Invariance analysis and exact solutions of some sixth-order difference equations

Volume 10, Issue 12, pp 6262--6273 http://dx.doi.org/10.22436/jnsa.010.12.11

Publication Date: December 09, 2017 Submission Date: May 23, 2017

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Authors

Darlison Nyirenda - School of Mathematics, University of the Witwatersrand, 2050, Johannesburg, South Africa. Mensah Folly-Gbetoula - School of Mathematics, University of the Witwatersrand, 2050, Johannesburg, South Africa.

Abstract

We perform a full Lie point symmetry analysis of difference equations of the form \[ u_{n+6}=\frac{u_nu_{n+4}}{u_{n+2}(A _n + B _n u_nu_{n+4})}\ , \] where the initial conditions are non-zero real numbers. Consequently, we obtain four non-trivial symmetries. Eventually, we get solutions of the difference equation for random sequences \((A_n)\) and \((B_n)\). This work is a generalization of a recent result by Khaliq and Elsayed [A. Khaliq, E. M. Elsayed, J. Nonlinear Sci. Appl., \({\bf 9}\) (2016), 1052--1063].

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ISRP Style

Darlison Nyirenda, Mensah Folly-Gbetoula, Invariance analysis and exact solutions of some sixth-order difference equations, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 12, 6262--6273

AMA Style

Nyirenda Darlison, Folly-Gbetoula Mensah, Invariance analysis and exact solutions of some sixth-order difference equations. J. Nonlinear Sci. Appl. (2017); 10(12):6262--6273

Chicago/Turabian Style

Nyirenda, Darlison, Folly-Gbetoula, Mensah. "Invariance analysis and exact solutions of some sixth-order difference equations." Journal of Nonlinear Sciences and Applications, 10, no. 12 (2017): 6262--6273

Keywords

Difference equation
symmetry
group invariant solutions

MSC

39A10
39A99
39A13

References

[1] E. M. Elsayed, T. F. Ibrahim, Periodicity and solutions for some systems of nonlinear rational difference equations, Hacet. J. Math. Stat., 44 (2015), 1361–1390.

[2] G. Feng, X.-J. Yang, H. M. Srivasta, Exact traveling-wave solutions for linear and nonlinear heat-transfer equations, Therm. Sci., 00 (2016), 6 pages.
- Google Scholar

[3] M. Folly-Gbetoula , Symmetry, reductions and exact solutions of the difference equation \(u_{n+2} = au_n/(1+bu_nu_{n+1})\), J. Difference Equ. Appl., 23 (2017), 1017–1024.

[4] M. Folly-Gbetoula, A. H. Kara , Symmetries, conservation laws, and ‘integrability’ of difference equations, Adv. Difference Equ., 2014 (2014), 14 pages.

[5] P. E. Hydon, Symmetries and first integrals of ordinary difference equations, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 456 (2000), 2835–2855.

[6] P. E. Hydon, Difference equations by differential equation methods, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge (2014)
- Google Scholar

[7] T. F. Ibrahim , Solving a class of three-order max-type difference equations , Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 21 (2014), 333–342.

[8] T. F. Ibrahim, M. A. El-Moneam, Global stability of a higher-order difference equation, Iran. J. Sci. Technol. Trans. A Sci., 41 (2017), 51–58.

[9] A. Khaliq, E. M. Elsayed , The dynamics and solution of some difference equations, J. Nonlinear Sci. Appl., 9 (2016), 1052–1063.
- MathSciNet
- Google Scholar

[10] A. Q. Khan, Q. Din, M. N. Qureshi, T. F. Ibrahim , Global behavior of an anti-competitive system of fourth-order rational difference equations, Comput. Ecol. Softw., 4 (2014), 35–46.
- Google Scholar

[11] D. Levi, L. Vinet, P. Winternitz, Lie group formalism for difference equations, J. Phys. A, 30 (1997), 633–649.

[12] S. Lie , Classification und Integration von gewhnlichen Differentialgleichungen zwischen xy, die eine Gruppe von Transformationen gestatten, (German) Math. Ann., 22 (1888), 213–281.
- View Article
- Google Scholar

[13] R. Morris, A. H. Kara, A. Chowdhury, A. Biswas, Soliton solutions, conservation laws, and reductions of certain classes of nonlinearwave equations, Z. Naturforsch. A, 67 (2012), 613–620.
- View Article
- Google Scholar

[14] E. Noether, Invariante variationsprobleme, (German) Nachr. D. König. Gesellsch. D. Wiss. Zu Göttingen, Math-phys. Klasse, 1918 (1918), 235–257.

[15] G. R. W. Quispel, R. Sahadevan, Lie symmetries and the integration of difference equations, Phys. Lett. A, 184 (1993), 64–70.

[16] W. Sarlet, F. Cantrijn, Generalizations of Noether’s theorem in classical mechanics, SIAM Rev., 23 (1981), 467–494.

[17] X.-J. Yang, F. Gao, H. M. Srivastava, Exact traveling wave solutions for the local fractional two-dimensional Burgers-type equations, Comput. Math. Appl., 73 (2017), 203–210.

[18] X.-J. Yang, J. A. Tenreiro Machado, D. Baleanu , Exact traveling-wave solution for local fractional Boussinesq equation in fractal domain, Fractals, 25 (2017), 7 pages.

[19] X.-J. Yang, J. A. Tenreiro Machado, D. Baleanu, C. Cattani, On exact traveling-wave solutions for local fractional Korteweg-de Vries equation, Chaos, 26 (2016), 5 pages.