Fractional Fourier Series Research Articles

Chirp-wave expansion of the electron wavefunctions in atoms

The description of the electron wavefunctions in atoms is generalized to the fractional Fourier series. This method introduces a continuous and infinite number of chirp basis sets with linear variation of the frequency to expand the wavefunctions, in which plane-waves are a special case. The chirp characteristics of each basis set can be adjusted through a single parameter. Thus, the basis set cutoff can be optimized variationally. The approach is tested with the expansion of the electron wavefunctions in atoms, and it is shown that chirp basis sets substantially improve the convergence in the description of the electron density. We have found that the natural oscillations of the electron core states are efficiently described in chirp-waves.

Iterative Solution of Nonlinear Dynamical System of an Electrostatically Actuated Micro-Cantilever

This paper presents an iterative method to solve the nonlinear dynamical system ofan electrostatically actuated micro-cantilever in MEMS. Different from someother iteration methods for nonlinear dynamical systems, no additionalprocedures have to be carried out to eliminate the so-called secular terms. Inaddition, an efficient approach is proposed to expand fractional functions intotruncated Fourier series. That makes it possible to implement the iterativeprocedures. Importantly, all the iteration equations are purely linear. Numericalexamples show that the solutions provided by the method are in excellentagreement with numerical ones. The effects of applied voltage, cubic nonlinearstiffness and squeeze film damping on vibration responses are discussed indetails.

Analysis of Fractal Wave Equations by Local Fractional Fourier Series Method

The fractal wave equations with local fractional derivatives are investigated in this paper. The analytical solutions are obtained by using local fractional Fourier series method. The present method is very efficient and accurate to process a class of local fractional differential equations.

Mappings for Special Functions on Cantor Sets and Special Integral Transforms via Local Fractional Operators

The mappings for some special functions on Cantor sets are investigated. Meanwhile, we apply the local fractional Fourier series, Fourier transforms, and Laplace transforms to solve three local fractional differential equations, and the corresponding nondifferentiable solutions were presented.

1-D heat conduction in a fractal medium: A solution by the local fractional Fourier series method

In this communication 1-D heat conduction in a fractal medium is solved by the local fractional Fourier series method. The solution developed allows relating the basic properties of the fractal medium to the local heat transfer mechanism.

Local Fractional Fourier Series with Application to Wave Equation in Fractal Vibrating String

We introduce the wave equation in fractal vibrating string in the framework of the local fractional calculus. Our particular attention is devoted to the technique of the local fractional Fourier series for processing these local fractional differential operators in a way accessible to applied scientists. By applying this technique we derive the local fractional Fourier series solution of the local fractional wave equation in fractal vibrating string and show the fundamental role of the Mittag‐Leffler function.

Sampling of fractional bandlimited signals associated with fractional Fourier transform

Abstract Fractional Fourier transform (FRFT) plays an important role in many fields of optics and signal processing. This paper considers the problem of reconstructing a fractional bandlimited signal with FRFT. We propose a novel reconstruction method for fractional bandlimited signals using the fractional Fourier series (FRFS). The advantage is that the sampling expansion can be deduced directly not based on the Shannon theorem. By utilizing the generalized form of Parseval’s relation for complex FRFS, we obtain the sampling expansion for fractional bandlimited signals with FRFT. We show that the sampling expansion for fractional bandlimited signals with FRFT is a special case of Parseval’s relation for complex FRFS.

Gibbs phenomenon for fractional Fourier series

It is well known that the partial sums of a Fourier series of a non-periodic analytic function on a finite interval exhibit spurious oscillations near the interval boundaries. This phenomenon is known as the Gibbs effect. The authors show that a similar phenomenon is observed for the fractional Fourier series (FrFS) of a function with jump discontinuities. The convergence of FrFS is discussed and proved in a theorem. Specifically, the present work proves the uniform convergence of the FrFS for a non-periodic analytic function in the smooth region. The maximum amplitude of the oscillations for the FrFS remains constant (the Gibbs constant), similar to that for a classical Fourier series expansion. Finally, three numerical examples are investigated to demonstrate that the Gibbs constant for an FrFS is the same as for a Fourier series.

Research progress on discretization of fractional Fourier transform

As the fractional Fourier transform has attracted a considerable amount of attention in the area of optics and signal processing, the discretization of the fractional Fourier transform becomes vital for the application of the fractional Fourier transform. Since the discretization of the fractional Fourier transform cannot be obtained by directly sampling in time domain and the fractional Fourier domain, the discretization of the fractional Fourier transform has been investigated recently. A summary of discretizations of the fractional Fourier transform developed in the last nearly two decades is presented in this paper. The discretizations include sampling in the fractional Fourier domain, discrete-time fractional Fourier transform, fractional Fourier series, discrete fractional Fourier transform (including 3 main types: linear combination-type; sampling-type; and eigen decomposition-type), and other discrete fractional signal transform. It is hoped to offer a doorstep for the readers who are interested in the fractional Fourier transform.

Fractional Fourier series expansion for finite signals and dual extension to discrete-time fractional Fourier transform

Conventional Fourier analysis has many schemes for different types of signals. They are Fourier transform (FT), Fourier series (FS), discrete-time Fourier transform (DTFT), and discrete Fourier transform (DFT). The goal of this article is to develop two absent schemes of fractional Fourier analysis methods. The proposed methods are fractional Fourier series (FRFS) and discrete-time fractional Fourier transform (DTFRFT), and they are the generalizations of Fourier series (FS) and discrete-time Fourier transform (DTFT), respectively.