Timeline for Analytic solution of a system of linear, hyperbolic, first order, partial differential equations
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Jun 18, 2014 at 14:31 | vote | accept | FraSchelle | ||
| Jun 17, 2014 at 23:28 | answer | added | Robert Israel | timeline score: 3 | |
| Jun 17, 2014 at 19:24 | answer | added | Robert Bryant | timeline score: 2 | |
| Jun 17, 2014 at 14:53 | comment | added | Igor Khavkine | Without looking at the Courant-Lax article, I would guess that the Dyson series arises by treating $\mathbf{B}(t)$ as a perturbation. If that term is absent, then the system decouples and can be solved by the methods I described above. I don't know if you'd be happy with a series solution, but unless $\mathbf{B}(t)$ has some very special algebraic structure, this might be the best you can do. | |
| Jun 17, 2014 at 12:51 | comment | added | FraSchelle | @IgorKhavkine Thanks for your comment. Actually, it is impossible to find a $t$-independent diagonalisation of $B$ which will keep $A$ diagonal. It seems to me that Courant and Lax propose a solution for this problem in section 3 of the cited paper, in a form which seems to me a kind of Dyson series. But I have difficulties to see if I'm correct about that. Also, in my problem the boundary conditions are known, they are some initial boundary conditions. | |
| Jun 17, 2014 at 12:47 | history | edited | FraSchelle | CC BY-SA 3.0 |
added 237 characters in body; edited title
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| Jun 16, 2014 at 19:26 | comment | added | Igor Khavkine | If you can find a time independent similarity transformation that simultaneously diagonalizes both $\mathbf{A}$ and $\mathbf{B}(t)$, then your system decouples into four first order, scalar, linear equations. Each of those can be solved by the method of characteristics. Perhaps that is enough. If not, one possibility is that $B(t)$ could be split into two parts, one part would allow for simultaneous diagnalization, the other might be treated as a perturbation. If the equations do decouple, perhaps that will give you an idea of what kind of boundary/initial conditions would be allowed. | |
| Jun 16, 2014 at 18:41 | review | First posts | |||
| Jun 16, 2014 at 18:41 | |||||
| Jun 16, 2014 at 18:23 | history | asked | FraSchelle | CC BY-SA 3.0 |