On solving a second-order scalar differential equation involving symmetry/reflection of argument: exact solution

Laila F. Seddek; Essam R. El-Zahar; Abdelhalim Ebaid; Laila F. Seddek; Essam R. El-Zahar; Abdelhalim Ebaid

doi:10.3934/math.20261152

AIMS Mathematics

2026, Volume 11, Issue 9: 28971-28990. doi: 10.3934/math.20261152

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On solving a second-order scalar differential equation involving symmetry/reflection of argument: exact solution

1.
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
2.
Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia

Received: 18 June 2026 Revised: 31 July 2026 Accepted: 13 August 2026 Published: 09 September 2026
MSC : 34B15, 47H11

This paper applies a reduction approach to solve a second-order scalar model with reflection of argument. Such types of models arise in modeling the transmission of nerve impulses in neuroscience. The proposed approach is capable of eliminating the reflection term with nonlocal classic behavior. Accordingly, an equivalent fourth-order model is deduced. The converted model is then exactly solved, basically in exponential and hyperbolic forms. Domains of valid parameters for obtaining several forms of the solution are specified. Moreover, various special cases are studied including repeated roots for the characteristic equation in which different types of solutions are determined. Furthermore, periodic solutions are derived under particular constraints. The proposed analysis reveals that the suggested approach can be extended to handle higher-order reflected models.
- reduction approach,
- scalar differential equation,
- reflection of argument,
- exact solution
Citation: Laila F. Seddek, Essam R. El-Zahar, Abdelhalim Ebaid. On solving a second-order scalar differential equation involving symmetry/reflection of argument: exact solution[J]. AIMS Mathematics, 2026, 11(9): 28971-28990. doi: 10.3934/math.20261152

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Abstract

This paper applies a reduction approach to solve a second-order scalar model with reflection of argument. Such types of models arise in modeling the transmission of nerve impulses in neuroscience. The proposed approach is capable of eliminating the reflection term with nonlocal classic behavior. Accordingly, an equivalent fourth-order model is deduced. The converted model is then exactly solved, basically in exponential and hyperbolic forms. Domains of valid parameters for obtaining several forms of the solution are specified. Moreover, various special cases are studied including repeated roots for the characteristic equation in which different types of solutions are determined. Furthermore, periodic solutions are derived under particular constraints. The proposed analysis reveals that the suggested approach can be extended to handle higher-order reflected models.

References

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