Second-kind Volterra integral equations with weakly singular kernels typically have solutions which are nonsmooth near the initial point of the interval of integration. Using an adaptation of the analysis originally developed for nonlinear weakly singular Fredholm integral equations, we present a complete discussion of the optimal (global and local) order of convergence of piecewise polynomial ...

The two-point time and temperature dependent correlation functions for the XX0 one-dimensional model in constant magnetic field are represented (in the thermodynamical limit) as Fredholm determinants of linear integral operators.

Surrogate modelling of solutions of integral equations by neural networks is investigated theoretically. Estimates of speed of convergence of suboptimal surrogate solutions to solutions described by Fredholm theorem are derived.

In this paper, an efficient method is presented for solving two dimensional Fredholm and Volterra integral equations of the second kind. Chebyshev polynomials are applied to approximate a solution for these integral equations. This method transforms the integral equation to algebraic equations with unknown Chebyshev coefficients. The high accuracy of this method is verified through some numeric...

A Fredholm second kind integral equation (SKIE) formulation is constructed for the Dirichlet problem of the biharmonic equation in three dimensions. A fast numerical algorithm is developed based on the constructed SKIE. Its performance is illustrated via several numerical examples.

We propose a new scheme of discretization for solving Fredholm integral equations of the first kind and show that for some classes of equations this scheme is order-optimal in the sense of amount of used Galerkin information.

A general integral formula for the spectral flow of a path of unbounded selfadjoint Fredholm operators subject to certain summability conditions is derived from the interpretation of the spectral flow as a winding number.