Solution Methods for Nonlinear Ordinary Differential Equations Using Lie Symmetry Groups

Disanayakage Hashan Sanjaya Perera; Dilruk Gallage

doi:10.21467/ajgr.13.1.37-61

Authors

Disanayakage Hashan Sanjaya Perera University of Colombo, Sri Lanka https://orcid.org/0009-0007-4256-8023
Dilruk Gallage University of Colombo, Sri Lanka https://orcid.org/0000-0002-3622-9152

DOI:

https://doi.org/10.21467/ajgr.13.1.37-61

Abstract

For formulating mathematical models, engineering problems and physical problems, Nonlinear ordinary differential equations(NODEs) are used widely. Nevertheless, explicit solutions can be obtained for very few NODEs, due to lack of techniques to obtain explicit solutions. Therefore methods to obtain approximate solution for NODEs are used widely. Although symmetry groups of ordinary differential equations (ODEs) can be used to obtain exact solutions however, these techniques are not widely used. The purpose of this paper is to present applications of Lie symmetry groups to obtain exact solutions of NODEs . In this paper we connect different methods,theorems and definitions of Lie symmetry groups from different references and we solve first order and second order NODEs. In this analysis three different methods are used to obtain exact solutions of NODEs. Using applications of these symmetry methods, drawbacks and advantages of these different symmetry methods are discussed and some examples have been illustrated graphically. Focus is first placed on discussing about the notion of symmetry groups of the ODEs. Focus is then changed to apply them to find general solutions for NODEs under three different methods. First we find suitable change of variables that convert given first order NODE into variable separable form using these symmetry groups. As another method to find general solutions for first order NODEs, we find particular type of solution curves called invariant solution curves under Lie symmetry groups and we use these invariant solution curves to obtain the general solutions. We find general solutions for the second order NODEs by reducing their order to first order using Lie symmetry groups.

Keywords:

differential equations, nonlinear, Lie symmetry groups

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References

P. J. Olver, Applications of Lie groups to differential equations, vol. 107. Springer Science & Business Media, 2000.

M. R. Ali, W.-X. Ma, and R. Sadat, ‘Lie symmetry analysis and wave propagation in variable-coefficient nonlinear physical phenomena’, East Asian J. Appl. Math., vol. 12, no. 1, pp. 201–212, 2022.

S. Kumar and S. K. Dhiman, ‘Lie symmetry analysis, optimal system, exact solutions and dynamics of solitons of a (3+ 1)-dimensional generalised BKP--Boussinesq equation’, Pramana, vol. 96, no. 1, p. 31, 2022.

D. V. Tanwar and M. Kumar, ‘On Lie symmetries and invariant solutions of Broer--Kaup--Kupershmidt equation in shallow water of uniform depth’, Journal of Ocean Engineering and Science, 2022.

N. Raza, F. Salman, A. R. Butt, and M. L. Gandarias, ‘Lie symmetry analysis, soliton solutions and qualitative analysis concerning to the generalized q-deformed Sinh-Gordon equation’, Communications in Nonlinear Science and Numerical Simulation, vol. 116, p. 106824, 2023.

S.-F. Tian, ‘Lie symmetry analysis, conservation laws and solitary wave solutions to a fourth-order nonlinear generalized Boussinesq water wave equation’, Applied Mathematics Letters, vol. 100, p. 106056, 2020.

L. Xia and X. Ge, ‘Lie Symmetry Analysis and Conservation Laws of the Axially Loaded Euler Beam’, Mathematics, vol. 10, no. 15, p. 2759, 2022.

A. Paliathanasis, ‘Lie symmetries and similarity solutions for rotating shallow water’, Zeitschrift für Naturforschung A, vol. 74, no. 10, pp. 869–877, 2019.

D. H. Sattinger and O. L. Weaver, Lie groups and algebras with applications to physics, geometry, and mechanics, vol. 61. Springer Science & Business Media, 2013.

P. E. Hydon, ‘Introduction to symmetry methods in the solution of differential equations that occur in chemistry and chemical biology’, International journal of quantum chemistry, vol. 106, no. 1, pp. 266–277, 2006.

M. Edwards and M. C. Nucci, ‘Application of Lie group analysis to a core group model for sexually transmitted diseases’, Journal of Nonlinear Mathematical Physics, vol. 13, no. 2, pp. 211–230, 2006.

