Timeline for Solving Fractional Laplacian Equations with Boundary Condition
Current License: CC BY-SA 4.0
17 events
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| Apr 14, 2019 at 15:35 | vote | accept | Timothy Chu | ||
| Apr 4, 2019 at 14:54 | comment | added | Timothy Chu | By that, I mean a ball, unioned with any single point outside the ball. | |
| Apr 4, 2019 at 8:33 | comment | added | Mateusz Kwaśnicki | I do not think there are closed-form formulae for these. However, I do not understand what you mean by an "extra point". | |
| Apr 4, 2019 at 8:03 | comment | added | Timothy Chu | Thank you! Do you happen to know of Greens function for the One dimensional Fractional Laplacian in the union of two disjoint balls of differing radius? (Or more simply, the ball and one extra point.) Thanks for all your helpful and informative answers thus far! | |
| Apr 4, 2019 at 7:28 | comment | added | Mateusz Kwaśnicki | I am afraid I do not. "Generalized fractional Riesz potential/derivative" seem to be good search phrases. I know one paper by Igor Podlubny, Riesz Potential and Riemann-Liouville Fractional Integrals and Derivatives of Jacobi Polynomials, which turned out to be closely related to what I was trying to prove. It deals with the asymmetric case, but this seems to be a good way to distinguish between dimension 1 (where things typically work well without symmetry) and higher dimensions (where symmetry is sort of necessary). | |
| Apr 4, 2019 at 6:15 | comment | added | Timothy Chu | Do you have any good pointers to the literature on the one-dimensional case? | |
| Apr 4, 2019 at 6:14 | vote | accept | Timothy Chu | ||
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| Apr 1, 2019 at 15:31 | history | edited | Timothy Chu | CC BY-SA 4.0 |
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| Apr 1, 2019 at 15:19 | vote | accept | Timothy Chu | ||
| Apr 1, 2019 at 15:19 | |||||
| Apr 1, 2019 at 8:39 | comment | added | Mateusz Kwaśnicki | It just came to my mind that one can give an explicit solution in an integral form. I wrote up an answer below. I am in a rush, so please excuse me all typos and errors. | |
| Apr 1, 2019 at 8:38 | answer | added | Mateusz Kwaśnicki | timeline score: 3 | |
| Mar 31, 2019 at 16:58 | comment | added | Mateusz Kwaśnicki | One more comment: there is a huge literature on the one-dimensional case, that I do not know at all. In this case the fractional Laplacian is the composition of two one-sided fractional derivatives, and this often helps. | |
| Mar 31, 2019 at 16:56 | comment | added | Mateusz Kwaśnicki | Well, there's no closed-form expression for the Green function, and one has to live with it. What I meant is given as Thm 3.1 in Claudia Bucur's paper you mentioned. Another expression involves the hypergeometric function $_2F_1$. For your particular $f$ there might be a simpler expression for the solution, as this is essentially the Mellin transform of the Green function. I once worked with the fractional Laplacian and Mellin transforms (here); I do not remember anything similar to your question, though. | |
| Mar 31, 2019 at 16:07 | comment | added | Timothy Chu | Hi, the Green's function for the Fractional Laplacian is what I'm looking for. From what I know, Green's function for 1 dimensional ball is written in terms of a definite integral with the Poisson kernel. (Reference: arxiv.org/pdf/1502.06468.pdf, Definition 1.9, or web.ma.utexas.edu/mediawiki/index.php/…). A closed form without definite integrals would be useful for my application. Also, I can't find a copy of your survey online. Pasting the formula you mention would be great, especially if it also works for -1/2 < s < 0. | |
| Mar 31, 2019 at 9:07 | comment | added | Mateusz Kwaśnicki | Not sure if I understand the question correctly; the Green's function for the fractional Laplacian in a ball (and in particular, for an integral) is known since late 50's, and in fact it goes back to Riesz's 1938 paper. I recently wrote a survey on the frational Laplacian (M. Kwaśnicki, Fractional Laplace Operator and its Properties, in: A. Kochubei, Y. Luchko, Handbook of Fractional Calculus with Applications. Volume 1: Basic Theory, De Gruyter Reference, De Gruyter, Berlin, 2019), see Theorem 3.4 there. I can copy-and-paste the formula here if this is what you are looking for. | |
| Mar 31, 2019 at 4:41 | history | edited | Timothy Chu |
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| Mar 31, 2019 at 3:44 | history | asked | Timothy Chu | CC BY-SA 4.0 |