Developed analytical approach for a special kind of differential-difference equation: Exact solution

Volume 33, Issue 4, pp 381--389 https://dx.doi.org/10.22436/jmcs.033.04.05

Publication Date: January 28, 2024 Submission Date: October 25, 2023 Revision Date: November 18, 2023 Accteptance Date: December 19, 2023

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Authors

L. F. Seddek - Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, P.O. Box 83, Al-Kharj 11942, Saudi Arabia. - Department of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig University, Zagazig, 44519, Egypt. A. Ebaid - Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia. E. R. El-Zahar - Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, P.O. Box 83, Al-Kharj 11942, Saudi Arabia. - Department of Basic Engineering Science, Faculty of Engineering, Menoufia University, Shebin El-Kom 32511, Egypt.

Abstract

This paper investigates a differential-difference equation with a variable coefficient of exponential order in the form \( \phi'(t)=\alpha \phi(t)+\beta e^{\sigma t} \phi(- t)\). In literature, periodic solution has been obtained at the special case \(\sigma=0\). In this paper, an effective approach is developed to determine the exact solution in terms of exponential and trigonometric functions. In addition, the exact solution is expressed in terms of exponential and hyperbolic functions under specific conditions of the involved parameters. Exact solutions of several special cases are derived and found in full agreement with the corresponding results in the relevant literature. Some theoretical results are presented and proved which can be generalized to include other complex models. The behavior of the obtained shows periodicity in the absence of \(\sigma\) while the damped oscillations are shown graphically when \(\sigma\) is assigned to negative values.

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Keywords

• Ansatz method
• differential-difference equation
• exact solution

MSC

•  34A99
•  34A30

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