Extrapolation methods for the numerical solution of nonlinear Fredholm integral equations

In this paper, we want to exemplify the use of extrapolation methods (namely Shanks transformations, the recursive algorithms for their implementation, and the freely available corresponding matlab software) in the solution of nonlinear Fredholm integral equations of the second kind. Extrapolations methods are well–known in some domains of numerical analysis and applied mathematics, but, unfortunately, they are not frequently used in other domains. Thus, after presenting the most simple iterative method for the solution of Fredholm equations, we will show how the sequence it produces can be accelerated (under some assumptions) and also how the underlying system of nonlinear equations generated by it can be solved quite efficiently by a restarting method. Numerical examples and comparisons with other methods demonstrate the usefulness of these procedures.