On the numerical solution of Urysohn integral equation using Legendre approximation

Urysohn integral equation is one of the most applicable topics in both pure and applied mathematics. The main objective of this paper is to solve the Urysohn type Fredholm integral equation. To do this, we approximate the solution of the problem by substituting a suitable truncated series of the well known Legendre polynomials instead of the known function. After discretization of the problem on the given integral interval, by using the proposed procedure the original integral equation is converted to a linear algebraic system. Now, the solution of the resulting system yields the unknown Legendre coefficients. Finally, two numerical examples are given to show the effectiveness of the proposed method.