Lie Symmetry Analysis and Conservation Laws of a Generalized Time Fractional Foam Drainage Equation

Wang, Li; Tian, Shou‐Fu; Zhao, Zhentao; Song, Xiaoqiu

doi:10.1088/0253-6102/66/1/035

Cited by 14 publications

(5 citation statements)

References 41 publications

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“…, 2013), the generalized homotopy analysis method (Iyiola et al. , 2013), the Lie group analysis method (Wang et al. , 2016), and the new fractional Riccati equation rational expansion method (Zhang and Feng, 2013) are used to deal with the Foam drainage equation.…”

Section: Introductionmentioning

confidence: 99%

Fractional-order view analysis of Fisher’s and foam drainage equations within Aboodh transform

Alshehry,

Yasmin,

Shah

et al. 2024

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PurposeThe purpose of this study is to solve two unique but difficult partial differential equations: the foam drainage equation and the nonlinear time-fractional fisher’s equation. Through our methods, we aim to provide accurate solutions and gain a deeper understanding of the intricate behaviors exhibited by these systems.Design/methodology/approachIn this study, we use a dual technique that combines the Aboodh residual power series method and the Aboodh transform iteration method, both of which are combined with the Caputo operator.FindingsWe develop exact and efficient solutions by merging these unique methodologies. Our results, presented through illustrative figures and data, demonstrate the efficacy and versatility of the Aboodh methods in tackling such complex mathematical models.Originality/valueOwing to their fractional derivatives and nonlinear behavior, these equations are crucial in modeling complex processes and confront analytical complications in various scientific and engineering contexts.

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Section: Introductionmentioning

confidence: 99%

Fractional-order view analysis of Fisher’s and foam drainage equations within Aboodh transform

Alshehry,

Yasmin,

Shah

et al. 2024

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“…There are many methods to construct the exact solutions of FPDEs such as similarity reduction method [17][18][19][20][21][22][23][24][25], ¢ G G method [26][27][28], tanh method [29,30], Extended sinh-Gordon equation expansion method (EShGeem) [31], Jacobi elliptic function method [32,33] and Kudryashov method [34,35] etc. The Lie symmetry reduction method is an efficient method to obtain the exact solutions of NLPDEs.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation analysis, chaotic analysis and diverse optical soliton solutions of time-fractional (2+1) dimensional generalized Camassa-Holm Kadomtsev-Petviashvili equation arising in shallow water waves

Gupta

2023

Phys. Scr.

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This paper presents a study on (2+1) generalized Camassa-Holm Kadomtsev-Petviashvili equation, which is used to describe the behavior of shallow water waves in nonlinear media. The considered equation provides a more accurate description of wave behavior compared to linear wave equations and can account for wave breaking and other nonlinear effects. This model can be used to describe and study the behavior of nonlinear waves such as rogue waves in complex fluid dynamics scenarios. This includes the behavior of waves in stratified fluids, nonlinear dispersive media and wave interactions in fluid flows with varying velocities and densities. Bifurcation analysis of the governing equation has been performed using the planar dynamical system method. The chaotic behavior of the dynamical system has been examined by utilizing various techniques such as time series analysis and construction of 2D and 3D phase space trajectories. Furthermore, introduction of perturbed term has resulted in the observation of chaotic and quasi-periodic behaviors across a range of parameter values. The considered equation has been reduced to ordinary differential equation by performing symmetry reduction. Kudryashov method has been used to obtain exact solution of reduced equation. Single soliton solution of governed equation has been obtained by using Hirota method and impact of fractional parameter on the obtained solution has been studied using graphical representation. The extended sinh-Gordon equation expansion method and modified generalized exponential rational function method have been exploited to obtain dark, bright and singular soliton of considered equation. The motivation for this study arises from need to understand and analyze the complex dynamics of shallow water waves in nonlinear media with a particular focus on governing equation. By performing symmetry reduction and applying various analytical methods, we aim to unravel the intricate behavior and soliton solutions of considered equation, contributing to broader understanding of nonlinear wave phenomena. 

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“…Furthermore, there are many methods to construct solutions for FPDEs. [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] Here, we considered the three-dimensional non-linear time-fractional generalized Z-K equation in the following form:…”

Section: Introductionmentioning

confidence: 99%

“…These definitions enable us to construct the partial differential equations of fractional order (FPDEs), which are used to modulate the non‐linear phenomena in different fields supported by explaining their behaviors and physical properties. Furthermore, there are many methods to construct solutions for FPDEs 18–34 …”

Section: Introductionmentioning

confidence: 99%

The time‐fractional generalized Z‐K equation: Analysis of Lie group, similarity reduction, conservation laws, and explicit solutions

Al-Denari

Ahmed

Tharwat

2022

Math Methods in App Sciences

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In this paper, we consider the three-dimensional non-linear time-fractional generalized Zakharov-Kuznetsov (Z-K) equation that describes the dust-ion acoustic solitary waves in a magnetized dusty plasma. Based on the definition of the Riemann-Liouville (R-L) fractional derivatives, the analysis of the Lie group method can be applied to find the symmetries for the considered equation. After that, these symmetries enable us to transform the equation under study into a two-dimensional non-linear ordinary differential equation of fractional order in the sense of ErdLélyi-Kober (E-K) fractional operator. Also, we obtain a set of new analytical solutions for the time-fractional generalized Z-K equation via the power series and the fractional sub-equation methods. Furthermore, we compute the conservation laws for this equation with a detailed derivation based on the formal Lagrangian.

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Lie Symmetry Analysis and Conservation Laws of a Generalized Time Fractional Foam Drainage Equation

Cited by 14 publications

References 41 publications

Fractional-order view analysis of Fisher’s and foam drainage equations within Aboodh transform

Fractional-order view analysis of Fisher’s and foam drainage equations within Aboodh transform

Bifurcation analysis, chaotic analysis and diverse optical soliton solutions of time-fractional (2+1) dimensional generalized Camassa-Holm Kadomtsev-Petviashvili equation arising in shallow water waves

The time‐fractional generalized Z‐K equation: Analysis of Lie group, similarity reduction, conservation laws, and explicit solutions

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