In this paper, we will give some results for developing the two-dimensional triangular orthogonal functions (2D-TFs) for numerical solution of the linear two-dimensional Fredholm integral equations of the second kind. The product of 2D-TFs and some formulas for calculating definite integral of them are derived and utilized to reduce the solution of two-dimensional Fredholm integral equation to the solution of algebraic equations. Also a theorem is proved for convergence analysis. Numerical examples are presented and results are compared with analytical solution to demonstrate the validity and applicability the method.

The main purpose of this paper is to demonstrate that using modified Simpson's quadrature rule for solving linear Fredholm integral equations of the second kind. We convert the integral equations to a system of linear equations, and by using numerical examples we show our estimation have a good degree of accuracy.