A single-point iterative method to solve boundary value problems with quadratic nonlinearity based on second-order Fréchet derivative

Newton method is a classical method for solving system of nonlinear equations and offers quadratic convergence. The order of convergence of the Newton method is optimal as it requires one evaluation for the system of nonlinear equations and second for the Jacobian. Many boundary value problems in nature have quadratic non-linearity and the corresponding system of nonlinear equations associated with their discrete formulation has constant second order Fréchet derivatives. We try to get benefit from this information and develop a single-point iterative method to solve such system of nonlinear equations with quadratic nonlinearity. In our proposed single-point iterative method, we perform one evaluation of system of nonlinear equations and second for Jacobian. In total, there are two functional evaluations and we do not count the evaluation of second-order Fréchet derivative as it is constant in all the iterations of the method. The convergence order (CO) of our proposed method is four. The efficiency index of our method is 4^{1∕2} = 2 which is higher than that of Newton method 2^{1∕2} = 1.4142. To quantify the functionality of our proposed algorithm, we have performed extensive numerical testing on a collection of test problems that have quadratic nonlinearity.