Solving Nonlinear Difference Equations: Insights from Three-Dimensional Systems and Numerical Examples

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DOI: https://doi.org/10.28924/2291-8639-22-2024-191
Published: Oct 22, 2024
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Najmeddine Attia, Ahmed Ghezal, Solving Nonlinear Difference Equations: Insights from Three-Dimensional Systems and Numerical Examples, Int. J. Anal. Appl., 22 (2024), 191.

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Najmeddine Attia, Ahmed Ghezal

Abstract

This paper presents a study on nonlinear difference equation systems of 6k + 3 order. The equations are of the form pn+1 =pn−(6k+2) / (±1±qn−2k rn−(4k+1) pn−(6k+2)), qn+1 =qn−(6k+2) / (±1±rn−2kpn−(4k+1) qn−(6k+2)),rn+1 =rn−(6k+2) / (±1± pn−2kqn−(4k+1) rn−(6k+2)), k ≥ 0 where n is a non-negative integer (belonging to the set N0 = N ∪ {0}) and the starting values p−l , q−l , r−l , l ∈ {0, 1, . . . , 6k+2} are arbitrary nonzero real numbers. We propose a systematic approach to solve this system, introducing a novel technique to find explicit solutions. The main outcomes of our study are the explicit solutions derived from the considered system. The study examines four different cases of this system and provides numerical examples to illustrate the results. The numerical examples demonstrate the behavior of the system for various initial conditions. The study is concluded with graphical representations of the solutions for each case, providing insights into the behavior of the systems.

