Solving Two-Dimensional Nonlinear Fredholm Integral Equations Using Rationalized Haar Functions in the Complex Plane

As far as we are aware, no research has been published about two-dimensional integral equations in the complex plane by using Haar bases or any other kinds of wavelets. We introduce a numerical method to solve two-dimensional Fredholm integral equations, using Haar wavelet bases. To attain this purpose, first, an operator and then an orthogonal projection should be defined. Regarding the characteristics of Haar wavelet, we solve an integral equation without using common mathematical methods. We prove the convergence and an upper bound that mentioned in the method by employing the Banach fixed point theorem. Moreover, the rate of convergence our method is O(n(2 q) n ). We present several examples of different kinds of functions and solve them by this method in this study. © 2019, Springer Nature India Private Limited.