Numerical solvability of a class of Volterra-Hammerstein integral equations with noncompact kernels. (English) Zbl 1099.65135

The authors consider the numerical solvability of a class of nonlinear weakly singular integral equations of Volterra-Hammerstein type, with noncompact kernels. The equations take the form, \(y(t)=f(t)+\int_0^t k\left(\frac{t}{s}\right)g(s, y(s))\frac{1}{s}\,ds\) where \(g\) and \(f\) are smooth given functions. A brief review of current literature relating to this type of equation is given.
The authors present and prove a theorem relating to the existence and uniqueness of a solution to the equation provided that five stated conditions are satisfied. They investigate the approximation of the solution by two numerical schemes, an Euler-type method and the product trapezoidal method, restricting their investigation to the two cases where the elements of the spaces are piecewise constant functions and continuous linear polynomials.
Theorems giving the convergence properties of the Euler and the product trapezoidal methods are stated and proved. Numerical examples are included to illustrate these results. Maximum absolute errors and predicted convergence orders are given for each example. Mathematica programming has been used to perform the computations.
Reviewer: Pat Lumb (Chester)


65R20 Numerical methods for integral equations
45G05 Singular nonlinear integral equations


Full Text: DOI EuDML