# Finite difference method for a Volterra equation with a power-type nonlinearity

## TL;DR

This paper proves the convergence of explicit finite-difference methods for nonlinear Volterra equations with power-type nonlinearities, overcoming challenges posed by non-Lipschitz kernels using discrete Gronwall's lemmas.

## Contribution

It introduces a convergent explicit finite-difference scheme for nonlinear Volterra equations with non-Lipschitz kernels, including convergence analysis and order bounds.

## Key findings

- Convergent explicit finite-difference methods for the problem.
- Upper bounds on the convergence order.
- Application to example problems.

## Abstract

In this work we prove that a family of explicit numerical finite-difference methods is convergent when applied to a nonlinear Volterra equation with a power-type nonlinearity. In that case the kernel is not of Lipschitz type, therefore the classical analysis cannot be applied. We indicate several difficulties that arise in the proofs and show how they can be remedied. The tools that we use consist of variations on discreet Gronwall's lemmas and comparison theorems. Additionally, we give an upper bound on the convergence order. We conclude the paper with a construction of a convergent method and apply it for solving some examples.