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References

  1. N. Attia, A. Ghezal, Global Stability and Co-Balancing Numbers in a System of Rational Difference Equations, Elec. Res. Arch. 32 (2024), 2137–2159. https://doi.org/10.3934/era.2024097.
  2. R. Abo-Zeid, Global Behavior of Two Third Order Rational Difference Equations with Quadratic Terms, Math. Slovaca 69 (2019), 147–158. https://doi.org/10.1515/ms-2017-0210.
  3. R. Abo-Zeid, Global Behavior of a Higher Order Difference Equation, Math. Slovaca 64 (2014), 931–940. https://doi.org/10.2478/s12175-014-0249-z.
  4. R. Abo-Zeid, C. Cinar, Global Behavior of the Difference Equation xn+1 = Axn−1/B−Cxnxn−2 , Bol. Soc. Paran. Mat. 31 (2011), 43–49. https://doi.org/10.5269/bspm.v31i1.14432.
  5. M. Berkal, J.F. Navarro, Qualitative Study of a Second Order Difference Equation, Turk. J. Math. 47 (2023), 516–527. https://doi.org/10.55730/1300-0098.3375.
  6. M. Berkal, R. Abo-Zeid, On a Rational (P + 1)th Order Difference Equation with Quadratic Term, Univ. J. Math. Appl. 5 (2022), 136–144. https://doi.org/10.32323/ujma.1198471.
  7. M. Berkal, K. Berehal, N. Rezaiki, Representation of Solutions of a System of Five-Order Nonlinear Difference Equations, J. Appl. Math. Inf. 40 (2022), 409–431. https://doi.org/10.14317/JAMI.2022.409.
  8. L. Brand, A Sequence Defined by a Difference Equation, Amer. Math. Mon. 62 (1955), 489–492. https://doi.org/10.2307/2307362.
  9. E.M. Elsayed, F. Alzahrani, I. Abbas, N.H. Alotaibi, Dynamical Behavior and Solution of Nonlinear Difference Equation via Fibonacci Sequence, J. Appl. Anal. Comp. 10 (2020), 282–296. https://doi.org/10.11948/20190143.
  10. E.M. Elsayed., H.S. Gafel, On Some Systems of Three Nonlinear Difference Equations, J. Comp. Anal. Appl. 29 (2021), 86–108.
  11. E.M. Elsayed, Solution for Systems of Difference Equations of Rational Form of Order Two, Comp. Appl. Math. 33 (2014), 751–765. https://doi.org/10.1007/s40314-013-0092-9.
  12. E.M. Elsayed, Solutions of Rational Difference Systems of Order Two, Math. Comp. Model. 55 (2012), 378–384. https://doi.org/10.1016/j.mcm.2011.08.012.
  13. E.M. Elsayed, Solutions of Rational Difference Systems of Order Two, Math. Comp. Model. 55 (2012), 378–384. https://doi.org/10.1016/j.mcm.2011.08.012.
  14. A. Ghezal, Note on a Rational System of (4k + 4)-Order Difference Equations: Periodic Solution and Convergence, J. Appl. Math. Comp. 69 (2023), 2207–2215. https://doi.org/10.1007/s12190-022-01830-y.
  15. A. Ghezal, Higher-Order System of p-Nonlinear Difference Equations Solvable in Closed-Form with Variable Coefficients, Bol. Soc. Paran. Mat. 41 (2022), 1–14. https://doi.org/10.5269/bspm.63529.
  16. A. Ghezal, I. Zemmouri, On the Markov-Switching Autoregressive Stochastic Volatility Processes, SeMA J. 81 (2024), 413–427. https://doi.org/10.1007/s40324-023-00329-1.
  17. A. Ghezal, Spectral Representation of Markov-Switching Bilinear Processes, São Paulo J. Math. Sci. 18 (2024), 459–479. https://doi.org/10.1007/s40863-023-00380-w.
  18. T.F. Ibrahim, A.Q. Khan, B. O ˘gul, D. ¸Sim¸sek, Closed-Form Solution of a Rational Difference Equation, Math. Probl. Eng. 2021 (2021), 3168671. https://doi.org/10.1155/2021/3168671.
  19. T.F. Ibrahim, Asymptotic Behavior of a Difference Equation Model in Exponential Form, Math. Meth. Appl. Sci. 45 (2022), 10736–10748. https://doi.org/10.1002/mma.8415.
  20. M. Kara, Y. Yazlik, Solvable Three-Dimensional System of Higher-Order Nonlinear Difference Equations, Filomat 36 (2022), 3449–3469. https://doi.org/10.2298/FIL2210449K.
  21. M. Kara, Y. Yazlik, On a Solvable Three-Dimensional System of Difference Equations, Filomat 34 (2020), 1167–1186. https://doi.org/10.2298/FIL2004167K.
  22. A.S. Kurbanlı, C. Çinar, ˙I. Yalçinkaya, On the Behavior of Positive Solutions of the System of Rational Difference Equations xn+1 = xn−1/ynxn−1+1 , yn+1 = yn−1/xn yn−1+1 , Math. Comp. Model. 53 (2011), 1261–1267. https://doi.org/10.1016/j.mcm.2010.12.009.
  23. A.S. Kurbanli, On the Behavior of Solutions of the System of Rational Difference Equations xn+1 = xn−1/ynxn−1−1 , yn+1 = yn−1/xn yn−1−1 , zn+1 = zn−1/ynzn−1−1 , Adv. Diff. Equ. 2011 (2011), 40. https://doi.org/10.1186/1687-1847-2011-40.
  24. B. Ogul, D. ¸Sim¸sek, T.F. Ibrahim, Solution of Rational Difference Equation, Dyn. Contin. Discr. Impuls. Syst. Ser. B: Appl. Algor. 28 (2021), 125-141.
  25. G. Papaschinopoulos, B.K. Papadopoulos, On the Fuzzy Difference Equation xn+1 = A + B/xn, Soft Comp. 6 (2002), 456–461. https://doi.org/10.1007/s00500-001-0161-7.
  26. S. Stevic, Representations of Solutions to Linear and Bilinear Difference Equations and Systems of Bilinear Difference Equations, Adv. Diff. Equ. 2018 (2018), 474. https://doi.org/10.1186/s13662-018-1930-2.
  27. S. Stevic, B. Iricanin, W. Kosmala, Z. Smarda, Note on the Bilinear Difference Equation with a Delay, Math. Meth. the Appl. Sci. 41 (2018), 9349–9360. https://doi.org/10.1002/mma.5293.
  28. S. Stevic, B. Iricanin, Z. Smarda, On a Symmetric Bilinear System of Difference Equations, Appl. Math. Lett. 89 (2019), 15–21. https://doi.org/10.1016/j.aml.2018.09.006.