The numerical solution of second-order boundary-value problems by collocation method with the Haar wavelets

Abstract

An efficient numerical method based on uniform Haar wavelets is proposed for the numerical solution of second-order boundary-value problems (BVPs) arising in the mathematical modeling of deformation of beams and plate deflection theory, deflection of a cantilever beam under a concentrated load, obstacle problems and many other engineering applications. The Haar wavelet basis permits to enlarge the class of functions used so far in the collocation framework. The performance of the Haar wavelets is compared with the Walsh wavelets, semi-orthogonal B-spline wavelets, spline functions, Adomian decomposition method (ADM), finite difference method, and Runge–Kutta method coupled with nonlinear shooting method. A more accurate solution can be obtained by wavelet decomposition in the form of a multi-resolution analysis of the function which represents the solution of a given problem. Through this analysis the solution is found on the coarse grid points, and then refined towards higher accuracy by increasing the level of the Haar wavelets. Neumann’s boundary conditions which are problematic for most of the numerical methods are automatically coped with. The main advantage of the Haar wavelet based method is its efficiency and simple applicability for a variety of boundary conditions. The convergence analysis of the proposed method alongside numerical procedure for multi-point boundary-value problems are given to test wider applicability and accuracy of the method.