Main Article Content
Difference equations are widely utilized to describe some phenomena arising in nonlinear sciences. In particular, systems of difference equations play an important role in investigating most nonlinear applications. Future behaviors of such phenomena can be sometimes known and understood by using exact solutions of systems of difference equations. Therefore, this article investigates the exact solutions of fourth order systems of difference equations. We use successive iterations and Padovan numbers to obtain the exact solutions in the form of rational functions. The stability of the considered systems are analyzed using Jacobian matrix. Real equilibrium points are found saddle. Under some selected parameters, we plot some 2D figures to show the behavior of the obtained solutions. The used methods can be successfully applied for high order systems of difference equations.
- J.D. Murray, Mathematical Biology: I. An Introduction, 3rd Ed., Springer-Verlag, New York, 2001. https://doi.org/10.1007/b98868.
- D.T. Tollu, Y. Yazlik, N. Taskara, On the Solutions of Two Special Types of Riccati Difference Equation via Fibonacci Numbers, Adv. Differ. Equ. 2013 (2013), 174. https://doi.org/10.1186/1687-1847-2013-174.
- Y. Yazlik, D.T. Tollu, N. Taskara, On the Solutions of Difference Equation Systems with Padovan Numbers, Appl. Math. 04 (2013), 15–20. https://doi.org/10.4236/am.2013.412a002.
- M.B. Almatrafi, Solutions Structures for Some Systems of Fractional Difference Equations, Open J. Math. Anal. 3 (2019), 52–61. https://doi.org/10.30538/psrp-oma2019.0032.
- M.B. Almatrafi, M.M. Alzubaidi, Analysis of the Qualitative Behaviour of an Eighth-Order Fractional Difference Equation, Open J. Discret. Appl. Math. 2 (2019), 41–47. https://doi.org/10.30538/psrp-odam2019.0010.
- C. Çinar, On the Positive Solutions of the Difference Equation System xn+1 = 1 yn , yn+1 = yn yn−1xn−1 , Appl. Math. Comput. 158 (2004), 303–305. https://doi.org/10.1016/j.amc.2003.08.073.
- E.M. Elsayed, Solutions of Rational Difference Systems of Order Two, Math. Computer Model. 55 (2012), 378–384. https://doi.org/10.1016/j.mcm.2011.08.012.
- H.S. Alayachi, A.Q. Khan, M.S.M. Noorani, On the Solutions of Three-Dimensional Rational Difference Equation Systems, J. Math. 2021 (2021), 2480294. https://doi.org/10.1155/2021/2480294.
- H.S. Alayachi, A.Q. Khan, M.S.M. Noorani, A. Khaliq, Displaying the Structure of the Solutions for Some Fifth-Order Systems of Recursive Equations, Math. Probl. Eng. 2021 (2021), 6682009. https://doi.org/10.1155/2021/6682009.
- H.S. Alayachi, M.S.M. Noorani, A.Q. Khan, M.B. Almatrafi, Analytic Solutions and Stability of Sixth Order Difference Equations, Math. Probl. Eng. 2020 (2020), 1230979. https://doi.org/10.1155/2020/1230979.
- S. Elaydi, An Introduction to Difference Equations, Springer-Verlag, New York, 2005. https://doi.org/10.1007/0-387-27602-5.
- M.B. Almatrafi, E.M. Elsayed, F. Alzahrani, Qualitative Behavior of Two Rational Difference Equations, Fund. J. Math. Appl. 1 (2018), 194–204. https://doi.org/10.33401/fujma.454999.
- M.B. Almatrafi, E.M. Elsayed, Solutions And Formulae For Some Systems Of Difference Equations, MathLAB J. 1 (2018), 356-369.
- M.B. Almatrafi, E.M. Elsayed, F. Alzahrani, Qualitative Behavior of a Quadratic Second-Order Rational Difference Equation, Int. J. Adv. Math. 2019 (2019), 1-14.
- M.B. Almatrafi, Exact Solutions and Stability of Sixth Order Difference Equations, Electron. J. Math. Anal. Appl. 10 (2022), 209-225.
- M.B. Almatrafi, Abundant Traveling Wave and Numerical Solutions for Novikov-Veselov System With Their Stability and Accuracy, Appl. Anal. (2022). https://doi.org/10.1080/00036811.2022.2027381.