[Paper Review] On the nature of the conformable derivative and its applications to physics

Douglas R. Anderson, Evan Camrud+1 more|arXiv (Cornell University)|Oct 3, 2018

Fractional Differential Equations SolutionsMathematics36 references50 citations

TL;DR

The paper shows that Khalil and Katugampola conformable derivatives are equivalent to the simple variable change u = x^α/α for differentiable functions, and explores the resulting mathematical structure and physical interpretations, focusing on quantum mechanics, self-adjointness, Sturm–Liouville problems, and integral transforms.

ABSTRACT

The purpose of this work is to show that the Khalil and Katagampoula conformable derivatives are equivalent to the simple change of variables $x$ $ ightarrow $ $x^{α}/α,$ where $α$ is the order of the derivative operator, when applied to differential functions. Although this means no extquotedblleft new mathematics extquotedblright\ is obtained by working with these derivatives, it is a second purpose of this work to argue that there is still significant value in exploring the mathematics and physical applications of these derivatives. This work considers linear differential equations, self-adjointness, Sturm-Liouville systems, and integral transforms. A third purpose of this work is to contribute to the physical interpretation when these derivatives are applied to physics and engineering. Quantum mechanics serves as the primary backdrop for this development.

Motivation & Objective

• Motivate the study of conformable derivatives as a useful tool for applying standard differential equation techniques to non-integer order calculus.
• Demonstrate that the conformable derivative is equivalent to a change of variables and that this yields no new mathematics but offers valuable physical interpretation and applications.
• Develop the calculus, self-adjointness, and spectral theory framework for conformable differential equations.
• Investigate conformable integral transforms, especially the Fourier transform, and discuss physical units and space interpretations in quantum mechanics.

Proposed method

• Define the conformable derivative D^α as x^{1-α} d/dx and relate it to the change of variables u = x^α/α via the chain rule.
• Translate ordinary second-order differential equations into conformable form using the u-variable, yielding a recipe that maps SOLDEs to conformable SOLDEs.
• Derive the self-adjoint conformable operator Â_{2α} = d/dx [x^{1-α} d/dx] and study its Sturm–Liouville eigenproblem with boundary conditions y(0)=y(1)=0.
• Construct and analyze conformable analogues of Bessel and confluent hypergeometric equations and present explicit conformable forms and solutions.
• Introduce conformable Sturm–Liouville theory and discuss spectra, eigenfunctions, and scaling properties of the generalized J_n^{(α)} functions.
• Develop conformable transforms, especially the Fourier transform, and outline the associated inverse transform and relation to conformable Laplace transform.

Experimental results

Research questions

• RQ1 Does the conformable derivative D^α equal the ordinary differential structure under a simple x-to-u change of variable for differentiable functions?
• RQ2 How does the conformable framework alter self-adjointness, Sturm–Liouville problems, and spectral theory?
• RQ3 What are the conformable counterparts of classical special functions (e.g., Bessel, Airy) and how do their properties (zeros, scaling) behave with α?
• RQ4 How can conformable integral transforms (Fourier, Laplace) be formulated and interpreted physically, especially in quantum mechanics?

Key findings

• The conformable derivative is equivalent to the simple change of variable u = x^α/α for differentiable functions.
• A self-adjoint conformable operator Â_{2α} = d/dx [x^{1-α} d/dx] provides a natural conformable Sturm–Liouville system with explicit eigenfunctions and eigenvalues.
• The conformable Bessel equation leads to a conformable Bessel function expressed in terms of standard Bessel functions with modified arguments, and its SOLDE corresponds to an x^{2}-type independent variable transformation.
• A family of orthonormal conformable eigenfunctions J_n^{(α)}(x) generalizes sine functions and forms a complete basis on 0≤x≤1, with zeros and scaling dependent on α.
• Conformable analogues of Fourier-Bessel expansions and Kummer/0F1 relations relate the new basis to classical special functions.
• Conformable Sturm–Liouville operators and boundary-value problems yield solvable eigenvalue problems with explicit forms, illustrating how parameter α modulates the spectrum and eigenfunctions.
• A conformable Fourier transform framework is developed, enabling a symmetrical forward and inverse transform compatible with quantum-mechanical interpretations.