Enhanced Symmetry Analysis of Two-Dimensional Burgers System

Kontogiorgis, Stavros; Popovych, Roman O.; Sophocleous, Christodoulos

doi:10.1007/s10440-018-0215-9

Cited by 19 publications

(14 citation statements)

References 47 publications

(173 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Some results on ordinary differential equations are presented in Leach 21 and Mahomed, 22 while there are various studies and for elliptic partial differential equations; for instance, see other works 23–25 and references therein. As far as the application of Lie symmetries on hyperbolic partial differential equations is concerned, there are various of studies in all areas of applied mathematics 26–29 . In theory of fluid dynamics Lie symmetries have played an important role on the determination of exact and analytic solutions.…”

Section: Introductionmentioning

confidence: 99%

Similarity solutions for two‐phase fluids models

Paliathanasis

2021

Math Methods in App Sciences

View full text Add to dashboard Cite

The algebraic properties of drift–flux two‐phase fluids models without gravitational and wall friction forces are studied. More precisely, for the two fluids we consider an equation of state of polytropic gases. We perform a classification scheme of the unknown parameters of the model to determine all the possible admitted Lie symmetries. We find that in the most general case the dynamical system of hyperbolic equations is invariant under the action of a four‐dimensional Lie algebra, while the larger number of admitted Lie symmetries is 6. For each admitted Lie algebra the one‐dimensional optimal system is derived which is applied for the determination of all the unique similarity transformations which lead to similarity solutions. Our results are compared with that of previous studies from where we see that most of the solutions presented in this study have not been found before in the literature.

show abstract

Section: Introductionmentioning

confidence: 99%

Similarity solutions for two‐phase fluids models

Paliathanasis

2021

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…As far as the application of Lie symmetries on hyperbolic partial differential equations is concerned, there are various of studies in all areas of applied mathematics [26][27][28][29]. In theory of fluid dynamics Lie symmetries have played an important role on the determination of exact and analytic solutions.…”

Section: Introductionmentioning

confidence: 99%

Similarity solutions for two-phase fluids models

Paliathanasis

2021

Preprint

View full text Add to dashboard Cite

The algebraic properties of drift-flux two-phase fluids models without gravitational and wall friction forces are studied. More precisely, for the two fluids we consider equation of states of polytropic gases.We perform a classification scheme of the unknown parameters of the model such that to determine all the possible admitted Lie symmetries. We find that in the most general case the dynamical system of hyperbolic equations is invariant under the action of a four dimensional Lie algebra, while the larger number of admitted Lie symmetries is six. For each admitted Lie algebra the one-dimensional optimal system is derived which is applied for the determination of all the unique similarity transformations which lead to similarity solutions. Our results are compared with that of previous studies from where we see that most of the solutions presented in this study have not found before in the literature.

show abstract

“…To comprehensively carry out Lie reductions of (1) to partial differential equations with two independent variables and to ordinary differential equations in Sections 3 and 4, respectively, we first classify both one-and two-dimensional subalgebras of the algebra g. We use the optimized Lie reduction technique that was developed in [26,27,47] and adapted for multidimensional Burgers-like models in [36]. The optimization is achieved due to a special selection of representatives of classes of inequivalent subalgebras and of the form of related ansatzes.…”

Section: Introductionmentioning

confidence: 99%

Extended symmetry analysis of two-dimensional degenerate Burgers equation

Vaneeva¹,

Popovych²,

Sophocleous³

2019

Preprint

Self Cite

View full text Add to dashboard Cite

We carry out enhanced symmetry analysis of a two-dimensional degenerate Burgers equation. Its complete point-symmetry group is found using the algebraic method, and its generalized symmetries are proved to be exhausted by its Lie symmetries. We also prove that the space of conservation laws of this equation is infinite-dimensional and is naturally isomorphic to the solution space of the (1+1)-dimensional backward linear heat equation. Lie reductions of the two-dimensional degenerate Burgers equation are comprehensively studied in the optimal way and its new Lie invariant solutions are constructed. We additionally consider solutions that also satisfy an analogous nondegenerate Burgers equation. In total, we construct four families of solutions of two-dimensional degenerate Burgers equation that are expressed in terms of solutions of the (1+1)-dimensional linear heat equation. Various kinds of hidden symmetries and hidden conservation laws (local and potential ones) are discussed as well. As by-product, we exhaustively describe generalized symmetries, cosymmetries and conservation laws of the transport equation, also called the inviscid Burgers equation, and construct new invariant solutions of the nonlinear diffusion equation with power nonlinearity of degree −1/2.

show abstract

Enhanced Symmetry Analysis of Two-Dimensional Burgers System

Cited by 19 publications

References 47 publications

Similarity solutions for two‐phase fluids models

Similarity solutions for two‐phase fluids models

Similarity solutions for two-phase fluids models

Extended symmetry analysis of two-dimensional degenerate Burgers equation

Contact Info

Product

Resources

About