Abstract
We derive the one-dimensional optimal system for a system of three partial differential equations, which describe the two-dimensional rotating ideal gas with polytropic parameter γ > 2 . The Lie symmetries and the one-dimensional optimal system are determined for the nonrotating and rotating systems. We compare the results, and we find that when there is no Coriolis force, the system admits eight Lie point symmetries, while the rotating system admits seven Lie point symmetries. Consequently, the two systems are not algebraic equivalent as in the case of γ = 2 , which was found by previous studies. For the one-dimensional optimal system, we determine all the Lie invariants, while we demonstrate our results by reducing the system of partial differential equations into a system of first-order ordinary differential equations, which can be solved by quadratures.
Highlights
A powerful mathematical treatment for the determination of exact solutions for nonlinear differential equations is the Lie symmetry analysis [1,2,3]
We focus on the classification of the one-dimensional optimal system for the two-dimensional rotating ideal gas system described by the following system of partial differential equations (PDEs) [16,17,18]: ht + x +y ut + uu x + vuy + hγ−2 h x − f v vt + uv x + vvy + h γ −2 hy + f u where u and v are the velocity components in the x and y directions, respectively, h is the density of the ideal gas, f is the Coriolis parameter, and γ is the polytropic parameter of the fluid
X ( F ) = 0, where F is a function, provides the Lie invariants where by replacing in the differential equation H A = 0, we reduce the number of the independent variables or the order of the differential equation (in the case of ordinary differential equations (ODEs))
Summary
A powerful mathematical treatment for the determination of exact solutions for nonlinear differential equations is the Lie symmetry analysis [1,2,3]. The reduction process is based on the existence of functions that are invariant under a specific group of point transformations When someone uses these invariants as new dependent and independent variables, the differential equation is reduced. Lie symmetries can be used to construct new similarity solutions for a given differential equation by applying the adjoint representation of the Lie group [10]. We focus on the classification of the one-dimensional optimal system for the two-dimensional rotating ideal gas system described by the following system of PDEs [16,17,18]: ht + (hu) x + (hv)y. We found that in total, there are twenty-three one-dimensional independent Lie symmetries and possible reductions, and the corresponding invariants are determined and presented in tables. In Appendix A, we present the tables, which include the results of our analysis
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