Invariant Solutions for a Class of Perturbed Nonlinear Wave Equations

Ahmed, Waheed A.; Zaman, F. D.; Saleh, Khairul

doi:10.3390/math5040059

Mathematics

2017

DOI: 10.3390/math5040059

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Invariant Solutions for a Class of Perturbed Nonlinear Wave Equations

Waheed A. Ahmed¹,

F. D. Zaman²,

Khairul Saleh³

Abstract: Approximate symmetries of a class of perturbed nonlinear wave equations are computed using two newly-developed methods. Invariant solutions associated with the approximate symmetries are constructed for both methods. Symmetries and solutions are compared through discussing the advantages and disadvantages of each method.

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Cited by 2 publications

References 21 publications

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New Approximate Symmetry Theorems and Comparisons With Exact Symmetries

Pakdemirli

2024

Qeios

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Three new approximate symmetry theories are proposed. The approximate symmetries are contrasted with each other and with the exact symmetries. The theories are applied to nonlinear ordinary differential equations for which exact solutions are available. It is shown that from the symmetries, approximate solutions as well as exact solutions in some restricted cases can be retrievable. Depending on the specific approximate theory and the equations considered, the approximate symmetries may expand the Lie Algebra of the exact symmetries, may be a perturbed form of the exact symmetries or may be a subalgebra of the exact symmetries. Exact and approximate solutions are retrieved using the symmetries.

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New Approximate Symmetry Theorems and Comparisons With Exact Symmetries

Pakdemirli

2024

Qeios

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Relationship between Unstable Point Symmetries and Higher-Order Approximate Symmetries of Differential Equations with a Small Parameter

Tarayrah

Cheviakov

2021

Symmetry

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The framework of Baikov–Gazizov–Ibragimov approximate symmetries has proven useful for many examples where a small perturbation of an ordinary or partial differential equation (ODE, PDE) destroys its local exact symmetry group. For the perturbed model, some of the local symmetries of the unperturbed equation may (or may not) re-appear as approximate symmetries. Approximate symmetries are useful as a tool for systematic construction of approximate solutions. For algebraic and first-order differential equations, to every point symmetry of the unperturbed equation, there corresponds an approximate point symmetry of the perturbed equation. For second and higher-order ODEs, this is not the case: a point symmetry of the original ODE may be unstable, that is, not have an analogue in the approximate point symmetry classification of the perturbed ODE. We show that such unstable point symmetries correspond to higher-order approximate symmetries of the perturbed ODE and can be systematically computed. Multiple examples of computations of exact and approximate point and local symmetries are presented, with two detailed examples that include a fourth-order nonlinear Boussinesq equation reduction. Examples of the use of higher-order approximate symmetries and approximate integrating factors to obtain approximate solutions of higher-order ODEs are provided.

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Invariant Solutions for a Class of Perturbed Nonlinear Wave Equations

Cited by 2 publications

References 21 publications

New Approximate Symmetry Theorems and Comparisons With Exact Symmetries

New Approximate Symmetry Theorems and Comparisons With Exact Symmetries

Relationship between Unstable Point Symmetries and Higher-Order Approximate Symmetries of Differential Equations with a Small Parameter

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