A numerical method for two-dimensional Hammerstein integral equations

"In this paper we investigate a collocation method for the approximate solution of Hammerstein integral equations in two dimensions. As in [8], col- location is applied to a reformulation of the equation in a new unknown, thus reducing the computational cost and simplifying the implementation. We start with a special type of piecewise linear interpolation over triangles for a refor- mulation of the equation. This leads to a numerical integration scheme that can then be extended to any bounded domain in R2, which is used in collocation. We analyze and prove the convergence of the method and give error estimates. As the quadrature formula has a higher degree of precision than expected with linear interpolation, the resulting collocation method is superconvergent, thus requiring fewer iterations for a desired accuracy. We show the applicability of the proposed scheme on numerical examples and discuss future research ideas in this area."

Nontrivial Solutions of Systems of Perturbed Hammerstein Integral Equations with Functional Terms

We discuss the solvability of a fairly general class of systems of perturbed Hammerstein integral equations with functional terms that depend on several parameters. The nonlinearities and the functionals are allowed to depend on the components of the system and their derivatives. The results are applicable to systems of nonlocal second order ordinary differential equations subject to functional boundary conditions, this is illustrated in an example. Our approach is based on the classical fixed point index.

Positive solutions of systems of perturbed Hammerstein integral equations with arbitrary order dependence

Motivated by the study of systems of higher-order boundary value problems with functional boundary conditions, we discuss, by topological methods, the solvability of a fairly general class of systems of perturbed Hammerstein integral equations, where the nonlinearities and the functionals involved depend on some derivatives. We improve and complement earlier results in the literature. We also provide some examples in order to illustrate the applicability of the theoretical results. This article is part of the theme issue ‘Topological degree and fixed point theories in differential and difference equations’.

A high order approach for nonlinear Volterra-Hammerstein integral equations

<abstract><p>Here a scheme for solving the nonlinear integral equation of Volterra-Hammerstein type is given. We combine the related theories of homotopy perturbation method (HPM) with the simplified reproducing kernel method (SRKM). The nonlinear system can be transformed into linear equations by utilizing HPM. Based on the SRKM, we can solve these linear equations. Furthermore, we discuss convergence and error analysis of the HPM-SRKM. Finally, the feasibility of this method is verified by numerical examples.</p></abstract>