On the Invalidity of Fourier Series Expansions of Fractional Order

Massopust, Peter; Zayed, Ahmed I.

doi:10.1515/fca-2015-0087

Cited by 7 publications

(3 citation statements)

References 20 publications

(11 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Existence of the exact solution of the nonlinear Sawada-Kotera equation modeled with Caputo time fractional derivative is proven in this section. Recall that similar analysis was performed in other works [22,23]. We focus our attention on the following Sawada-Kotera model expressed as…”

Section: Analysis Using the Caputo Fractional Derivative (Cd)mentioning

confidence: 94%

Shallow Water Wave Models with and without Singular Kernel: Existence, Uniqueness, and Similarities

Goufo¹,

Kumar²

2017

Mathematical Problems in Engineering

View full text Add to dashboard Buy / Rent full text

After the recent introduction of the Caputo-Fabrizio derivative by authors of the same names, the question was raised about an eventual comparison with the old version, namely, the Caputo derivative. Unlike Caputo derivative, the newly introduced Caputo-Fabrizio derivative has no singular kernel and the concern was about the real impact of this nonsingularity on real life nonlinear phenomena like those found in shallow water waves. In this paper, a nonlinear Sawada-Kotera equation, suitable in describing the behavior of shallow water waves, is comprehensively analyzed with both types of derivative. In the investigations, various fixed-point theories are exploited together with the concept of Piccard K-stability. We are then able to obtain the existence and uniqueness results for the models with both versions of derivatives. We conclude the analysis by performing some numerical approximations with both derivatives and graphical simulations being presented for some values of the derivative order γ. Similar behaviors are pointed out and they concur with the expected multisoliton solutions well known for the Sawada-Kotera equation. This great observation means either of both derivatives is suitable to describe the motion of shallow water waves.

show abstract

Section: Analysis Using the Caputo Fractional Derivative (Cd)mentioning

confidence: 94%

Shallow Water Wave Models with and without Singular Kernel: Existence, Uniqueness, and Similarities

Goufo¹,

Kumar²

2017

Mathematical Problems in Engineering

View full text Add to dashboard Buy / Rent full text

show abstract

“…The ideal solution of the Equation ( 15) is 𝑈 (𝑥, 𝑡) = 𝐸 , (𝑥 − 𝑡 /2). The function 𝐸 , (𝑧) , is called the Mittag-Leffler function [39] and is described as 𝐸 , (𝑧) = Figure 2. A 1D plot of the absolute error between approximate (fx) and exact (sol) solutions is depicted on the left-hand for t = x changed in the solution, Equation ( 14).…”

Section: Technique For Approximating Solutionsmentioning

confidence: 99%

“…However, in Ref. [39] the authors state that, in fractional calculus, this kind of trigonometry identity does not hold. In this example, we have computationally proven that the above identity is no longer valid in fractional calculus, i.e.,…”

Section: Example 2: Consider Another Example Of Fractional-order Line...mentioning

confidence: 99%

Technique to Solve Linear Fractional Differential Equations Using B-Polynomials Bases

Bhatti¹,

Rahman²

2021

Fractal Fract

View full text Add to dashboard Buy / Rent full text

A multidimensional, modified, fractional-order B-polys technique was implemented for finding solutions of linear fractional-order partial differential equations. To calculate the results of the linear Fractional Partial Differential Equations (FPDE), the sum of the product of fractional B-polys and the coefficients was employed. Moreover, minimization of error in the coefficients was found by employing the Galerkin method. Before the Galerkin method was applied, the linear FPDE was transformed into an operational matrix equation that was inverted to provide the values of the unknown coefficients in the approximate solution. A valid multidimensional solution was determined when an appropriate number of basis sets and fractional-order of B-polys were chosen. In addition, initial conditions were applied to the operational matrix to seek proper solutions in multidimensions. The technique was applied to four examples of linear FPDEs and the agreements between exact and approximate solutions were found to be excellent. The current technique can be expanded to find multidimensional fractional partial differential equations in other areas, such as physics and engineering fields.

show abstract

On the Invalidity of Fourier Series Expansions of Fractional Order

Abstract: 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 e inωx are replaced by the Mittag-Leffler functions

Cited by 7 publications

References 20 publications

Shallow Water Wave Models with and without Singular Kernel: Existence, Uniqueness, and Similarities

Shallow Water Wave Models with and without Singular Kernel: Existence, Uniqueness, and Similarities

Technique to Solve Linear Fractional Differential Equations Using B-Polynomials Bases

Contact Info

Product

Resources

About