How does one solve a fourth-order PDE of the form $\frac{\partial^4y}{\partial x^4}=c^2\frac{\partial^2y}{\partial t^2}$? It looks like a one dimensional wave equation, but I'm unfortunately very bad at PDEs.
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asked Sep 11, 2012 at 4:01
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$\begingroup$ Well, you can use the good old method of se $\endgroup$– RougeSegwayUserSep 11, 2012 at 4:10
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1$\begingroup$ As Chris was saying, just suppose $y(x,t)=X(x)T(t)$ and subsitute that into the problem to obtain $X''''T=c^2XT''$. Divide by $XT$ etc... $\endgroup$– James S. CookSep 11, 2012 at 4:17
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1 Answer
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You do almost the same thing as people explained in your other question. Unfortunately, you can only factor the operator into
$$ \left(\frac{\partial^2}{\partial x^2} - c\frac{\partial}{\partial t}\right) \left(\frac{\partial^2}{\partial x^2} + c\frac{\partial}{\partial t}\right)y = 0. $$
Then you have to solve a heat-equation like equation.
If your domain is finite, you should try separation of variables.
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$\begingroup$ Why does the finitiness of the domain matter for the separation of variables? $\endgroup$– LurcoJul 21, 2015 at 13:35
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1$\begingroup$ When the domain is finite, it may be possible to find a countable complete orthogonal basis that consists of functions that satisfy given homogeneous boundary conditions. $\endgroup$– TunococAug 10, 2015 at 7:07
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$\begingroup$ I see, so if the domain is not bounded then the separation of variables still works, but the solutions obtained by it do not generate the whole space of solutions? $\endgroup$– LurcoAug 10, 2015 at 10:29
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$\begingroup$ @Lurco There are many things that won't work. The most obvious reason is probably that when the number of solutions is uncountable, summation does not work. People have tried to use integration instead, and that yields two common alternatives to Fourier series: Laplace transform (half real line) and Fourier transform (whole real line). Of course there are many other alternatives (even for the finite case), but I think you get the picture. $\endgroup$– TunococAug 11, 2015 at 3:24
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$\begingroup$ You write, "Then you have to solve a heat-equation like equation." Can you be a bit more specific. How would you go about solving this? How would one use separation of variables on your equation?? $\endgroup$ May 17, 2022 at 4:49