Title says it all. How do I go about finding inverse Laplace transform of that expression? If it were complete exponents, I would have used partial fractions. But what to do with non integer exponents?

  • $\begingroup$ Where did the expression come from? Did you perform a Laplace transform first, simplify and now need to do the inverse Laplace transform? $\endgroup$ – John Habert Feb 7 '14 at 18:29
  • $\begingroup$ @JohnHabert: I am solving coupled partial differential equations(space and time) with a complicated boundary conditions. So, I solved it in Laplace domain and now I am trying to invert the solution. $\endgroup$ – tumchaaditya Feb 8 '14 at 19:46

$$\frac{1}{s^2-As^{1.5}} = -\frac{1}{A^3(\sqrt{s}+A)} + \frac{1}{A^3\sqrt{s}}-\frac{1}{A^2s}+\frac{1}{As^{1.5}}$$

The inverse Laplace transform of the first term doesn't have a closed form solution:


The rest of the terms are easy.

  • $\begingroup$ erfc is considered a closed-form solution. Can you derive it yourself? $\endgroup$ – Ron Gordon Feb 7 '14 at 21:15
  • $\begingroup$ I suppose it depends on your definition of closed-form solution. math.stackexchange.com/questions/9199/… $\endgroup$ – Perry Elliott-Iverson Feb 7 '14 at 21:43
  • $\begingroup$ I have a specific definition that says an expression is "closed-form" when there exists algorithms that allow one to compute that expression significantly faster to within machine precision than via a numerical integration. For erf and erfc, this is absolutely the case. $\endgroup$ – Ron Gordon Feb 7 '14 at 21:52
  • $\begingroup$ Here is where I stated my definition of closed-form: math.stackexchange.com/questions/562769/… $\endgroup$ – Ron Gordon Feb 7 '14 at 21:57

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