# An explicit numerical algorithm to the solution of Volterra integral equation of the second kind

## TL;DR

This paper introduces a numerical algorithm for solving Volterra integral equations of the second kind, accommodating generalized functions like the Dirac delta, by approximating kernels and explicitly computing solutions.

## Contribution

It presents a novel algorithm that handles generalized input functions and provides explicit solutions using piecewise constant kernel approximations.

## Key findings

- The algorithm effectively approximates solutions with specified accuracy.
- It handles generalized functions such as the Dirac delta.
- Explicit solutions are obtained via convolution with known functions.

## Abstract

This paper considers a numeric algorithm to solve the equation \begin{align*} y(t)=f(t)+\int^t_0 g(t-\tau)y(\tau)\,d\tau \end{align*} with a kernel $g$ and input $f$ for $y$. In some applications we have a smooth integrable kernel but the input $f$ could be a generalised function, which could involve the Dirac distribution. We call the case when $f=\delta$, the Dirac distribution centred at 0, the fundamental solution $E$, and show that $E=\delta+h$ where $h$ is integrable and solve \begin{align*} h(t)=g(t)+\int^t_0 g(t-\tau)h(\tau)\,d\tau \end{align*} The solution of the general case is then \begin{align*} y(t)=f(t)+(h*f)(t) \end{align*} which involves the convolution of $h$ and $f$. We can approximate $g$ to desired accuracy with piecewise constant kernel for which the solution $h$ is known explicitly. We supply an algorithm for the solution of the integral equation with specified accuracy.