[Paper Review] Fourier Cosine and Sine Transform on fractal space

Guang-Sheng Chen|arXiv (Cornell University)|Oct 21, 2011

Mathematical Analysis and Transform Methods12 references3 citations

TL;DR

This paper introduces local fractional Fourier cosine and sine transforms on fractal spaces using local fractional calculus, extending classical transforms to handle continuous fractal functions. It establishes key properties like Parseval's identity and convolution theorems, and applies the transforms to solve local fractional differential equations, demonstrating the method with a second-order local fractional ODE whose solution is expressed in terms of Mittag-Leffler functions.

ABSTRACT

In this paper, we establish local fractional Fourier Cosine and Sine Transforms on fractal space, considered some properties of local fractional Cosine and Sine Transforms, show applications of local fractional Fourier Cosine and Sine transform to local fractional equations with local fractional derivative.

Motivation & Objective

• To develop local fractional Fourier cosine and sine transforms on fractal sets of dimension α using local fractional calculus.
• To establish fundamental properties such as Parseval's identity, convolution theorems, and inversion formulas for these transforms.
• To apply the transforms to solve local fractional differential equations involving local fractional derivatives.
• To demonstrate the utility of the transforms through analytical solutions of specific local fractional ODEs.

Proposed method

• Derives the local fractional Fourier cosine and sine integral formulas by exploiting the even and odd symmetry of cosine and sine functions on fractal sets.
• Defines the local fractional Fourier cosine transform via the integral $ F_{\alpha,c}\{f(x)\} = \frac{2}{\Gamma(1+\alpha)}\int_0^\infty f(x)\cos_\alpha(\omega^\alpha x^\alpha)(dx)^\alpha $.
• Defines the inverse local fractional Fourier cosine transform as $ f(x) = \frac{2}{(2\pi)^\alpha}\int_0^\infty f_{\omega,c}^{F,\alpha}(\omega)\cos_\alpha(\omega^\alpha x^\alpha)(d\omega)^\alpha $.
• Applies the local fractional Fourier sine transform to solve a second-order local fractional differential equation with initial condition.
• Uses the linearity and transformation rules of the local fractional sine transform, including the identity $ F_{\alpha,s}\{y^{(2\alpha)}(t)\} = -\omega^{2\alpha}F_{\alpha,s}\{y(t)\} + 2\omega^\alpha y(0) $.
• Employs partial fraction decomposition and inverse transform to express the solution in terms of Mittag-Leffler functions.

Experimental results

Research questions

• RQ1 How can the classical Fourier cosine and sine transforms be generalized to functions defined on fractal sets of dimension α using local fractional calculus?
• RQ2 What are the fundamental properties—such as Parseval’s identity and convolution theorems—of the local fractional Fourier cosine and sine transforms?
• RQ3 Can the local fractional Fourier sine transform be effectively applied to solve local fractional differential equations with local fractional derivatives?
• RQ4 What is the analytical solution of a second-order local fractional ODE using the local fractional sine transform method?

Key findings

• The local fractional Fourier cosine transform is defined as $ F_{\alpha,c}\{f(x)\} = \frac{2}{\Gamma(1+\alpha)}\int_0^\infty f(x)\cos_\alpha(\omega^\alpha x^\alpha)(dx)^\alpha $, with a corresponding inverse formula.
• The local fractional Fourier sine transform of $ E_\alpha(-at^\alpha) $ is $ \frac{2\omega^\alpha}{a^2 + \omega^{2\alpha}} $, and for the cosine transform it is $ \frac{2a}{a^2 + \omega^{2\alpha}} $.
• The Parseval identity for the local fractional sine transform is $ \int_0^\infty |f_{\omega,s}^{F,\alpha}(\omega)|^2 (d\omega)^\alpha = \frac{(2\pi)^\alpha}{\Gamma(1+\alpha)} \int_0^\infty |f(x)|^2 (dx)^\alpha $.
• The convolution theorem for the local fractional cosine transform is $ \int_0^\infty f_{\omega,s}^{F,\alpha}(\omega)g_{\omega,s}^{F,\alpha}(\omega)\cos_\alpha(\omega^\alpha x^\alpha)(d\omega)^\alpha = \frac{(2\pi)^\alpha}{2\Gamma(1+\alpha)} \int_0^\infty f(\xi)[g(\xi+x)+g(\xi-x)](d\xi)^\alpha $.
• The solution to the local fractional ODE $ y^{(2\alpha)} - 9y(t) = 50E_\alpha(-2t^\alpha) $ with $ y(0) = y_0 $ is $ y(t) = (y_0 + 10)E_\alpha(-3t^\alpha) - 10E_\alpha(-2t^\alpha) $.
• The method successfully solves the differential equation by transforming it into an algebraic equation in the transform domain, then inverting using known transform pairs.