Timeline for Space of solutions to a fourth order wave equation
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 28, 2021 at 15:02 | vote | accept | Jojo | ||
| Jul 28, 2021 at 15:02 | comment | added | Jojo | 1) ok thanks I'll try to come up with a more clear question and make a new post 2) I see, thanks very much | |
| Jul 27, 2021 at 21:01 | comment | added | Igor Khavkine | @Joe 1) I don't know what to say about orthogonality, except the usual Fourier inversion formula. If it is not what you need, you might have to formulate your question more sharply. 2) Expand all $k$-derivatives. Any term like $h(k) \delta^{(n>1)}(k^2)$ will not be annihilated by $k^4$, unless $h(k)\propto (k^2)^{n-1}$, but then $k^2\delta^{n}(k^2) = -n\delta^{(n-1)}(k^2)$, etc. So these higher order derivatives of $\delta$ are redundant. 3) About separability, the issue is just in the terminology. | |
| Jul 27, 2021 at 17:43 | comment | added | Jojo | When I said the equation appears to me not to be separable, I meant that eg. in 2D taking $\phi(x,t) = X(x)T(t)$, we get $X'''' T - 2 X'' T'' + X T''' =0$, and I can't see how to seperate out $T$ and $X$ here. Like you say though, I think this is not important given your Fourier space analysis | |
| Jul 27, 2021 at 17:41 | comment | added | Jojo | 2) Using that $u^3 \delta''(u) = 0$, I can also solve the equation in Fourier space by $\hat{\phi}(k) = \Box_k \left[H(k) \delta(k^2)\right]$, because the $\Box_k$ acting on the delta function gives me a factor of $k^2$ additional to the $k^4$ coming from the original PDE. It looks to me like I could potentially continue this process using $u^{n+1} \delta^{(n)}(u) = 0$ to get more and more solutions. Did you use some kind of principle to select only the solution involving $\delta'(k^2)$ and not these higher solutions? | |
| Jul 27, 2021 at 17:35 | comment | added | Jojo | Thanks for a great answer, I can see this is definitely a good way to think about the problem. I have two questions; 1) Suppose $\phi_1,\phi_2 \in L^2$, then I can expand them in a basis of plane waves. Can I think of $a\cdot x \phi_2$ as having an expansion in terms of some basis, (eg. $(a\cdot x) e^{i k \cdot x}$ or similar), such that these basis elements are orthogonal each other and to the plane waves? | |
| Jul 26, 2021 at 22:32 | history | answered | Igor Khavkine | CC BY-SA 4.0 |