Timeline for Method of characteristics for higher order PDEs in more than two variables
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Mar 21 at 23:22 | answer | added | wijohns777 | timeline score: 1 | |
| Oct 31, 2023 at 4:30 | history | became hot network question | |||
| Oct 31, 2023 at 3:16 | comment | added | Willie Wong | If you set $y_1 = x_1$, $y_2 = x_2 + x_3$ and $y_3 = x_2 - x_3$, the specific example you asked about can be seen to just be the wave equation (with $y_2$ being the time direction). The fundamental solution of this is well-known, and you can see based on it that a method of characteristics solution is not really possible. | |
| Oct 31, 2023 at 3:12 | answer | added | Willie Wong | timeline score: 19 | |
| Oct 31, 2023 at 0:55 | comment | added | Deane Yang | In general, there is no way to solve a 2nd order PDE in 3 variables using the method of characteristics. If there were, you would have learned about it, and we would not be struggling so much with PDE theory. The question is whether there are examples of 2nd order PDEs in 3 variables that can be solved by such a method, and how does one detect such a PDE. | |
| Oct 30, 2023 at 23:13 | comment | added | Puzzled | @ThomasKojar. In the reference you gave there are just examples with first order PDEs. | |
| Oct 30, 2023 at 23:10 | comment | added | Puzzled | I added an example to my question to make it more clear. | |
| Oct 30, 2023 at 23:08 | history | edited | Puzzled | CC BY-SA 4.0 |
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| Oct 30, 2023 at 22:59 | comment | added | Willie Wong | To OP: when you say more than two variables to you mean having a domain with more than two dimensions, or having more than 2 unknowns? It is a bit of folklore that method of characteristics is impossible for vector-valued equations in higher dimensions, and similarly for higher order scalar equations in higher dimensions; I'll try to see if I can find a reference or a heuristic argument for you. | |
| Oct 30, 2023 at 21:56 | history | edited | Puzzled | CC BY-SA 4.0 |
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| Oct 30, 2023 at 20:30 | history | asked | Puzzled | CC BY-SA 4.0 |