layer due to the discontinuity. Accordingly, we have utilized a special type of piecewise uniform mesh called a Shishkin mesh to resolve the layer behaviour of the solution. As the Haar functions are discontinous, the approximate solution is obtained with the integration approach. The second-order derivative is approximated by the linear combinations of Haar functions and then integrated to obtain the numerical approximations. The convergence analysis of the numerical method proposed is carried out, showing that the proposed method is of order two. The adaptability of the proposed method is established by numerical results on two test problems. Even at lower resolution levels, the proposed method provides high accuracy. In any programming language, the proposed method can be easily implemented and is computationally efficient.