[Paper Review] Generalized Fractional Order Derivatives, Its Properties and Applications

Ali Karcı|arXiv (Cornell University)|Jun 20, 2013

Fractional Differential Equations Solutions5 references3 citations

TL;DR

This paper redefines fractional order derivatives (FOD) using classical derivative principles and L'Hôpital's rule to resolve inconsistencies in existing methods like Riemann-Liouville and Caputo. It demonstrates that traditional FOD formulations are essentially curve-fitting approximations, and proposes a generalized FOD definition that reduces to the classical derivative at order 1, ensuring mathematical consistency and completeness across polynomial, exponential, trigonometric, and logarithmic functions.

ABSTRACT

The concept of fractional order derivative can be found in extensive range of many different subject areas. For this reason, the concept of fractional order derivative should be examined. After giving different methods mostly used in engineering and scientific applications, the omissions or errors of these methods will be discussed in this study. The mostly used methods are Euler, Riemann-Liouville and Caputo which are fractional order derivatives. The applications of these methods to constant and identity functions will be given in this study. Obtained results demonstrated that all of three methods have errors and deficiencies. In fact, the obtained results demonstrated that the methods given as fractional order derivatives are curve fitting methods or curve approximation methods. In this paper, we redefined fractional order derivative (FOD) by using classical derivative definition and L Hospital method, since classical derivative definition concluded in indefinite limit such as 0/0. The obtained definition is same as classical derivative definition in case of fractional order is equal to 1. This implies that definitions and theorems in this paper sound and complete. After defining the FOD concept, the applications of FOD for polynomial, exponential, trigonometric and logarithmic functions were handled in this study. The properties of FOD for positive monotonic increasing/decreasing functions were demonstrated in this paper. Another important point is that the relation between derivatives and complex functions were also verified in this paper.

Motivation & Objective

• To identify and correct fundamental errors and omissions in widely used fractional order derivative methods such as Euler, Riemann-Liouville, and Caputo.
• To redefine fractional order derivatives using classical derivative definitions and L'Hôpital’s rule to resolve indeterminate forms like 0/0.
• To ensure the new FOD definition reduces to the classical derivative when the order equals 1, guaranteeing consistency and completeness.
• To analyze the properties of the generalized FOD for positive monotonic functions and verify its behavior with complex functions.
• To demonstrate the applicability and correctness of the new FOD formulation across polynomial, exponential, trigonometric, and logarithmic functions.

Proposed method

• Reformulate fractional order derivatives by applying the classical derivative definition and resolving the 0/0 indeterminate form using L'Hôpital’s rule.
• Define the generalized FOD such that it matches the classical derivative when the fractional order is 1, ensuring continuity and consistency.
• Apply the new FOD definition to constant, identity, polynomial, exponential, trigonometric, and logarithmic functions to verify correctness.
• Analyze the behavior of the generalized FOD for positive monotonic increasing and decreasing functions to establish its theoretical properties.
• Verify the relationship between the generalized FOD and complex functions to extend its theoretical foundation.
• Use mathematical proofs to show that the new definition avoids the deficiencies of existing methods, which are shown to be curve-fitting approximations.

Experimental results

Research questions

• RQ1 Why do traditional fractional order derivative methods like Riemann-Liouville and Caputo produce inconsistent results when applied to constant and identity functions?
• RQ2 Can a generalized fractional order derivative be defined that reduces to the classical derivative at order 1 and maintains mathematical consistency?
• RQ3 Are existing fractional derivative formulations fundamentally curve-fitting methods rather than true generalizations of differentiation?
• RQ4 How does the proposed generalized FOD behave across different function classes, including polynomials, exponentials, trigonometric, and logarithmic functions?
• RQ5 What is the relationship between the generalized FOD and complex functions, and does it preserve essential analytical properties?

Key findings

• The proposed generalized FOD definition is mathematically consistent and reduces exactly to the classical derivative when the fractional order is 1.
• Traditional methods such as Riemann-Liouville and Caputo are shown to be flawed and better interpreted as curve-fitting or approximation techniques rather than exact derivatives.
• The new FOD formulation correctly differentiates constant and identity functions, resolving key inconsistencies present in prior approaches.
• The generalized FOD maintains proper behavior for polynomial, exponential, trigonometric, and logarithmic functions, demonstrating its broad applicability.
• The method preserves monotonicity properties for positive increasing and decreasing functions, confirming its theoretical soundness.
• The relationship between the generalized FOD and complex functions is verified, supporting its use in advanced analytical contexts.