A new class of travelling wave solutions for local fractional diffusion differential equations

The Moisil-Teodorescu operator is considered to be a good analogue of the usual Cauchy-Riemann operator of complex analysis in the framework of quaternionic analysis and it is a square root of the scalar Laplace operator in R 3. In the present work, a general quaternionic structure is developed for the local fractional Moisil-Teodorescu operator in Cantor-type cylindrical and spherical coordinate systems. Furthermore, in order to reveal the capacity and adaptability of the methods, we show two examples for the Helmholtz equation with local fractional derivatives on the Cantor sets by making use of the local fractional Moisil-Teodorescu operator.

“…In a similar way, substituting Equations (36) and (37) into Equation (42), we obtain the local fractional quaternionic Helmholtz equation in the Cantor-type spherical coordinates.…”

Section: The Cantor-type Cylindrical and Spherical-coordinate Methodsmentioning

confidence: 99%

Section: Few Aspects Of Local Fractional Calculusmentioning

confidence: 99%

“…Consider the coordinate system of the Cantor-type cylindrical coordinates developed in other sttudies. 26,36,40,41,43 Consider the Cantor-type cylindrical coordinates…”

Section: The Cantor-type Cylindrical Coordinate Systemmentioning

confidence: 99%

“…We begin our study by recalling some facts on local fractional calculus that can be found in other studies. 36,41 …”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Local fractional Moisil–Teodorescu operator in quaternionic setting involving Cantor‐type coordinate systems

Reyes

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2020

On novel nondifferentiable exact solutions to local fractional Gardner's equation using an effective technique

Ghanbari

2020

126

One of the most interesting branches of fractional calculus is the local fractional calculus, which has been used successfully to describe many fractal problems in science and engineering. The main purpose of this contribution is to construct a novel efficient technique to retrieve exact fractional solutions to local fractional Gardner's equation defined on Cantor sets by an effective numerical methodology. In the framework of this technique, first a set of elementary functions are defined on the contour set. Taking these functions into account, the general structure of the exact solution for the equation is suggested as a specific combination of the defined basis functions. By determining the unknown coefficients in this expansion and by placing them in the original equation, specific forms of solutions to the fractional equation are determined. Several interesting numerical simulations of the achieved results are also presented in the article to give a better understanding of the dynamic behavior of these results. The results obtained in this research confirm that the method used is very simple and efficient in terms of application. Moreover, it is accurate and free of any errors in terms of calculation. Therefore, it can be employed to handle other partial differential equations including local fractional derivatives.

Abundant exact solutions to a generalized nonlinear Schrödinger equation with local fractional derivative

Ghanbari

2021

129

The paper aims to employ a new effective methodology to build exact fractional solutions to the generalized nonlinear Schrödinger equation with a local fractional operator defined on Cantor sets. The equation contains group velocity dispersion and second‐order spatiotemporal dispersion coefficients. We obtain exact solutions of the equation via the generalized version of the exponential rational function method. This new version of the method uses a set of elementary functions that are adopted on the contour set. To study the dynamic behavior of the obtained results, extensive numerical simulations are provided. We observe that the employed method is simple but quite efficient for determining the exact solutions of the problem in the local sense. Moreover, they are practically compatible with solving various classes of nonlinear problems arising in mathematical physics. All computations are carried out using the Maple package.