Numerical Solution of Weakly Regular Volterra Integral Equations of the First Kind

TL;DR

This paper proposes a numerical method using mid-rectangular quadrature to solve weakly regular Volterra integral equations of the first kind, characterized by kernels with jump discontinuities, achieving an accuracy of order 1/N.

Contribution

It introduces a new numerical approach employing mid-rectangular quadrature for weakly regular Volterra equations with discontinuous kernels.

Findings

Achieves numerical solution with accuracy order O(1/N).

Effectively handles kernels with jump discontinuities.

Provides a practical method for weakly regular Volterra equations.

Abstract

The numerical method for solution of the weakly regular scalar Volterra integral equation of the 1st kind is proposed. The kernels of such equations have jump discontinuities on the continuous curves which starts at the origin. The mid-rectangular quadrature rule is employed. The accuracy of proposed numerical method is $\mathcal{O}(1/N).$