Recent content by Hoplite

  1. H

    A tricky inverse Laplace transform

    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.
  2. H

    A tricky inverse Laplace transform

    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)}...
  3. H

    A tricky inverse Laplace transform

    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...
  4. H

    A tricky inverse Laplace transform

    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...
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    A tricky inverse Laplace transform

    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...
  6. H

    A scalar on a semi-infinite domain with source and sink

    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...
  7. H

    A scalar on a semi-infinite domain with source and sink

    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...
  8. H

    A scalar on a semi-infinite domain with source and sink

    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...
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    A scalar on a semi-infinite domain with source and 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...
  10. H

    Looking for a modified Poisson distribution

    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...
  11. H

    Approximating unsolvable recursion relations

    That's correct. In fact my equation is S''''+(a+bx^2)S''+(c+dx^2)S=0, with some inhomogenious boundary conditions.
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    Approximating unsolvable recursion relations

    Oops, sorry. I had the wrong equation for S. I've fixed it now.
  13. H

    Approximating unsolvable recursion relations

    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...
  14. H

    Why doesn't this method work? (Re: Simultaneous ODEs)

    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...
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