On Computational Efficiency of Nine Iterative Algorithms for Hammerstein Equations

Abstract

Many fixed point iterative methods have been proposed. Their authors provide, theoretically, the rate of convergence which indicates how much the error changes after one iteration step. Of practical importance are the CPU time required by each method and their accuracy. This has never been investigated. In this work we investigate nine iterative processes and compare their CPU time and accuracy. The schemes are the Picard, Mann, Ishikawa, Noor, Argawal, Abbas-Nazir, Thakur, Ullah, and S* methods. By applying a recently proposed fourth order quadrature rule to the integral equation and replacing the analytical operator with the discretized one, the iterative schemes are derived. They are tested on five Hammerstein equations. The results show that (i) all the methods converge to the exact solution at the correct discretization order of convergence, (ii) the Picard scheme is the fastest and also has good accuracy, and (iii) the S* and Abbas-Nazir schemes are the least efficient. It is concluded that for a contractive problem the Picard scheme is sufficient but if the operator is non-expansive, then the Mann scheme is sufficient.