Abstract
A complete symmetry group classification for the system of shallow water equations with the horizontal temperature gradient, also known as Ripa system, is presented. A rigorous and systematic procedure based on the general invariants of the adjoint representation is used to construct the one-dimensional optimal system of the Lie algebra. The complete inequivalence class of the group invariant solutions are obtained by using the one-dimensional optimal system. One such solution of the Ripa system is used to study the evolutionary behaviour of the discontinuity wave.
References
[1] G. W. Bluman and S. Anco, Symmetry and integration methods for differential equations, vol. 154, Springer Science & Business Media, New York, 2008.Suche in Google Scholar
[2] P. J. Olver, Applications of Lie groups to differential equations, Springer, New York, 1993.10.1007/978-1-4612-4350-2Suche in Google Scholar
[3] P. J. Olver and P. Rosenau, Group-invariant solutions of differential equations, SIAM J. Appl. Math. 47(2) (1987), 263–278.10.1137/0147018Suche in Google Scholar
[4] L. V. Ovsiannikov, Group analysis of differential equations, Academic, New York, 1982.10.1016/B978-0-12-531680-4.50012-5Suche in Google Scholar
[5] N. H. Ibragimov, Lie group analysis of differential equations, CRC Press, Boca Raton, 1994.Suche in Google Scholar
[6] K. S. Chou, G. X. Li and C. Qu, A note on optimal systems for the heat equation, J. Math. Anal. Appl. 261(2) (2001), 741–751.10.1006/jmaa.2001.7579Suche in Google Scholar
[7] X. Hu, Y. Li and Y. Chen, A direct algorithm of one-dimensional optimal system for the group invariant solutions, J. Math. Phys. 56(5) (2015), 053504.10.1063/1.4921229Suche in Google Scholar
[8] S. GoshHajra, S. Kandel, and S. P. Pudasaini, Optimal systems of Lie subalgebras for a two-phase mass flow, Int. J. Non-Linear Mech. 88 (2017), 109–121.10.1016/j.ijnonlinmec.2016.10.005Suche in Google Scholar
[9] Z. Zhao and B. Han, Lie symmetry analysis of the Heisenberg equation, Commun. Nonl. Sci. Numer. Simu. 45 (2017), 220–234.10.1016/j.cnsns.2016.10.008Suche in Google Scholar
[10] P. Satapathy and T. Raja Sekhar, Optimal system, invariant solutions and evolution of weak discontinuity for isentropic drift flux model, Appl. Math. Comput. 334 (2018), 107–116.10.1016/j.amc.2018.03.114Suche in Google Scholar
[11] C. Zoppou and S. Roberts, Catastrophic collapse of water supply reservoirs in urban areas, J. Hydraul. Eng. 125 (1999), 686–695.10.1061/(ASCE)0733-9429(1999)125:7(686)Suche in Google Scholar
[12] C. Synolakis, E. Okal and E. Bernard, The megatsunami of December 26, 2004, Bridge Natl. Acad. Eng. 35(2) (2005), 26–35.Suche in Google Scholar
[13] M. Pandey, Lie symmetries and exact solutions of Shallow water wquations with variable bottom, Int. J. Nonl. Sci. Numer. Simu. 16 (2015), 337–342.10.1515/ijnsns-2015-0093Suche in Google Scholar
[14] S. Dimas and D. Tsoubelis, SYM: A new symmetry – finding package for Mathematica, in: N. H. Ibragimov, C. Sophocleous and P. A. Damianou, editors, The 10th International Conference in Modern Group Analysis, pp. 64–70, Nicosia, 2005.Suche in Google Scholar
[15] M. Pandey, Group theoretic method for analyzing interaction of a discontinuity wave with a strong shock in an ideal gas, Z. Angew. Math. Phys. 61 (2010), 87–94.10.1007/s00033-009-0030-2Suche in Google Scholar
[16] T. Nath, R. K. Singh and L. P. Singh, Evolution of weak shock waves in non-ideal magnetogasdynamics, Acta Astronaut. 133 (2017), 397–402.10.1016/j.actaastro.2016.10.029Suche in Google Scholar
