Numerical solution of nonlinear Hammerstein integral equations by using Legendre-Bernstein basis

In this study a numerical method is developed to solve the Hammerstein integral equations. To this end the kernel has been approximated using the leastsquares approximation schemes based on Legender-Bernstein basis. The Legender polynomials are orthogonal and these properties improve the accuracy of the approximations. Also the nonlinear unknown function has been approximated by using the Bernstein basis. The useful properties of Bernstein polynomials help us to transform Hammerstein integral equation to solve a system of nonlinear algebraic equations.