Efficient Analytical Solutions for Linear and Nonlinear Difference Equations Using the Discrete Putzer Algorithm

Muna E. Abdulhafed; Aafaf E. Abduelhafid; Abdalftah  Elbori

doi:10.36602/jsba.2025.20.27

Authors

Muna E. Abdulhafed Department of Mathematics / Faculty of Science, Azzaytuna University, Tarhuna-Libya
Aafaf E. Abduelhafid Department of Mathematics /Faculty of Education, Azzaytuna University, Tarhuna-Libya
Abdalftah Elbori Department of Mathematics / Faculty of Science, Azzaytuna University, Tarhuna-Libya

DOI:

https://doi.org/10.36602/jsba.2025.20.27

Keywords:

Discrete Putzer Algorithm, Difference Equations, Cayley-Hamilton Theorem, Eigenvalue Analysis, Analytical Solutions.

Abstract

The discrete Putzer algorithm is examined in this work as an effective technique for resolving linear and nonlinear difference equations. The approach simplifies calculations and offers analytical answers by utilizing matrix theory and eigenvalue analysis, particularly for higher-order and non-homogeneous systems. Its efficacy is demonstrated by examples such as Fibonacci sequences and population dynamics. The Cayley-Hamilton theorem is applied to increase the algorithm's usefulness and make it a potent tool for dynamic systems in both theoretical and practical settings.

References

[1] A. Cayley, “A memoir on the theory of matrices,” Philosophical Transactions of the Royal Society of London, vol. 148, pp. 17–37, 1858.

[2] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1985.

[3] W. G. Kelley and A. C. Peterson, Difference Equations: An Introduction with Applications, Academic Press, 2001.

[4] A. F. Neto, “Extending Putzer’s representation to all analytic matrix functions via Omega matrix calculus,” Electronic

Journal of Differential Equations, vol. 2021, no. 97, pp. 1–18, 2021.

[5] E. J. Putzer, “Avoiding the computation of powers of a matrix by iteration,” SIAM Journal on Numerical Analysis, vol. 3, no. 1,pp. 68–74, 1966.

[6] E. J. Putzer, “Avoiding the Jordan Canonical Form in the Discussion of Linear Systems with Constant Coefficients,” American Mathematical Monthly, vol. 73, no. 1, pp. 2–7, 1966.

[7] F. Zhang and L. Feng, “Discrete dynamic systems and matrix theory: A combined approach to solving difference equations,” Applied Mathematics Letters, vol. 98, pp.30–39, 2019.