Multi-dimensional quasi-simple waves in weakly dissipative flows

Rajaee, L.; Eshraghi, Homayoon; Popovych, Roman O.

doi:10.1016/j.physd.2007.09.010

Cited by 13 publications

(17 citation statements)

References 17 publications

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“…Another related problem, is to derive exact solutions for these two classes of partial differential equations using non-Lie ansatzes. Such ansatz was used in [20] to derive exact solutions for the two-dimensional Burgers equation (3). Furthermore Lie and non-Lie ansatzes can be applied to three-dimensional Burgers equation.…”

Section: Resultsmentioning

confidence: 99%

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Group analysis of(2+1)- and(3+1)-dimensional diffusion–convection e

Demetriou

Иванова

Sophocleous

2008

Journal of Mathematical Analysis and Applications

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Section: Resultsmentioning

confidence: 99%

“…Both approaches are examined in the following two subsections. The first approach was also employed in [9,20]. …”

Section: Similarity Reductions For (2 + 1)-dimensional Burgers Equationmentioning

confidence: 99%

Group analysis of(2+1)- and(3+1)-dimensional diffusion–convection e

Demetriou

Иванова

Sophocleous

2008

Journal of Mathematical Analysis and Applications

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“…Such procedure is often called antireduction. Thus for arbitrary values of A and B u = p(t, y)x + r(t, y) + m(t, y) x + n(t, y) Partial cases of this substitution with m = n = 0 or p = r = 0 for the constant coefficient Burgers equation have been studied in [34] and [4] correspondingly. Consider the constant-coefficient Burgers equation…”

Section: Non-lie Solutionsmentioning

confidence: 99%

“…In [34], a partial case of substitution (3.9) with m = n = 0 has been proposed for (3.11) and some solutions of this form were found, in particular,…”

Section: Non-lie Solutionsmentioning

confidence: 99%

Lie group analysis of two-dimensional variable-coefficient Burgers equation

Иванова

Sophocleous

Tracinà

2010

Z. Angew. Math. Phys.

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The modern group analysis of differential equations is used to study a class of two-dimensional variable coefficient Burgers equations. The group classification of this class is performed. Equivalence transformations are also found that allow us to simplify the results of classification and to construct the basis of differential invariants and operators of invariant differentiation. Using equivalence transformations, reductions with respect to Lie symmetry operators and certain non-Lie ansätze, we construct exact analytical solutions for specific forms of the arbitrary elements. Finally, we classify the local conservation laws.

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“…which was derived in [40] as an equation for the wave phase of two-dimensional sound simple waves in weakly dissipative flows. Therein, symmetry analysis of this equation was carried out, which included the first exhaustive study of its Lie reductions in an optimized way and the construction of several new families of its exact solutions.…”

Section: Introductionmentioning

confidence: 99%

Enhanced Symmetry Analysis of Two-Dimensional Burgers System

2018

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We carry out enhanced symmetry analysis of a two-dimensional Burgers system. The complete point symmetry group of this system is found using an enhanced version of the algebraic method. Lie reductions of the Burgers system are comprehensively studied in the optimal way and new Lie invariant solutions are constructed. We prove that this system admits no local conservation laws and then study hidden conservation laws, including potential ones. Various kinds of hidden symmetries (continuous, discrete and potential ones) are considered for this system as well. We exhaustively describe the solution subsets of the Burgers system that are its common solutions with its inviscid counterpart and with the two-dimensional Navier-Stokes equations. Using the method of differential constraints, which is particularly efficient for the Burgers system, we construct a number of wide families of solutions of this system that are expressed in terms of solutions of the (1+1)-dimensional linear heat equation although they are not related to the well-known linearizable solution subset of the Burgers system.

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Multi-dimensional quasi-simple waves in weakly dissipative flows

Cited by 13 publications

References 17 publications

Group analysis of(2+1)- and(3+1)-dimensional diffusion–convection e

Group analysis of(2+1)- and(3+1)-dimensional diffusion–convection e

Lie group analysis of two-dimensional variable-coefficient Burgers equation

Enhanced Symmetry Analysis of Two-Dimensional Burgers System

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