Timeline for Generalized functional for solution of PDEs
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Apr 16, 2022 at 16:30 | comment | added | Deane Yang | It depends on what your assumptions on $L$ are. For example, if $L$ is linear and has constant coefficients, then the existence of $L^{-1}$ was proved by Leon Ehrenpreis. References can be found here: mathoverflow.net/questions/186779/… | |
| Apr 16, 2022 at 11:01 | comment | added | Igor Khavkine | A simple, but case-by-case approach: Extend your solution $\Psi$ by 0 outside your domain. The derivatives in $L[\Psi]$ produce $\delta$ functions and their derivatives on the boundary. The structure of the coefficients of these $\delta$ functions encode the boundary conditions and the boundary data. Pick a source term $J$ that has $\delta$ functions on the boundary, whose coefficients have the same structure as the above $L[\Psi]$. Then solve $L[\Psi] = J$ with the only boundary condition being that vanishes outside your domain. Keywords: Duhamel's Principle, Multi-Layer Potentials. | |
| Apr 15, 2022 at 21:31 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Minor Formatting and Math Jaxing
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| Apr 15, 2022 at 16:00 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals from title, changed tag
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| Apr 15, 2022 at 15:48 | comment | added | Ben McKay | For a general linear differential operator, we don't even know if there are local solutions. | |
| Apr 15, 2022 at 15:43 | history | asked | Connor Dolan | CC BY-SA 4.0 |