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On the Invalidity of Fourier Series Expansions of Fractional Order

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Veröffentlicht/Copyright: 5. Dezember 2015
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Fractional Calculus and Applied Analysis
Aus der Zeitschrift Fractional Calculus and Applied Analysis Band 18 Heft 6

Abstract

The purpose of this short paper is to show the invalidity of a Fourier series expansion of fractional order as derived by G. Jumarie in a series of papers. In his work the exponential functions einωx are replaced by the Mittag-Leffler functions Eα (i(nωx)α) , over the interval [0,Mα/ω] where 0 < ω < ∞ and Mα > 0 is the period of the function Eα (ixα) , i.e., Eα (ixα) = Eα (i(x +Mα)α) .

He showed that any smooth periodic function f with period Mα/ω can be expanded in a Fourier-type series. We will show that the only possible period of the function Eα (ixα) is Mα = 0; hence the invalidity of any Fourier-type series expansion of f.

Keywords: fractional derivative; Fourier series of fractional order; Mittag-Leffler function

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Received: 2015-7-23
Revised: 2015-8-10
Published Online: 2015-12-5
Published in Print: 2015-12-1

© Diogenes Co., Sofia

Artikel in diesem Heft

  1. Contents – Fcaa, Vol. 18, No 6 (2015)
  2. FCAA Related News, Events and Books (Fcaa–Volume 18–6–2015)
  3. Nonexistence of Solutions of Some Non-Linear Non-Local Evolution Systems on the Heisenberg Group
  4. Necessary Conditions to Solve Fractional Order Wave Equations Using Traditional Laplace Transforms
  5. Identification and Fractional Super-Twisting Robust Control of IPMC Actuators
  6. Controllability of Abstract Systems of Fractional Order
  7. Dissipativity and Stability Analysis for Fractional Functional Differential Equations
  8. A Convergent Algorithm for Solving Higher-Order Nonlinear Fractional Boundary Value Problems
  9. Sliding Mode Control for a Class of Sub-Systems with Fractional Order Varying Trajectory Dynamics
  10. Implicit Difference Scheme of the Space-Time Fractional Advection Diffusion Equation
  11. Multiplicity of Nontrivial Solutions for Boundary Value Problem for Impulsive Fractional Differential Inclusions Via Nonsmooth Critical Point Theory
  12. Global Padé Approximations of the Generalized Mittag-Leffler Function and its Inverse
  13. On the Invalidity of Fourier Series Expansions of Fractional Order
  14. Fractional State Space Analysis of Temperature Time Series
https://doi.org/10.1515/fca-2015-0087
Schlagwörter für diesen Artikel
fractional derivative; Fourier series of fractional order; Mittag-Leffler function

Artikel in diesem Heft

  1. Contents – Fcaa, Vol. 18, No 6 (2015)
  2. FCAA Related News, Events and Books (Fcaa–Volume 18–6–2015)
  3. Nonexistence of Solutions of Some Non-Linear Non-Local Evolution Systems on the Heisenberg Group
  4. Necessary Conditions to Solve Fractional Order Wave Equations Using Traditional Laplace Transforms
  5. Identification and Fractional Super-Twisting Robust Control of IPMC Actuators
  6. Controllability of Abstract Systems of Fractional Order
  7. Dissipativity and Stability Analysis for Fractional Functional Differential Equations
  8. A Convergent Algorithm for Solving Higher-Order Nonlinear Fractional Boundary Value Problems
  9. Sliding Mode Control for a Class of Sub-Systems with Fractional Order Varying Trajectory Dynamics
  10. Implicit Difference Scheme of the Space-Time Fractional Advection Diffusion Equation
  11. Multiplicity of Nontrivial Solutions for Boundary Value Problem for Impulsive Fractional Differential Inclusions Via Nonsmooth Critical Point Theory
  12. Global Padé Approximations of the Generalized Mittag-Leffler Function and its Inverse
  13. On the Invalidity of Fourier Series Expansions of Fractional Order
  14. Fractional State Space Analysis of Temperature Time Series
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