The repeated midpoint rule for weakly singular Volterra integral equations of the first kind with perturbed data

TL;DR

This paper investigates the effectiveness of the repeated midpoint rule in stably solving weakly singular Volterra integral equations of the first kind with noisy data, emphasizing regularization and stability.

Contribution

It introduces a regularization approach using the repeated midpoint rule with correction weights, analyzing stability via Banach algebra techniques, and provides numerical validation.

Findings

The method achieves stable solutions with perturbed data.

Correction weights improve initial condition handling.

Numerical results confirm theoretical stability and accuracy.

Abstract

In the present paper we consider the regularizing properties of the repeated midpoint rule for the stable solution of weakly singular Volterra integral equations of the first kind with perturbed right hand sides. The H\"older continuity of the solution and its derivative is carefully taken into account, and correction weights are considered to get rid of initial conditions. The proof of the inverse stability of the quadrature weights relies on Banach algebra techniques. Finally, numerical results are presented.