This work investigated a scalar differential equation involving proportional arguments of the form ${\varphi }^{\prime }(t)=\alpha \varphi (\gamma t)+\beta \varphi (-\gamma t)$. The terms $\varphi (\gamma t)$ and $\varphi (-\gamma t)$ 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.

This paper introduced a new aftertreatment technique for solving a fractional nonlinear Susceptible-Infectious-Recovered (SIR)-epidemic model. The proposed approach reformulated the power series solution via incorporating the Laplace transform and its inverse. Basically, it applied the Laplace transform to convert the series solution into different Pad $\acute{e}$-approximants involving Laplace's parameter with arbitrary order. This procedure facilitated the method of deriving the inverse Laplace transform explicitly, as a final step, using basic special functions in fractional calculus. Our analysis was capable of obtaining a sequence of closed-form approximations in terms of the Mittag-Leffler functions. The current results may be provided for the first time regarding the Mittag-Leffler solution of the fractional SIR-model. Additionally, the outcome of this analysis revealed that our approach was not only effective but also applicable to a wide range of fractional differential equations and systems.