Exploration of Two Modified L1 Formulae to Approximate the Caputo Fractional Derivative

Abstract

In this work, we revisit two modified L1 interpolation approximations to the values of the Caputo fractional derivative of order $\alpha \in (0, 1)$ at the midpoints of the mesh. These approximations are distinct from the classical L1 formula which typically yields interpolation approximations at the mesh points themselves. Although these two formulae on uniform meshes have been proposed in previous literature (Dong et al. in Commun Appl Math Comput 5(4): 1446–1468, 2023; Osman in Numerical solution methods for fractional partial differential equations. PhD thesis, University of Southern Queensland, 2017), given their practical applications in solving fractional differential systems with possible weak singularities and the inherent simplicity of the L1-type formulae compared to the existing higher-order numerical formulae such as the L2-$1_{\sigma }$ formula and L2-type formulae, it is more essential to consider the corresponding results on non-uniform meshes. To this end, two modified L1 interpolation formulae on non-uniform meshes, subsequently referred to as Formula I and Formula II, respectively, are explored in the current work. On the basis of deriving the formulae, the properties of coefficients in the formulae and the truncation errors under the graded mesh are analyzed in detail. It is worth noting that Formula I is reduced to the classical central difference quotient formula as $\alpha \rightarrow 1^{-}$, which is superior to Formula II and the classical L1 formula. Furthermore, we present several numerical examples to demonstrate the efficacy of these two formulae, as well as the effectiveness of their applications in solving the fractional sub-diffusion equation. The unique solvability and stability of the resultant schemes with respect to the initial values are briefly mentioned. As far as we know, there has been hardly any previous systematic investigation for this type of formulae on non-uniform meshes.

Keywords

Caputo fractional derivative / Modified L1 formula / Nonuniform mesh / Truncation error / Sub-diffusion / 26A33 / 35R11 / 65M06 / 65M12

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