[17] B. Bera, T. Raja Sekhar and G. P. Raja Sekhar, Collision of characteristic shock with weak discontinuity in non-ideal magnetogasdynamics, Comput. Math. Appl. 75 (2018), 3873–3883.10.1016/j.camwa.2018.02.034Suche in Google Scholar
[18] R. Touma and C. Klingenberg, Well-balanced central finite volume methods for the Ripa system, Appl. Numer. Math. 97 (2015), 42–68.10.1016/j.apnum.2015.07.001Suche in Google Scholar
[19] Mai Duc Thanh, The Riemann problem for the shallow water equations with horizontal temperature gradients, Appl. Math. Comput. 325 (2018), 159–178.10.1016/j.amc.2017.12.031Suche in Google Scholar
[20] M. Pandey, R. Radha and V. D. Sharma, Symmetry analysis and exact solutions of magnetogasdynamic equations, Q. J. Mech. Appl. Math. 61 (2008), 291–310.10.1093/qjmam/hbn011Suche in Google Scholar
[21] T. Raja Sekhar and P. Satapathy, Group classification for isothermal drift flux model of two phase flows, Comput. Math. Appl. 72(5) (2016), 1436–1443.10.1016/j.camwa.2016.07.017Suche in Google Scholar
[22] A. Jeffrey, Quasilinear hyperbolic systems and waves, Pitman, London, 1976.Suche in Google Scholar
[23] V. D. Sharma, Quasilinear hyperbolic systems, compressible flows, and waves, CRC Press, New York, 2010.10.1201/9781439836910Suche in Google Scholar
[24] T. Ruggeri, Interaction between a discontinuity wave and a shock wave: critical time for the fastest transmitted wave, example of the polytropic fluid, Appl. Anal. 11 (1980), 103–112.10.1080/00036818008839323Suche in Google Scholar
[25] Ch. Radha, V. D. Sharma and A. Jeffrey, Interaction of shock waves with discontinuities, Appl. Anal. 50 (1993), 145–166.10.1080/00036819308840191Suche in Google Scholar
[26] M. Pandey and V. D. Sharma, Interaction of a characteristic shock with a weak discontinuity in a non-ideal gas, Wave Motion 44 (2007), 346–354.10.1016/j.wavemoti.2006.12.002Suche in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Research article
- Improved Results on Adaptive Control Approach for Projective Synchronization of Neural Networks with Time-Varying Delay
- A Method to Solve the Reaction-Diffusion-Chemotaxis System
- Two-Phase Flow of Fluid-Particle Interaction over a Stretching Sheet in the Presence of Magnetic Field
- An Algorithm for the Approximate Solution of the Fractional Riccati Differential Equation
- Additional Extension of the Mathematical Model for BCG Immunotherapy of Bladder Cancer and Its Validation by Auxiliary Tool
- Extended Transformed Rational Function Method to Nonlinear Evolution Equations
- Approximation of p-Biharmonic Problem using WEB-Spline based Mesh-Free Method
- Lie Symmetries, One-Dimensional Optimal System and Group Invariant Solutions for the Ripa System
- On the Numerical Computation of the Mittag–Leffler Function
Artikel in diesem Heft
- Frontmatter
- Research article
- Improved Results on Adaptive Control Approach for Projective Synchronization of Neural Networks with Time-Varying Delay
- A Method to Solve the Reaction-Diffusion-Chemotaxis System
- Two-Phase Flow of Fluid-Particle Interaction over a Stretching Sheet in the Presence of Magnetic Field
- An Algorithm for the Approximate Solution of the Fractional Riccati Differential Equation
- Additional Extension of the Mathematical Model for BCG Immunotherapy of Bladder Cancer and Its Validation by Auxiliary Tool
- Extended Transformed Rational Function Method to Nonlinear Evolution Equations
- Approximation of p-Biharmonic Problem using WEB-Spline based Mesh-Free Method
- Lie Symmetries, One-Dimensional Optimal System and Group Invariant Solutions for the Ripa System
- On the Numerical Computation of the Mittag–Leffler Function