Analytical and numerical treatment of a nonlinear Fredholm integral equation in two dimensions

In this article, we have analytically proven that the integral equation in two dimensions (IE-TD) of the second type has at least one solution by using Tychonoff’s fixed point theorem. Then, the Adomian decomposition method (ADM) was used to solve the IE-TD, and the convergence of the method was also proved. Moreover, we utilized the Homotopy analysis method (HAM) to get the solution of the IE-TD, and the method’s convergence was investigated. The effectiveness and preciseness of the proposed approaches were illustrated using numerical examples, and the error estimate for each case was calculated. Finally, conclusions are obtained.