A correction to “Rationalized Haar wavelet bases to approximate solution of nonlinear Fredholm integral equations with error analysis [Applied Mathematics and Computation 265 (2015) 304–312]”

摘要

In the recent paper, “Rationalized Haar wavelet bases to approximate solution of nonlinear Fredholm integral equations with error analysis [Applied Mathematics and Computation 265 (2015) 304–312]”, The authors have approximated the solutions of nonlinear Fredholm integral equations (NFIEs) of the second type by using the successive approximations method. In any stage, the rationalized Haar wavelets (RHWs) and the corresponding operational matrices were applied to approximate the integral operator. In Theorem 4.2, page 307 of the reference [4], the authors introduced an upper bound for the error and explicitly stated that the rate of convergence of ui to u is O(qi), in which i is the number of iterations, q is the contraction constant, u is the exact solution, and ui is the approximate solution in ith iteration. This statement is not true and we prove carefully that the rate of convergence will be O(iqi).