Exact Solutions of Newell-Whitehead-Segel Equations Using Symmetry Transformations

Bibi, Khudija; Ahmad, Khalid

doi:10.1155/2021/6658081

Cited by 12 publications

(5 citation statements)

References 15 publications

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“…The applications of Lie symmetry analysis have been used in many research for solving nonlinear partial dierential equations found in physical problems [2,26,4,7,3,6,32,8] and for mathematical Models such as SIR models [33]. But in this project, we put a signicant emphasis on discussing the idea of symmetries of ODEs and demonstrate how their features can be used to obtain the exact solutions.…”

Section: −2ξmentioning

confidence: 99%

Solution Methods for Nonlinear Ordinary Differential Equations Using Lie Symmetry Groups

Perera

Gallage²

2023

Adv. J. Grad. Res.

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For formulating mathematical models, engineering problems and physical problems, Nonlinear ordinary differential equations(NODEs) are used widely. Nevertheless, explicit solutions can be obtained for very few NODEs, due to lack of techniques to obtain explicit solutions. Therefore methods to obtain approximate solution for NODEs are used widely. Although symmetry groups of ordinary differential equations (ODEs) can be used to obtain exact solutions however, these techniques are not widely used. The purpose of this paper is to present applications of Lie symmetry groups to obtain exact solutions of NODEs . In this paper we connect different methods,theorems and definitions of Lie symmetry groups from different references and we solve first order and second order NODEs. In this analysis three different methods are used to obtain exact solutions of NODEs. Using applications of these symmetry methods, drawbacks and advantages of these different symmetry methods are discussed and some examples have been illustrated graphically. Focus is first placed on discussing about the notion of symmetry groups of the ODEs. Focus is then changed to apply them to find general solutions for NODEs under three different methods. First we find suitable change of variables that convert given first order NODE into variable separable form using these symmetry groups. As another method to find general solutions for first order NODEs, we find particular type of solution curves called invariant solution curves under Lie symmetry groups and we use these invariant solution curves to obtain the general solutions. We find general solutions for the second order NODEs by reducing their order to first order using Lie symmetry groups.

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Section: −2ξmentioning

confidence: 99%

Solution Methods for Nonlinear Ordinary Differential Equations Using Lie Symmetry Groups

Perera

Gallage²

2023

Adv. J. Grad. Res.

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“…[1]. Recently, several methods have been used to find exact solutions of nonlinear model equations like Cahn-Allen equation [2], ð2 + 1Þ -dimensional Date-Jimbo-Kashiwara-Miwa (DJKM) equation [3], Newell-Whitehead-Segel (NWS) equations [4], the Chaffee-Infante equation [5], DNA Peyrard-Bishop equation [6], Burger's equation [7], the ð2 + 1Þ-dimensional nonlinear Sharma-Tasso-Olver equation [8], and Ablowitz-Kaup-Newell-Segur water wave equation [9]. Recently, a number of concrete techniques have been recognized for finding accurate and comprehensible solutions of nonlinear physical models with the help of computer algebra, such as Maple, MATLAB, and Mathematica.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, a number of concrete techniques have been recognized for finding accurate and comprehensible solutions of nonlinear physical models with the help of computer algebra, such as Maple, MATLAB, and Mathematica. These include power ondex method [7,9], lie symmetry groups [3,4], new extended direct algebraic method, and the generalized Kudryashov method [10].…”

Section: Introductionmentioning

confidence: 99%

New Exact Solutions of Landau-Ginzburg-Higgs Equation Using Power Index Method

Ahmad

Bibi

Arif

et al. 2023

Journal of Function Spaces

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In the present study, the optimality approach is applied to find the exact solution of the Landau-Ginzburg-Higgs Equation (LGHE) using new transformations. This method is a direct algebraic method for obtaining exact solutions of nonlinear differential equations. We find suitable solutions of the LGHE in terms of elliptic Jacobi functions by applying transformations of basic functions. Exact solutions of the equations are obtained with the help of symbolic software (Maple) which allows the computation of equations with parameter constants. It is exposed that PIM is influential, suitable, and shortest and offers an exact solution of LGHE.

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“…In [9] and [10] Khalil and Khadija used the power index method to find new exact solutions of the Ablowitz-Kaup-Newell-Segur water wave equation and Burger's equation. Transformations of variables can sometimes be found that transform a nonlinear PDE into a nonlinear ODE such as Lie Symmetry Transformations [11], [12]. The phenomenon of a soliton with constant shape and speed was proposed by Russell in 1834, which he called a translational wave [8].…”

Section: Introductionmentioning

confidence: 99%

New Explicit Solutions of Zoomeron Equation Using Optimal Indexing Approach

Ahmad,

Naeem

2023

Preprint

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This paper presents a nonlinear physical model called the (2+1)-dimensional Zomeron equation, which is one of the equations that describes latent evolution. We apply the Power Index Method (PIM) to find analytical solutions of equations using function transformations with optimal indexing of one variable. After selecting index of the variable, then we adopt function transform of another variable that make sure all coefficient terms to change in new variable. PIM is a straightforward mathematical technique in which the indices of the independent variables are adjusted to our selected similarity transformations to effectively reduce the partial differential equation to an Ordinary Differential Equation (ODE). Several exact solutions of the nonlinear Zoumron equation have been built and are presented graphically for some values of the constant parameters. The proposed method has been effectively used to find new exact solutions of nonlinear PDE. Using these transforms, we can easily produce infinitesimal generators of the PDE.

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Exact Solutions of Newell-Whitehead-Segel Equations Using Symmetry Transformations

Cited by 12 publications

References 15 publications

Solution Methods for Nonlinear Ordinary Differential Equations Using Lie Symmetry Groups

Solution Methods for Nonlinear Ordinary Differential Equations Using Lie Symmetry Groups

New Exact Solutions of Landau-Ginzburg-Higgs Equation Using Power Index Method

New Explicit Solutions of Zoomeron Equation Using Optimal Indexing Approach

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