[Paper Review] A new fractional derivative and its fractional integral with some example

Fahed Zulfeqarr, Amit Ujlayan+1 more|arXiv (Cornell University)|Apr 26, 2017

Fractional Differential Equations Solutions1 references3 citations

TL;DR

This paper introduces a new fractional derivative called the deformable derivative, defined via a limit involving a parameter α ∈ [0,1], which generalizes both the function and its ordinary derivative. It establishes a linear relationship between the deformable derivative and the ordinary derivative, proves analogues of Rolle’s, Mean Value, and Taylor’s theorems, and defines a corresponding fractional integral; the framework successfully solves linear fractional differential equations, demonstrating equivalence to classical ODEs under transformation.

ABSTRACT

A new derivative, called deformable derivative, is introduced here which is equivalent to ordinary derivative in the sense that one implies other. The deformable derivative is defined using limit approach like that of ordinary one but with respect to a parameter varying over unit interval. Thus it could also be regarded as a fractional derivative. Reason of calling it as deformable derivative is because of its intrinsic property of continuously deforming function to derivative. This is substantiated by its linear connection to function and its derivative. Besides discussing some of its basic properties, we discover the forms of Rolle's, Mean Value and Taylor's theorems. The fundamental theorem of calculus for this fractional derivative could be taken as definition of its fractional integral. As a theoretical application some fractional differential equations are solved.

Motivation & Objective

• To introduce a new fractional derivative, the deformable derivative, that smoothly interpolates between a function and its ordinary derivative.
• To establish a rigorous mathematical framework for the deformable derivative, including its relationship to classical derivatives.
• To extend classical calculus theorems—Rolle’s, Mean Value, and Taylor’s—to the deformable derivative context.
• To define a corresponding fractional integral operator and prove its fundamental role in solving fractional differential equations.
• To demonstrate the practical utility of the deformable derivative by solving linear fractional differential equations with analytical solutions.

Proposed method

• Define the deformable derivative D^αf(t) as the limit: lim_{ε→0} [(1+εβ)f(t+αε) - f(t)] / ε, where α+β=1.
• Derive the key identity: D^αf(t) = βf(t) + αDf(t), showing equivalence between α-differentiability and ordinary differentiability.
• Prove that existence of the ordinary derivative implies existence of the deformable derivative, and vice versa.
• Introduce the fractional integral operator: I^α_a f(t) = (1/α) e^{(-β/α)t} ∫_a^t e^{(β/α)x} f(x) dx.
• Establish the fundamental theorem of calculus for the deformable derivative, linking D^α and I^α_a as inverse operators.
• Apply the framework to solve linear fractional differential equations by transforming them into standard first-order ODEs.

Experimental results

Research questions

• RQ1 What is the geometric and physical interpretation of the deformable derivative as α varies from 0 to 1?
• RQ2 How does the deformable derivative relate to classical fractional derivatives like Riemann-Liouville or Caputo?
• RQ3 Can classical theorems such as Rolle’s, Mean Value, and Taylor’s be generalized to the deformable derivative framework?
• RQ4 What is the precise relationship between the deformable derivative and the ordinary derivative?
• RQ5 How can the deformable derivative be used to solve fractional differential equations, and what are the analytical solutions?

Key findings

• The deformable derivative satisfies D^αf(t) = βf(t) + αDf(t), proving that α-differentiability is equivalent to ordinary differentiability.
• The deformable derivative smoothly interpolates between the function (at α=0) and its ordinary derivative (at α=1), with a linear dependence on both.
• Analogues of Rolle’s, Mean Value, and Taylor’s theorems are established for the deformable derivative, extending classical calculus to this fractional setting.
• The fractional integral operator I^α_a f(t) is defined such that it inverts the deformable derivative, satisfying the fundamental theorem of calculus.
• Solutions to linear fractional differential equations are derived by transforming them into standard ODEs; for example, D^{1/2}y + y = te^{-t} yields y(t) = Ce^{-3t} + (t - 1/2)e^{-t}.
• For the equation D^{α₂}[D^{α₁}y] = 0, the general solution is y = C₁e^{-(β₁/α₁)t} + C₂e^{-(β₂/α₂)t} for distinct roots, and (C₁ + C₂t)e^{-(β/α)t} for repeated roots.