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Journals / Turkish Journal of Mathematics and Computer Science / 2026 / Volume: 16 Issue: 1

On a General Non-Linear Difference Equation of Third-Order

Merve Kara(KARAMANOĞLU MEHMETBEY ÜNİVERSİTESİ)0000-0001-8081-0254
Pages
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Publication Date
2024-06-30
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⚡ TL;DR(Too Long; Didn't Read)

Finally, numerical applications are given.

📌 3 Key Takeaways

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Objective & Hypothesis:In this paper, we investigate the following general difference equations \begin{equation*} x_{n+1}=h^{-1}\left( h\left( x_{n}\right) \frac{Ah\left( x_{n-1}\right)+Bh\left( x_{n-2}\right) }{Ch\left( x_{n-1}\right)+Dh\left( x_{n-2}\right)}\right) ,\ n\in \mathbb{N}_{0}, \end{equation*} where the parameters $A, B, C, D$ and the initial values $x_{-\Phi}$, for $\Phi=\overline{0,2}$ are real numbers, $h$ is a continuous and strictly monotone function, $h\left( \mathbb{R}\right) =\mathbb{R}$, $h\left( 0\right) =0$.
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Key Findings & Data:In addition, we obtain closed-form solutions of aforementioned difference equations.
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Conclusion & Impact:Finally, numerical applications are given.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Riccati difference equation · solution · closed form

Abstract

In this paper, we investigate the following general difference equations \begin{equation*} x_{n+1}=h^{-1}\left( h\left( x_{n}\right) \frac{Ah\left( x_{n-1}\right)+Bh\left( x_{n-2}\right) }{Ch\left( x_{n-1}\right)+Dh\left( x_{n-2}\right)}\right) ,\ n\in \mathbb{N}_{0}, \end{equation*} where the parameters $A, B, C, D$ and the initial values $x_{-\Phi}$, for $\Phi=\overline{0,2}$ are real numbers, $h$ is a continuous and strictly monotone function, $h\left( \mathbb{R}\right) =\mathbb{R}$, $h\left( 0\right) =0$. In addition, we obtain closed-form solutions of aforementioned difference equations. Finally, numerical applications are given.

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Cite This Article

Kara, M. (2024). On a General Non-Linear Difference Equation of Third-Order. Turkish Journal of Mathematics and Computer Science, Volume: 16 Issue: 1. https://doi.org/10.47000/tjmcs.1366596

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