Oh yeah, I didn't look closely enough at that function. That inverse Laplace transform I posted does work too though, which is odd. I guess it's probably just a rearranged form of the other inverse Laplace transform though.
I think the difference relates to the process of rearranging the function. If we first make the substitution
$$ A=\pm \alpha, \qquad \alpha > 0.$$
Then the left-hand-side becomes
$$ \frac{\sqrt{B+s}\pm \alpha}{B+s-\alpha^2} =\frac{\sqrt{B+s}\pm \alpha}{(\sqrt{B+s}+ \alpha)(\sqrt{B+s}- \alpha)}...
I tried that, and it gives a function that can only be rearranged into the function I'm looking for if ##A<0##.
However, this did lead me to find a solution. First rearrange into
$$ \frac{1}{-A+\sqrt{B+s}} = \sum_{n=0}^\infty \frac{A^n}{(B+s)^{(n+1)/2 }}.$$
Then take the inverse Laplace...
Hi Ray. Thanks for your response. However, if I rearrange the function as you've suggested, and then ask Mathematica to find the inverse Laplace transform, it also gives the function you've written there, but only as a conditional expression for the case where ##A<0##. The problem is that my...
Homework Statement
I want to invert a function from Laplace transform space to normal space.
Homework Equations
In Laplace transform space, the function takes the form $$ \bar f (s) = \frac{\exp\left[ x (-a +\sqrt{a^2+ b +c s} )\right]}{-a +\sqrt{a^2+ b +c s}}.
$$
Here, ##s## is the Laplace...
Because if we integrate both sides over ##a-\epsilon <t< a+\epsilon## (then taking ##\epsilon \rightarrow 0##), the left-hand-side will appear to be zero (because ##f(t)## is incorrectly assumed to continuous with no singularities), while the right-hand-side equals 1.
I say appears to be zero...
I see what you mean about the integration method for equations with delta functions, Orodruin. It works in this instance because there's a double derivative in the equation. However, if we were to try to use it to solve, for example $$f'(t) = \delta (t-s),$$ it wouldn't work. So it's not a very...
Hi Orodruin, thanks for your response. Yes, it's not the heat equation. I just mentioned heat as an example of a possible scalar quantity. I can see no reason why the time derivative couldn't be removed from the heat equation if the system is assumed to be steady state though.
As for sink...
Hi everyone,
I've been looking at a problem that seems simple at first, but appears to be deceptively difficult (unless I'm missing something).
1. Homework Statement
I've been looking at a problem that involves the diffusion of a scalar quantity, ##q(x)##, on the semi-infinite domain, ##\leq...
I'm looking to model a system in which events are nearly perfectly randomly distributed but with a slight tendency for events to avoid each other. As you know, if the system were perfectly random, I could use a Poisson distribution. The probability distribution for the number of events would...
I have a complicated recursion replation, which I'm sure is unsolvable. (By "unsolvable" I mean that there is no closed form solution expressing \xi_1, \xi_2, \xi_3, etc. in terms of \xi_0.) It goes
\frac{(k+4)!}{k!}\xi_{k+4} +K_1 (k+2)(k+1)\xi_{k+2}+ [ K_2 k(k-1) +K_3] \xi_{k} +K_4...
I have been working on a derivation in which the following simultateous ordinary differential equations have appeared:
f^{(4)}(x)-2 a^2 f''(x)+a^4 f(x)+b(g''(x)-a^2 g(x))=0,
g^{(4)}(x)-2 a^2 g''(x)+a^4 g(x)-b(f''(x)-a^2 f(x))=0,
where a and b are constants. I figured that I could solve...