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:
(1) A t g ( t ) + B g ( t ) = f ( t ) ,
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 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 ( t ) = C e B t / A + 1 A t e B ( t t ) / A f ( t ) d t ,
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 ( t ) = B 1 f ( t ). 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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Answers (1)

pralkammj
Answered 2022-05-22 Author has 7 answers
I presume A > 0 and B > 0. A change of variables s = ( t t ) / A in the integral gives you
g ( t ) = C e B t / A + 0 e B s f ( t A s ) d s
Now as A 0 +, f ( t A s ) f ( t ) if f is continuous. Assuming f is bounded, we can use the Dominated Convergence Theorem and this integral goes to
f ( t ) 0 e B s d s = B 1 f ( t )
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