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.
Wave Equation - like 4th Order PDE
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pde
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1As 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... – 2012-09-11
1 Answers
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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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0@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. – 2015-08-11