A comparison of Newton–Raphson method with Newton–Krylov generalized minimal residual (GMRes) method for solving one and two dimensional nonlinear Fredholm integral equations

In this work, Newton–Raphson and Newton–Krylov GMRes methods are compared in the CPU time and accuracy points of view in solving of one and two dimensional nonlinear Fredholm integral equations of second kind. Since applying shifted Legendre collocation method and utilizing Gauss–Legendre integration rule on nonlinear Fredholm integral equations reduce the equations to solve a system of nonlinear algebraic equations, the solvers of Newton–Raphson and Newton–Krylov GMRes are applied on the obtained systems of nonlinear algebraic equations. The numerical results show that the use of Newton–Krylov GMRes is better than Newton–Raphson method. © 2019, Sociedad Española de Matemática Aplicada.