Solitary wave solutions of pZK equation using Lie point symmetries

Kumar, Dharmendra; Kumar, Sachin

doi:10.1140/epjp/s13360-020-00218-w

Cited by 54 publications

(15 citation statements)

References 37 publications

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“…The method can be assumed as a creative expansion of the well‐known generalized exponential rational function method (GERRM) 41–47 but with local derivatives. To exert the method in solving local fractional partial differential equations, the following process is required 40,48 : First, we consider the equation with partial derivatives of the following local type in terms of Λ ϵ ( ξ , τ ) as

scriptF false({normalΛ}_{ϵ} ((), ξ, τ), \frac{\partial^{ϵ} {normalΛ}_{ϵ} false(ξ, τ false)}{\partial τ^{ϵ}}, \frac{\partial^{ϵ} {normalΛ}_{ϵ} false(ξ, τ false)}{\partial ξ^{ϵ}}, \frac{\partial^{2 ϵ} {normalΛ}_{ϵ} false(ξ, τ false)}{\partial τ^{2 ϵ}}, \frac{\partial^{2 ϵ} {normalΛ}_{ϵ} false(ξ, τ false)}{\partial ξ^{2 ϵ}}, \dots false) = 0 .

With the aid of applying the transformations

{normalΛ}_{ϵ} ((), ξ, τ) = {normalΛ}_{ϵ} false(δ^{ϵ} false)

and

δ^{ϵ} = κ^{ϵ} ξ - ℏ^{ϵ} τ

in (), the following new differential equation with local derivative is then obtained:

scriptF false({normalΛ}_{ϵ} false(δ^{ϵ} false), - ℏ^{ϵ} \frac{d^{ϵ} {normalΛ}_{ϵ} false(δ^{ϵ} false)}{d δ^{ϵ}}, κ^{ϵ} d^{ϵ} {normalΛ}_{ϵ} false(<...>

…”

Section: The Main Methodsmentioning

confidence: 99%

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

Ghanbari

2021

Math Methods in App Sciences

129

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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.

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scriptF false({normalΛ}_{ϵ} ((), ξ, τ), \frac{\partial^{ϵ} {normalΛ}_{ϵ} false(ξ, τ false)}{\partial τ^{ϵ}}, \frac{\partial^{ϵ} {normalΛ}_{ϵ} false(ξ, τ false)}{\partial ξ^{ϵ}}, \frac{\partial^{2 ϵ} {normalΛ}_{ϵ} false(ξ, τ false)}{\partial τ^{2 ϵ}}, \frac{\partial^{2 ϵ} {normalΛ}_{ϵ} false(ξ, τ false)}{\partial ξ^{2 ϵ}}, \dots false) = 0 .

With the aid of applying the transformations

{normalΛ}_{ϵ} ((), ξ, τ) = {normalΛ}_{ϵ} false(δ^{ϵ} false)

and

δ^{ϵ} = κ^{ϵ} ξ - ℏ^{ϵ} τ

in (), the following new differential equation with local derivative is then obtained:

scriptF false({normalΛ}_{ϵ} false(δ^{ϵ} false), - ℏ^{ϵ} \frac{d^{ϵ} {normalΛ}_{ϵ} false(δ^{ϵ} false)}{d δ^{ϵ}}, κ^{ϵ} d^{ϵ} {normalΛ}_{ϵ} false(<...>

…”

Section: The Main Methodsmentioning

confidence: 99%

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

Ghanbari

2021

Math Methods in App Sciences

129

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“…The method used in this paper is inspired by the generalized exponential rational function method (GERRM), which is widely used to solve equations with standard partial derivatives 38‐44 . The only major difference is that all the steps of that method have been adapted to use local derivatives through some important properties outlined in Section 2.…”

Section: Description Of the New Methodsmentioning

confidence: 99%

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

Ghanbari

2020

Math Methods in App Sciences

126

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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.

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“…Recently, in Ref. [58], using two methods, Lie symmetry analysis and generalized exponential rational function. The authors present new exact solutions through optimal systems of one-dimensional Lie subalgebras; some solitary waves depict single soliton, multisoliton, and annihilation profiles and six other closed-form solutions.…”

Section: Introductionmentioning

confidence: 99%

Solutions of Generalized Fractional Perturbed Zakharov-Kuznetsov Equation Arising in a Magnetized Dusty Plasma

Akinyemi

2021

Rev. Mex. Fís.

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The generalized fractional perturbed (3+1)-dimensional Zakharov–Kuznetsov (PZK) equation which appear in the magnetized two-ion-temperature dusty plasma and quantum physics is considered. The sub-equation method in the conformable sense is proposed to obtained closed-form analytical solutions to this equation. The newly solutions obtained by the proposed method are kink-shape, multi-soliton, solitary wave, bell-shaped solitons, and periodic solutions that are substantial in the field of mathematical physics and can be of relevance in the field of plasma physics, also for future research.

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Solitary wave solutions of pZK equation using Lie point symmetries

Cited by 54 publications

References 37 publications

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

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

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

Solutions of Generalized Fractional Perturbed Zakharov-Kuznetsov Equation Arising in a Magnetized Dusty Plasma

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