# Approximation of Hamilton-Jacobi equations with Caputo time-fractional derivative

## TL;DR

This paper develops a numerical scheme for solving Hamilton-Jacobi equations involving Caputo time-fractional derivatives, demonstrating stability and convergence to the unique viscosity solution.

## Contribution

It introduces an explicit time discretization method for Caputo derivatives combined with a finite difference scheme for Hamiltonians, ensuring stability and convergence.

## Key findings

- The scheme is stable under specific discretization conditions.
- The scheme converges to the unique viscosity solution.
- Numerical experiments confirm theoretical results.

## Abstract

In this paper, we investigate the numerical approximation of Hamilton-Jacobi equations with the Caputo time-fractional derivative. We introduce an explicit in time discretization of the Caputo derivative and a finite difference scheme for the approximation of the Hamiltonian. We show that the approximation scheme so obtained is stable under an appropriate condition on the discretization parameters and converges to the unique viscosity solution of the Hamilton-Jacobi equation.