1
$\begingroup$

Consider the fractional differential equation \begin{align} D_{|x|}^\alpha u(x) +bu(x)=f(x) \end{align} with $0<\alpha<2$ on an unbounded domain. Instead of $D_{|x|}^\alpha$ one also often sees $\Delta^{\alpha/2}$, meaning the fractional (symmetric) Laplacian with Fourier transform \begin{align} \mathcal{F}\left( D_{|x|}^\alpha u(x)\right) = |\kappa |^\alpha \tilde u(\kappa). \end{align} To compute the Green's function $G(x)$ of the above problem, I need the inverse Fourier transform of $g(\kappa)$, given by \begin{align} g(\kappa) = \frac{1}{|\kappa|^\alpha +b}. \end{align} I know, that for \begin{align} h(\kappa) = \frac{1}{\kappa^\alpha +b} \end{align} the inverse Fourier transform is given by \begin{align} H(x) = x^{\alpha-1} E_{\alpha,\alpha}(-bx^\alpha) \end{align} where $E_{\alpha,\beta}(x)$ is the Mittag-Leffler function. Are there any references regarding this topic? I have also found numerous results for the Green's function on a domain that is a ball of size $R$ with $u(x)=u(|x|)$, meaning that we have a radial problem. However, this is not the case in my problem.

Any hint is very much appreciated.

Improve this question
$\endgroup$
5
  • $\begingroup$ I have a question about one of your statement. You said that for the function $g(\kappa)=\frac{1}{\kappa^\alpha +b}$, the inverse Fourier transform is given by $$ H(x) = x^{\alpha-1} E_{\alpha,\alpha}(-bx^\alpha) $$ where $E_{\alpha,\beta}(x)$ is the Mittag-Leffler function. As far as I know, it should be the inverse Laplace transform. $\endgroup$
    – CooLee
    May 17, 2015 at 12:08
  • $\begingroup$ Yes I think I made a mistake there, I will edit my question. Thanks! $\endgroup$
    – Thomas
    May 17, 2015 at 17:08
  • $\begingroup$ Do you know the H-function? Recently I studied this special function. I found it is very interesting because its Fourier transform is the Mittag-Lffler function. $\endgroup$
    – CooLee
    May 19, 2015 at 14:38
  • $\begingroup$ Yes I have read about the H-function. I have read the papers by Mainardi et al., where they consider the fundamental solution of $D_t^\beta u = D_x^\alpha u$ and tried to interpret my problem as the limit $\beta \rightarrow 0$, however, that did not really work out. $\endgroup$
    – Thomas
    May 20, 2015 at 10:12
  • $\begingroup$ Not sure what you mean by "computing" the Green's function, and not sure if two years later you are still interested, but the inverse Fourier transform $G(x)$ of $g(\kappa)$ is the resolvent (a.k.a. $b$-potential) kernel for the fractional Laplacian. Although there is no closed-form expression, you can get a lot of information about $G$; for example: $G(x) \approx \min(|x|^{-n-\alpha},|x|^{-n+\alpha})$ (where $n$ is the dimension). $\endgroup$
    – Mateusz Kwaśnicki
    Aug 23, 2017 at 21:52

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.