 # In studying a reflection-transmission problem involving exotic materials, I have come across the fol vestpn 2022-05-21 Answered
In studying a reflection-transmission problem involving exotic materials, I have come across the following linear first-order differential equation:
$\begin{array}{}\text{(1)}& A\frac{\mathrm{\partial }}{\mathrm{\partial }t}g\left(t\right)+Bg\left(t\right)=f\left(t\right),\end{array}$
where A and B are constants, g(t) is associated with the reflected wave, and f(t) is a (finite) driving function associated with the incident wave. Both A and B may be positive or negative. I am interested in the behavior of the solution in the limit that $A\to 0$
In studying a reflection-transmission problem involving exotic materials, I have come across the following linear first-order differential equation:A∂∂tg(t)+Bg(t)=f(t),(1)where A and B are constants, g(t) is associated with the reflected wave, and f(t) is a (finite) driving function associated with the incident wave. Both A and B may be positive or negative. I am interested in the behavior of the solution in the limit that A\rightarrow0.
I know there is an exact solution to Eq. (1), which is
$g\left(t\right)=C{e}^{-Bt/A}+\frac{1}{A}{\int }_{-\mathrm{\infty }}^{t}{e}^{-B\left(t-{t}^{\prime }\right)/A}f\left({t}^{\prime }\right)d{t}^{\prime },$
where C=0 because g(t)=0 if f(t)=0. However, I do not understand how this exact solution reduces to the case where A=0, which is $g\left(t\right)={B}^{-1}f\left(t\right)$. Any insight would be greatly appreciated.
I've seen a lot of documents discussing asymptotic analyses of linear differential equations, but they all start with second-order equations. Is this because there is inherently problematic with first-order?
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I presume $A>0$ and $B>0$. A change of variables $s=\left(t-{t}^{\prime }\right)/A$ in the integral gives you
$g\left(t\right)=C{e}^{-Bt/A}+{\int }_{0}^{\mathrm{\infty }}{e}^{-Bs}f\left(t-As\right)\phantom{\rule{thickmathspace}{0ex}}ds$
Now as $A\to 0+$, $f\left(t-As\right)\to f\left(t\right)$ if f is continuous. Assuming f is bounded, we can use the Dominated Convergence Theorem and this integral goes to
$f\left(t\right){\int }_{0}^{\mathrm{\infty }}{e}^{-Bs}\phantom{\rule{thickmathspace}{0ex}}ds={B}^{-1}f\left(t\right)$