Dynamical Analysis of Thirtieth-Order Difference Equations

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Tshenolo Thomas, Mensah Folly-Gbetoula


The main goal of this paper is to determine exact solutions of a family of thirtieth-order difference equations with variable coefficients. We use similarity variables obtained via symmetries to lower the order of the equations. We then reverse the transformations and obtain closed form solutions. We compare our solutions to those found in the literature for special cases. We investigate the periodic nature of the solutions and present some numerical examples to confirm the results. Finally, we analyze the stability of the equilibrium points. The method employed in this work can be applied to equations of higher order provided that they admit non zero characteristics.

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  1. L.S. Aljoufi, Qualitative Analysis of Nonlinear Difference Equations with a Thirty-order, Electron. J. Math. Anal. Appl. 11 (2023), 1–15.
  2. E.M. Elsayed, B.S. Alofi, The Periodic Nature and Expression on Solutions of Some Rational Systems of Difference Equations, Alexandria Eng. J. 74 (2023), 269–283. https://doi.org/10.1016/j.aej.2023.05.026.
  3. M. Folly-Gbetoula, Symmetry, Reductions and Exact Solutions of the Difference Equation un+2 = (aun)/(1 + bunun+1), J. Differ. Equ. Appl. 23 (2017), 1017–1024. https://doi.org/10.1080/10236198.2017.1308508.
  4. M. Folly-Gbetoula, K. Mkhwanazi, D. Nyirenda, On a Study of a Family of Higher-Order Recurrence Relations, Math. Probl. Eng. 2022 (2022), 6770105. https://doi.org/10.1155/2022/6770105.
  5. M. Folly-Gbetoula, Dynamics and Solutions of Higher-Order Difference Equations, Mathematics. 11 (2023), 3693. https://doi.org/10.3390/math11173693.
  6. M. Folly-Gbetoula, Solvability and Dynamical Analysis of Difference Equations, Int. J. Anal. Appl. 21 (2023), 122. https://doi.org/10.28924/2291-8639-21-2023-122.
  7. M. Folly-Gbetoula, On a Family of Higher Order Recurrence Relations: Symmetries, Formula Solutions, Periodicity and Stability Analysis, Arab. J. Math. 12 (2023), 541–551.
  8. N. Mnguni, Symmetry Lie Algebra and Exact Solutions of Some Fourth-Order Difference Equations, J. Nonlinear Sci. Appl. 11 (2018), 1262–1270. https://doi.org/10.22436/jnsa.011.11.06.
  9. P.E. Hydon, Difference Equations by Differential Equation Methods, Cambridge University Press, Cambridge, 2014.
  10. T.F. Ibrahim, A.Q. Khan, F.M. Alshehri, M.A. El-Moneam, Global Stability of a Second-Order Exponential-Type Difference Equation, Symmetry. 14 (2022), 1803. https://doi.org/10.3390/sym14091803.
  11. A. Khaliq, S. Sadiq, H.M.E. Ahmed, B.A.A. Mahmoud, B.R. Al-Sinan, T.F. Ibrahim, The Dynamical Behavior of a Three-Dimensional System of Exponential Difference Equations, Mathematics. 11 (2023), 1808. https://doi.org/10.3390/math11081808.
  12. S. MAEDA, The Similarity Method for Difference Equations, IMA J. Appl. Math. 38 (1987), 129–134. https://doi.org/10.1093/imamat/38.2.129.