Just need some clarification on a few things from this article that I am reading: Solutions of the Telegraph Equations using a Fractional Calculus Approach.
Quick note - since it isn't directly mentioned in the article - for those of you are wondering what the Telegrapher's Equation with loss: \begin{eqnarray} \frac{\partial^2 V}{\partial x^2} & = & RGV + (LG+RC)\frac{\partial V}{\partial t} + LC \frac{\partial^2 V}{\partial t^2} \end{eqnarray} and without losses (implying conductance $G$ and resistance $R$ are zero): \begin{eqnarray} \frac{\partial^2 V}{\partial x^2} & = & LC \frac{\partial^2 V}{\partial t^2} \end{eqnarray} Where $V=V(x,t)$ is our voltage across the transmission line.
The first case they talk about is the Fractional Time case, where our fractional time derivative is \begin{eqnarray} \frac{\partial^2 V}{\partial x^2} - \frac{LC}{(\sigma_t)^{2(1-\gamma)}} \frac{\partial^{2\gamma} V}{\partial t^{2\gamma}} & = & 0 \end{eqnarray} Where , $0 < \gamma \leq 1$.
I have many questions regarding their results and how they define certain things. For starters, they define $\sigma_t$ as a new parameter representing the fractional time component; however, I am not sure what exactly it is. Same goes for the spatial case when they define $\alpha_x$ as a new parameter for the fractional space.
I was curious as to how they knew the following particular solution to the Fractional Time Transmission line equation: \begin{eqnarray} V(x,t) = V_0e^{-ikx}u(t) \end{eqnarray}
Afterwards, they combine the result and the Fractional-Time Derivative to get $u(t)$ (which I also wonder how they got it). \begin{eqnarray} u(t) = E_{2\gamma}\{-\omega^2t^{2\gamma}\} \end{eqnarray}
As a result, the particular solution they obtained was
\begin{eqnarray} V(x,t) = V_0e^{-ikx}E_{2\gamma}\{-\omega^2t^{2\gamma}\} \end{eqnarray}
Note $E_{2\gamma}(t)$ is the Mittag-Leffler function defined by \begin{eqnarray} E_{2\gamma}(t) = \sum_{n=0}^{\infty}\frac{t^n}{\Gamma(2\gamma + 1)} \end{eqnarray}
I have similar questions for the Fractional-Spatial case, but I will focus on that later once I fully understood the Fractional-Time case.
I'm trying to understand this article to the best of my abilities for a brief report I am writing for my Fractional Calculus course. Hopefully I can. I do want to add I am fairly new to fractional calculus myself, this being my first semester.
I'm sorry for the long read but I thank you so much for taking the time to help me further my knowledge in fractional calculus (mathematics in general). I greatly appreciate any guidance you may give me. Thank you once again.