Exact solutions to Euler equation and Navier–Stokes equation

“…In recent six decades, scholars such as Ovsyannikov, Olver, Ibragimov, Miao and Hu et al [2][3][4][5][6] have sought to, but not limited to improve the way of searching optimal system. The impact of Lie symmetry's thought simultaneously spread more widely [7][8][9].…”

Section: Introductionmentioning

confidence: 99%

Lie Symmetry Analysis and Conservation Laws for the (2 + 1)-Dimensional Dispersionless B-Type Kadomtsev–Petviashvili Equation

Zhao

Wang

et al. 2022

J Nonlinear Math Phys

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The Lie symmetry analysis is adopted to the (2 + 1)-dimensional dispersionless B-type Kadomtsev–Petviashvili (dBKP) equation. The combination of symmetry analysis and symbolic computing methods proves that Lie algebra of infinitesimal symmetry of the dBKP equation depends on four independent arbitrary functions and one arbitrary parameter. The Lie algebra is reduced to four classes for deriving commutative relations, group invariant solutions of dBKP equation and conservation laws, and the optimal system of 1-dimensional subalgebras from one class is constructed. Based on the optimal system and other particular infinitesimal symmetries, plentiful symmetry reductions and invariant solutions are computed by using Lie group method. Six successive symmetries and conserved quantities of the dBKP equation are linked by the new conservation theorem. Besides, exact solution of the dBKP equation is constructed according to a conservation vector.

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“…In this context, we indicate that Lie symmetry named as nonlocal symmetry is a very useful tool. Partial invariance method, weak symmetry method, and conditional Lie‐Backlund symmetry method are some of the other branches of Lie symmetry. In order to solve fractional PDEs apart from usual methods, such as power series method and kernel Hilbert space method, one can also apply Lie symmetry technique .…”

Section: Introductionmentioning

confidence: 99%

Optimal classification, exact solutions, and wave interactions of Euler system with large friction

Sahoo

Sekhar

2020

Math Methods in App Sciences

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We derive exact solutions of one-dimensional Euler system that accounts for gravity together with large friction. Certain optimal classes of subalgebra using Lie symmetry analysis are obtained for this system. We apply the reduction procedure to reduce the Euler system to a system of ordinary differential equations in terms of new similarity variable for each class of subalgebras leading to invariant solutions. The evolution of characteristic shock and its interaction with the weak discontinuity by using one of the invariant solutions is studied. Further, the properties of reflected and transmitted waves and jump in acceleration influenced by the incident wave have been characterized. KEYWORDS euler system with large friction, invariant solution, lie symmetry analysis, optimal classification, wave interactions, weak discontinuity MSC CLASSIFICATION 35A30; 35C05; 35L60; 35Q35; 76D33 Math Meth Appl Sci. 2020;43:5744-5757. wileyonlinelibrary.com/journal/mma

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Exact solutions to Euler equation and Navier–Stokes equation

Cited by 19 publications

References 38 publications

Some Exact Solutions of Compressible and Incompressible Euler Equations

Some Exact Solutions of Compressible and Incompressible Euler Equations

Lie Symmetry Analysis and Conservation Laws for the (2 + 1)-Dimensional Dispersionless B-Type Kadomtsev–Petviashvili Equation

Optimal classification, exact solutions, and wave interactions of Euler system with large friction

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