In this work, we used the properties of the two-dimensional Haar wavelet; for this purpose, it is required to define the integral operator and obtain an operational matrix for our integral equation. Also, we used from 2D-Haar wavelet to approximate solutions of nonlinear two-dimensional Fredholm integral equations without solving a linear system. In section error analysis, we apply Banach fixed point theorem, and we proved my integral operator has a unique fixed point. In section four, numerical example, we choose one example, and we have compared my method with the other methods. It has been observed that the approximation solutions are obtained are very suitable. Moreover, the CPU runs times in seconds are presented. Also, we can expand this method to another type of 2D integral equation, such as Volterra or mixed of Volterra and Fredholm.

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M. Erfanian and A. Akrami, “Using of two dimensional HAAR wavelet for solving of two dimensional nonlinear fredholm integral equation,” International Journal of Applied and Physical Sciences, vol. 3, no. 2, pp. 55-60, 2017.