V. Torrisi and M. C. Nucci, ‘Application of Lie Group Analysis to a Mathematical Model’, in The Geometrical Study of Differential Equations: NSF-CBMS Conference on the Geometrical Study of Differential Equations, June 20-25, 2000, Howard University, Washington, 2001, vol. 285, p. 11.

M. B. Matadi, ‘Application of Lie Symmetry to a Mathematical Model that Describes a Cancer Sub-Network’, Symmetry, vol. 14, no. 2, p. 400, 2022.

M. C. Nuccia and P. G. L. Leach, ‘Singularity and symmetry analyses of mathematical models of epidemics’, South African Journal of Science, vol. 105, no. 3, pp. 136–146, 2009.

J. Yu, Y. Feng, and X. Wang, ‘Lie symmetry analysis and exact solutions of time fractional Black--Scholes equation’, International Journal of Financial Engineering, vol. 9, no. 04, p. 2250023, 2022.

S. Kontogiorgis and C. Sophocleous, ‘Lie symmetries and the constant elasticity of variance (CEV) model’, Partial Differential Equations in Applied Mathematics, vol. 5, p. 100290, 2022.

D. J. Arrigo, Symmetry analysis of differential equations: an introduction. John Wiley & Sons, 2015.

P. J. Olver, ‘Nonlinear Ordinary Differential Equations’.

F. Granström, ‘Symmetry methods and some nonlinear differential equations: Background and illustrative examples’. 2017.

P. E. Hydon and P. E. Hydon, Symmetry methods for differential equations: a beginner’s guide. Cambridge University Press, 2000.

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

P. Etingof, ‘Lie groups and Lie algebras’, arXiv preprint arXiv:2201. 09397, 2022.

D. H. S. Perera and D. Gallage, ‘Lie Groups of Symmetries for Nonlinear Ordinary Differential Equations’.

N. Dehmamy, R. Walters, Y. Liu, D. Wang, and R. Yu, ‘Automatic symmetry discovery with lie algebra convolutional network’, Advances in Neural Information Processing Systems, vol. 34, pp. 2503–2515, 2021.

G. A. Miller, ‘An Introduction to the Lie Theory of One-Parameter Groups, with applications to the solution of differential equations. By Abraham Cohen, Ph. D., Associate in Mathematics, Johns Hopkins University. Boston, DC Heath & Co. 1911. Pp. iv+ 248. Half leather’, Science, vol. 34, no. 887, pp. 924–924, 1911.

A. Hussain, S. Bano, I. Khan, D. Baleanu, and K. Sooppy Nisar, ‘Lie symmetry analysis, explicit solutions and conservation laws of a spatially two-dimensional Burgers--Huxley equation’, Symmetry, vol. 12, no. 1, p. 170, 2020.

J. Starrett, ‘Solving differential equations by symmetry groups’, The American Mathematical Monthly, vol. 114, no. 9, pp. 778–792, 2007.

E. S. Cheb-Terrab, L. G. S. Duarte, and L. Da Mota, ‘Computer algebra solving of first order ODEs using symmetry methods’, Computer physics communications, vol. 101, no. 3, pp. 254–268, 1997.

E. S. Cheb-Terrab, L. G. S. Duarte, and L. Da Mota, ‘Computer algebra solving of second order ODEs using symmetry methods’, Computer Physics Communications, vol. 108, no. 1, pp. 90–114, 1998.

M. M. Hassan, A. R. Shehata, and M. S. Abdel-Daym, ‘The investigation of exact solutions and conservation laws of the classical Boussinesq system via the Lie symmetry method’, Applied. Mathematics and Information Science, vol. 16, no. 2, pp. 177–185, 2022.

S. Kumar and I. Hamid, ‘Dynamics of closed-form invariant solutions and diversity of wave profiles of (2+ 1)-dimensional Ito integro-differential equation via Lie symmetry analysis’, Journal of Ocean Engineering and Science, 2022.

K. Bibi and K. Ahmad, ‘Exact solutions of Newell-Whitehead-Segel equations using symmetry transformations’, Journal of Function Spaces, vol. 2021, pp. 1–8, 2021.

J. Borgqvist, F. Ohlsson, and R. E. Baker, ‘Symmetries of systems of first order odes: symbolic symmetry computations, mechanistic model construction and applications in biology’, arXiv preprint arXiv:2202. 04935, 2022.

C. N. Retz, ‘Classical and Nonclassical Lie Symmetries of the K (m, N) Dispersion Equation’, University of North Carolina Wilmington, 2012.