Two new algorithms based on Haar wavelets are proposed. The first algorithm is proposed for the numerical solution of nonlinear Fredholm integral equations of the second kind, and the second for the numerical solution of nonlinear Volterra integral equations of the second kind. These methods are designed to exploit the special characteristics of Haar wavelets in both one and two dimensions. Formulae for calculating Haar coefficients without solving the system of equations have been derived. These formulae are then used in the proposed numerical methods. In contrast to other numerical methods, the advantage of our method is that it does not involve any intermediate numerical technique for evaluation of the integral present in integral equations. The methods are validated on test problems, and numerical results are compared with those from existing methods in the literature. © 2012 Elsevier B.V. All rights reserved.
Aziz, I., & Siraj-Ul-Islam. (2013). New algorithms for the numerical solution of nonlinear Fredholm and Volterra integral equations using Haar wavelets. Journal of Computational and Applied Mathematics, 239(1), 333–345. https://doi.org/10.1016/j.cam.2012.08.031