I'm interested in inverting the following integral transform:
$$f(t) = \int_0^{L} b(t+L-x) e^{\int^L_x \gamma(r) dr} dx$$
to find $\gamma(x)$ on $[0,L]$ given $f(t)\geq0$ and $b(t)\geq0$, where $f$ and $b$ agree sufficiently so that the system is not overdetermined. (I'm also interested in the conditions for which this system is overdetermined, e.g., it breaks if $f(t) =0$ but $b(t)$ isn't, but for the moment I'm considering systems where I know it isn't. Many exist.) Does anyone have thoughts, or can anyone recommend a resource for similar transforms?
Some possibly (but by no means necessarily) useful things are:
I. $f(t)$ can be understood as the mass of a population whose density satisfies the PDE:
$$u_t + u_x = -\gamma(x) u$$ with $b(t) = u(t,L)$.
II. Constraints on $\gamma(x)$ are allowed, including:
- $\gamma(x) > 0$ for all $x$.
- $\gamma'(x) > 0$ for all $x$.
- $\gamma(x)\rightarrow \infty$ as $x\rightarrow\infty$.
III. Condition (2) above allows for invertibility of $\gamma.$ With that assumption, I thought about making the change of variables:
$$yt= -\int_x^L \gamma(r)dr,$$
but this feels a bit like trying to force a square peg into a Laplace shaped hole. In the end I have an integrand of the form $e^{-yt}g(y,t)$ with $g(y,t)$ not clearly factoring into functions of $y$ and $t$.
IV. I can extend $f$ and $b$ to $\hat{f}$ and $\hat{b}$ so that they're zero for $t<0$, multiply the original integrand by the Heaviside functions $H(x)$ and $H(L-x)$, take a FT of the result, do the same thing with $f'(t)$, and then solve the two equations for $\gamma(x)$ on $[0,L]$ to get
$$ \gamma (x) = \frac{\mathcal{F}^{-1}\left(\frac{\mathcal{F}\left(\hat{f}'\right)}{\mathcal{F}\left(\hat{b}\right)}\right)}{\mathcal{F}^{-1}\left(\frac{\mathcal{F}\left(\hat{f}\right)}{\mathcal{F}\left(\hat{b}\right)}\right)}, $$
but something just feels a little wacky...
Thoughts?