Linearisation of a second-order nonlinear ordinary differential equation

Adhir Maharaj; Peter G. L. Leach; Megan Govender; David P. Day

doi:10.14311/AP.2023.63.0019

Authors

Adhir Maharaj Durban University of Technology, Steve Biko Campus, Department of Mathematics, Durban, 4000, Republic of South Africa
Peter G. L. Leach Durban University of Technology, Steve Biko Campus, Department of Mathematics, Durban, 4000, Republic of South Africa
Megan Govender Durban University of Technology, Steve Biko Campus, Department of Mathematics, Durban, 4000, Republic of South Africa
David P. Day Durban University of Technology, Steve Biko Campus, Department of Mathematics, Durban, 4000, Republic of South Africa

DOI:

https://doi.org/10.14311/AP.2023.63.0019

Keywords:

lie symmetries, integrability, linearisation

Abstract

We analyse nonlinear second-order differential equations in terms of algebraic properties by reducing a nonlinear partial differential equation to a nonlinear second-order ordinary differential equation via the point symmetry f(v)∂_v. The eight Lie point symmetries obtained for the second-order ordinary differential equation is of maximal number and a representation of the sl(3,R) algebra. We extend this analysis to a more general nonlinear second-order differential equation and we obtain similar interesting algebraic properties.

Downloads

Download data is not yet available.

References

B. Abraham-Shrauner, P. G. L. Leach, K. S. Govinder, G. Ratcliff. Hidden and contact symmetries of ordinary differential equations. Journal of Physics A: Mathematical and General 28(23):6707, 1995. https://doi.org/10.1088/0305-4470/28/23/020

K. Andriopoulos, P. G. L. Leach, A. Maharaj. On differential sequences. Applied Mathematics & Information Sciences 5(3):525–546, 2011.

C. Géronimi, M. R. Feix, P. G. L. Leach. Third order differential equation possessing three symmetries the two homogeneous ones plus the time translation. Tech. rep., SCAN-9905040, 1999.

A. K. Halder, A. Paliathanasis, P. G. L. Leach. Singularity analysis of a variant of the Painlevé–Ince equation. Applied Mathematics Letters 98:70–73, 2019. https://doi.org/10.1016/j.aml.2019.05.042

A. Maharaj, P. G. L. Leach. Properties of the dominant behaviour of quadratic systems. Journal of Nonlinear Mathematical Physics 13(1):129–144, 2006. https://doi.org/10.2991/jnmp.2006.13.1.11

A. Maharaj, K. Andriopoulos, P. G. L. Leach. Properties of a differential sequence based upon the Kummer-Schwarz equation. Acta Polytechnica 60(5):428–434, 2020. https://doi.org/10.14311/AP.2020.60.0428

K. Karmarkar. Gravitational metrics of spherical symmetry and class one. Proceedings of the Indian Academy of Sciences – Section A 27:56, 1948. https://doi.org/10.1007/BF03173443

A. V. Nikolaev, S. D. Maharaj. Embedding with Vaidya geometry. The European Physical Journal C 80(7):1–9, 2020. https://doi.org/10.1140/epjc/s10052-020-8231-0

P. Chunilal Vaidya. The external field of a radiating star in general relativity. Current Science 12:183, 1943.

F. M. Mahomed, P. G. L. Leach. Symmetry Lie algebras of nth order ordinary differential equations. Journal of Mathematical Analysis and Applications 151(1):80–107, 1990. https://doi.org/10.1016/0022-247X(90)90244-A

H. Stephani. Differential equations: Their solution using symmetries. Cambridge University Press, 1989.

G. W. Bluman, S. Kumei. Symmetries and differential equations, vol. 81. Springer Science & Business Media, 2013.

A. Maharaj, P. G. L. Leach. The method of reduction of order and linearization of the two-dimensional Ermakov system. Mathematical methods in the applied sciences 30(16):2125–2145, 2007. https://doi.org/10.1002/mma.919

A. Maharaj, P. G. L. Leach. Application of symmetry and singularity analyses to mathematical models of biological systems. Mathematics and Computers in Simulation 96:104–123, 2014. https://doi.org/10.1016/j.matcom.2013.06.005

P. G. L. Leach. Symmetry and singularity properties of a system of ordinary differential equations arising in the analysis of the nonlinear telegraph equations. Journal of mathematical analysis and applications 336(2):987–994, 2007. https://doi.org/10.1016/j.jmaa.2007.03.045

P. G. L. Leach, J. Miritzis. Analytic behaviour of competition among three species. Journal of Nonlinear Mathematical Physics 13(4):535–548, 2006. https://doi.org/10.2991/jnmp.2006.13.4.8

K. Andriopoulos, S. Dimas, P. G. L. Leach, D. Tsoubelis. On the systematic approach to the classification of differential equations by group theoretical methods. Journal of computational and applied mathematics 230(1):224–232, 2009. https://doi.org/10.1016/j.cam.2008.11.002

S. Dimas, D. Tsoubelis. SYM: A new symmetryfinding package for Mathematica. In Proceedings of the 10th international conference in modern group analysis, pp. 64–70. University of Cyprus Press, 2004.

S. Dimas, D. Tsoubelis. A new Mathematica-based program for solving overdetermined systems of PDEs. In 8th International Mathematica Symposium, pp. 1–5. Avignon, 2006.

S. Dimas. Partial differential equations, algebraic computing and nonlinear systems. Ph.D. thesis, University of Patras, Greece, 2008.

V. V. Morozov. Classification of nilpotent Lie algebras of sixth order. Izvestiya Vysshikh Uchebnykh Zavedenii Matematika 4:161–171, 1958.

G. M. Mubarakzyanov. On solvable Lie algebras. Izvestiya Vysshikh Uchebnykh Zavedenii Matematika (1):114–123, 1963.

G. M. Mubarakzyanov. Classification of real structures of Lie algebras of fifth order. Izvestiya Vysshikh Uchebnykh Zavedenii Matematika (3):99–106, 1963.

G. M. Mubarakzyanov. Classification of solvable Lie algebras of sixth order with a non-nilpotent basis element. Izvestiya Vysshikh Uchebnykh Zavedenii Matematika (4):104–116, 1963.

F. M. Mahomed, P. G. L. Leach. The linear symmetries of a nonlinear differential equation. Quaestiones Mathematicae 8(3):241–274, 1985. https://doi.org/10.1080/16073606.1985.9631915