Fourier transform of PDE

[bobred]

Hi

I have the following PDE

\(\displaystyle \frac{\partial f}{\partial t}=-A\frac{\partial f^4}{\partial x^4}+B\frac{\partial f^2}{\partial t^2}\)

Show if Fourier transform satisfies (which I have, using FT of derivatives)

\(\displaystyle \frac{\partial \tilde{f}}{\partial t}(k,t)=-(Ak^4+Bk^2)\tilde{f}(k,t)\)

Then the PDE for \(\displaystyle f\) is also satisfied.

Would taking the inverse show this?

 

[bobred]

Any ideas?

 

[Deleted member 4993]

Hi

I have the following PDE

\(\displaystyle \frac{\partial f}{\partial t}=-A\frac{\partial f^4}{\partial x^4}+B\frac{\partial f^2}{\partial t^2}\)

Show if Fourier transform satisfies (which I have, using FT of derivatives)

\(\displaystyle \frac{\partial \tilde{f}}{\partial t}(k,t)=-(Ak^4+Bk^2)\tilde{f}(k,t)\)

Then the PDE for \(\displaystyle f\) is also satisfied.

Would taking the inverse show this?

What are your thoughts?

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[bobred]

Hi

Using the Fourier transform of derivatives \(\displaystyle \tilde{f}_{n}(k)=\left(i k\right)^{n}\tilde{f}\left(k\right)\) I get

\(\displaystyle \frac{\partial \tilde{f}}{\partial t}(k,t)=-(Ak^4+Bk^2)\tilde{f}(k,t)\)

It is just how this satisfies the partial differential equation that I can't see.

James