Stability and convergence analysis of a class of continuous piecewise polynomial approximations for time fractional differential equations

TL;DR

This paper introduces and analyzes a class of numerical schemes using continuous piecewise polynomials to approximate Caputo fractional derivatives, focusing on their stability, convergence, and error properties for time fractional differential equations.

Contribution

It develops a new polynomial-based approximation method for fractional derivatives and provides a comprehensive theoretical analysis of its stability and convergence properties.

Findings

The schemes achieve favorable stability regions.

Error analysis confirms expected convergence rates.

Numerical experiments verify theoretical predictions.

Abstract

We propose and study a class of numerical schemes to approximate time fractional differential equations. The methods are based on the approximation of the Caputo fractional derivative by continuous piecewise polynomials, which is strongly related to the backward differentiation formulae for the integer-order case. We investigate their theoretical properties, such as the local truncation error and global error analyses with respect to a sufficiently smooth solution, and the numerical stability in terms of the stability region and $A(\frac{\pi}{2})$-stability by refining the technique proposed in \cite{LubichC:1986b}. Numerical experiments are given to verify the theoretical investigations.