Numerical solution of Volterra integral equations of the first kind with discontinuous kernels

TL;DR

This paper introduces numerical methods for solving Volterra integral equations of the first kind with discontinuous kernels, including linearization, quadrature, and iterative schemes, validated by numerical examples.

Contribution

It presents new direct quadrature and iterative methods tailored for equations with jump discontinuities in the kernels, improving accuracy and stability.

Findings

Methods achieve $ ext{O}(1/N)$ and $ ext{O}(1/N^2)$ accuracy.

Numerical examples demonstrate efficiency and applicability.

Proposed schemes handle nonlinear and discontinuous kernels effectively.

Abstract

We propose the numerical methods for solution of the weakly regular linear and nonlinear evolutionary (Volterra) integral equation of the first kind. The kernels of such equations have jump discontinuities along the continuous curves (endogenous delays) which starts at the origin. In order to linearize these equations we use the modified Newton-Kantorovich iterative process. Then for linear equations we propose two direct quadrature methods based on the piecewise constant and piecewise linear approximation of the exact solution. The accuracy of proposed numerical methods is $\mathcal{O}(1/N)$ and $\mathcal{O}(1/N^2)$ respectively. We also suggest a certain iterative numerical scheme enjoying the regularization properties. Furthermore, we adduce generalized numerical method for nonlinear equations. We employ the midpoint quadrature rule in all the cases. In conclusion we include several numerical examples in order to demonstrate the efficiency of proposed numerical methods