The Laplace transform method for a scalar differential equation with symmetry/reflection

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

doi:10.3934/math.2026790

AIMS Mathematics

2026, Volume 11, Issue 7: 19442-19456. doi: 10.3934/math.2026790

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The Laplace transform method for a scalar differential equation with symmetry/reflection

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: 20 March 2026 Revised: 20 May 2026 Accepted: 02 June 2026 Published: 02 July 2026
MSC : 34K06, 34K07, 65L03

This work investigated a scalar differential equation involving proportional arguments of the form $ \phi'(t) = \alpha \phi(\gamma t)+\beta \phi(-\gamma t) $. The terms $ \phi(\gamma t) $ and $ \phi(-\gamma t) $ represented proportional delay and reflection (time-reversal) effects, respectively, which arose in models exhibiting scaling and symmetry properties. A constructive analytical approach based on the Laplace transform was developed to derive a series representation of the solution. The method converted the original scalar equation into a recursive sequence of algebraic relations in the Laplace variable, allowing the systematic computation of successive terms. Explicit formulas for the series components were obtained, revealing a clear structure separating even and odd contributions. The convergence of the resulting series was rigorously established, showing that the solution existed globally and defined an entire function. Furthermore, the series representation was summed into a closed form expressed through hyperbolic functions, providing a compact analytical expression of the solution. Numerical illustrations confirmed the effectiveness of the derived formulation.
- scalar differential equation,
- Laplace transform,
- series,
- exact solution
Citation: Laila F. Seddek, Essam R. El-Zahar, Abdelhalim Ebaid. The Laplace transform method for a scalar differential equation with symmetry/reflection[J]. AIMS Mathematics, 2026, 11(7): 19442-19456. doi: 10.3934/math.2026790

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Abstract

This work investigated a scalar differential equation involving proportional arguments of the form $ \phi'(t) = \alpha \phi(\gamma t)+\beta \phi(-\gamma t) $. The terms $ \phi(\gamma t) $ and $ \phi(-\gamma t) $ represented proportional delay and reflection (time-reversal) effects, respectively, which arose in models exhibiting scaling and symmetry properties. A constructive analytical approach based on the Laplace transform was developed to derive a series representation of the solution. The method converted the original scalar equation into a recursive sequence of algebraic relations in the Laplace variable, allowing the systematic computation of successive terms. Explicit formulas for the series components were obtained, revealing a clear structure separating even and odd contributions. The convergence of the resulting series was rigorously established, showing that the solution existed globally and defined an entire function. Furthermore, the series representation was summed into a closed form expressed through hyperbolic functions, providing a compact analytical expression of the solution. Numerical illustrations confirmed the effectiveness of the derived formulation.

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