Similarity solutions to non-linear partial differential equation of physical phenomena represented by the Zakharov–Kuznetsov equation

Moussa, M. H. M.

doi:10.1016/s0020-7225(01)00016-7

Cited by 25 publications

(18 citation statements)

References 11 publications

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“…where µ is known function and v, p, z are arbitrary functions in their arguments. Formula (22) defines the Ansatz that maps (21) into a partial differential equation with two independent variables p and z, and w being the dependent variable. Below in case 1 of the reductions with respect to 1-dimensional subalgebras we consider this procedure in details.…”

Section: Reductions With Respect To 1-dimensional Subalgebrasmentioning

confidence: 99%

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Algorithmic framework for group analysis of differential equations and its application to generalized Zakharov–Kuznetsov equations

Huang

Иванова

2016

Journal of Differential Equations

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In this paper, we explain in more details the modern treatment of the problem of group classification of (systems of) partial differential equations (PDEs) from the algorithmic point of view. More precisely, we revise the classical Lie-Ovsiannikov algorithm of construction of symmetries of differential equations, describe the group classification algorithm and discuss the process of reduction of (systems of) PDEs to (systems of) equations with smaller number of independent variables in order to construct invariant solutions. The group classification algorithm and reduction process are illustrated by the example of the generalized Zakharov-Kuznetsov (GZK) equations of form u t + (F (u)) xxx + (G(u)) xyy + (H(u)) x = 0. As a result, a complete group classification of the GZK equations is performed and a number of new interesting nonlinear invariant models which have non-trivial invariance algebras are obtained. Lie symmetry reductions and exact solutions for two important invariant models, i.e., the classical and modified Zakharov-Kuznetsov equations, are constructed. The algorithmic framework for group analysis of differential equations presented in this paper can also be applied to other nonlinear PDEs.

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Section: Reductions With Respect To 1-dimensional Subalgebrasmentioning

confidence: 99%

“…As the computations are very similar to the ones from the previous subsection, we skip all technical details and summarize the results only. (Some of the reduced with respect to 1dimensional subalgebras equations together with their solutions can be found also in [22].) The table of commutation relations is…”

Section: Zakharov-kuznetsov Equationmentioning

confidence: 99%

Algorithmic framework for group analysis of differential equations and its application to generalized Zakharov–Kuznetsov equations

Huang

Иванова

2016

Journal of Differential Equations

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“…In this section, we apply the extended generalized Riccati equation mapping method to construct travelling wave solution of the (2+1)-dimensional Generalized Zakharov-Kuznetsov (GZK) equation given by [22][23],…”

Section: Applicationmentioning

confidence: 99%

Travelling wave solutions of the generalized Zakharov-Kuznetsov equation via the extended generalized Riccati equation mapping method

Kolebaje¹,

Akinyemi²,

Obende

2013

International Journal of Advanced Mathematical Sciences

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Exact hyperbolic, trigonometric and rational travelling wave solutions to the (2+1)-dimensional Generalized ZakharovKuznetsov (GZK) equation via the extended generalized Riccati equation mapping method are presented in this paper. The twenty one travelling wave solutions obtained were verified by putting them back into the GZK equation with the aid of Mathematica. This shows that the extended generalized Riccati equation mapping method is a powerful tool for finding exact solution to nonlinear partial differential equations in physics, mathematics and other applications.

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“…The symmetry method is an alternative and systematic approach for the determination of the solutions of linear and nonlinear, single and a system of partial differential equations. Originally developed by Sophus Lie [6], the technique has found an important place in the literature of group theoretic methods, see [7][8][9][10][11][12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Similarity reductions and similarity solutions for the generalized diffusion equation

Moussa

Al-Halim

2015

Math Methods in App Sciences

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Herein, the generalized diffusion equation that encompasses the nonlinear diffusion equation with a source term and the Boussinesq equation in hydrology as its particular form and appears in a wide variety of physical and engineering applications has been analyzed via symmetry method that was developed by Steinberg. According to physical situations, in each case, the similarity variables obtained have led us to an ordinary differential equation, and we acquire some new solutions by solving the ODEs. Copyright © 2015 John Wiley & Sons, Ltd

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Similarity solutions to non-linear partial differential equation of physical phenomena represented by the Zakharov–Kuznetsov equation

Cited by 25 publications

References 11 publications

Algorithmic framework for group analysis of differential equations and its application to generalized Zakharov–Kuznetsov equations

Algorithmic framework for group analysis of differential equations and its application to generalized Zakharov–Kuznetsov equations

Travelling wave solutions of the generalized Zakharov-Kuznetsov equation via the extended generalized Riccati equation mapping method

Similarity reductions and similarity solutions for the generalized diffusion equation

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