newton-krylov generalized minimal residual algorithm in solving nonlinear volterra-fredholm-hammerstein integral equations

In this paper, galerkin and collocation methods have been applied on nonlinear volterra- fredholm-hammerstein (vfh) integral equations, these methods are besed on shifted legendre polynomials, then methods transfer the finding solution of a nonlinear integral equation to finding the solution of nonlinear algebraic equations, in order to solve these nonlinear algebraic equations we use newton method composed by generalized minimal residual (ngmres) method, the iteration number and running time for implementation of ngmres method are important parameters that have been considered to solve this type of integral equations. these methods are applied on several various nonlinear vfh integral equations that confirm accuracy and efficiency of the methods.