Error of discretization of Caputo fractional derivative in weighted spaces

TL;DR

This paper derives uniform error bounds for the L1 discretization of the Caputo fractional derivative in weighted Sobolev spaces, demonstrating convergence and numerical validation.

Contribution

It establishes new uniform error bounds for the L1 scheme in weighted Sobolev spaces with Muckenhoupt weights, applicable to fractional ODEs.

Findings

Error bounds are uniform in weighted Sobolev spaces.

The L1 scheme converges for fractional ODEs under the studied conditions.

Numerical results confirm the theoretical error estimates.

Abstract

We establish uniform error bounds of the L1 discretization of the Caputo fractional derivative of the function from the weighted Sobolev space with weight belonging to the Mucknenhoupt class. We present how our framework works for several examples of weight, which belong to the Muckenhoupt class. As and application, we show the convergence of the L1 scheme for the Fractional ODE. Finally, we verify the theoretical results with numerical illustrations.