Superfast finite-difference algorithms for solving the fredholm equations of the 1$^{\mathrm{st}}$ and the 2$^{\mathrm{nd}}$ kind

A. A. Belov, Zh. O. Dombrovskaya

Peoples' Friendship University of Russia named after Patrice Lumumba, Moscow

Abstract: For the Fredholm equations of the first and second kind, difference schemes with superpower convergence are proposed. They are dramatically more accurate than previously known ones. For the equation of the first kind, a new method of regularization is proposed, based on adding a stabilizer directly to the matrix of the difference scheme. For a non-self-adjoint problem, the proposed approach reduces the complexity and improves the conditionality of the matrix of the linear system. A new procedure for selecting the regularization parameter is proposed. It is selected so that the systematic error introduced by the stabilizer and the random error due to rounding errors are comparable. A new calculation algorithm with precision control has been built, based on thickening grids and simultaneously increasing the number of digits. The proposed approaches have been verified on representative test problems with a known exact solution.

Keywords: Fredholm equations, grid method, super-power convergence.

Received: 17.03.2025
Revised: 16.06.2025
Accepted: 30.06.2025

DOI: 10.20948/mm-2026-01-09