[Paper Review] Generalized local Taylor's formula with local fractional derivative

Xiao‐Jun Yang|arXiv (Cornell University)|Jun 13, 2011

Fractional Differential Equations Solutions3 citations

TL;DR

This paper proposes a generalized local Taylor formula using local fractional derivatives (LFDs) within the framework of local fractional calculus (LFC), enabling approximation and series representation of fractal and non-differentiable functions. The key contribution is the derivation of a local fractional Taylor series for Mittag-Leffler type functions, extending classical Taylor expansions to fractal domains.

ABSTRACT

In the present paper, a generalized local Taylor formula with the local fractional derivatives (LFDs) is proposed based on the local fractional calculus (LFC). From the fractal geometry point of view, the theory of local fractional integrals and derivatives has been dealt with fractal and continuously non-differentiable functions, and has been successfully applied in engineering problems. It points out the proof of the generalized local fractional Taylor formula, and is devoted to the applications of the generalized local fractional Taylor formula to the generalized local fractional series and the approximation of functions. Finally, it is shown that local fractional Taylor series of the Mittag-Leffler type function is discussed.

Motivation & Objective

• To develop a generalized local Taylor formula based on local fractional calculus for functions defined on fractal sets.
• To address the limitation of classical Taylor expansions in handling continuously non-differentiable and fractal functions.
• To enable the representation of such functions via generalized local fractional series.
• To demonstrate the applicability of the formula through the analysis of Mittag-Leffler type functions.

Proposed method

• The paper employs local fractional calculus (LFC) to define local fractional derivatives (LFDs) applicable to fractal and non-differentiable functions.
• It formulates a generalized local Taylor expansion using higher-order LFDs to capture local behavior on fractal sets.
• The method relies on the theory of local fractional integrals and derivatives to ensure convergence in fractal domains.
• The derivation is grounded in fractal geometry, ensuring consistency with the structure of continuous but non-differentiable functions.
• The approach extends classical Taylor series by replacing standard derivatives with local fractional derivatives.
• The formula is applied to derive a local fractional Taylor series for the Mittag-Leffler function, a key function in fractional calculus.

Experimental results

Research questions

• RQ1 How can a generalized Taylor expansion be formulated for functions that are non-differentiable in the classical sense but defined on fractal sets?
• RQ2 What is the structure and convergence behavior of a local fractional Taylor series for special functions like the Mittag-Leffler function?
• RQ3 How do local fractional derivatives enable accurate approximation of fractal functions where standard derivatives fail?
• RQ4 What are the necessary conditions for the existence and validity of a generalized local fractional Taylor formula?
• RQ5 Can the generalized formula be systematically applied to represent and approximate other classes of fractal functions?

Key findings

• A generalized local Taylor formula is successfully derived using local fractional derivatives, extending classical Taylor expansions to fractal domains.
• The formula enables the representation of fractal and non-differentiable functions through generalized local fractional series.
• The Mittag-Leffler type function admits a local fractional Taylor series expansion, demonstrating the formula’s applicability to special functions in fractional calculus.
• The method provides a consistent framework for function approximation on sets with fractal geometry, where standard calculus fails.
• The results confirm the theoretical validity and practical utility of local fractional Taylor expansions in modeling complex, non-smooth phenomena.