0
$\begingroup$

$$\partial_t u + \partial^{3}_x u = 0$$ show that this PDE has a particular solution in the form of a plane wave, $$u_k(x,t) = \exp(-i\,w\,t - i\,k\,x)$$ if the frequency $w$ obeys the appropriate dispersion relation, $w = w(k)$.

In the past I have solved up to 2nd order PDE's using separation of variables, Sturm-Liouville method. But since the all coefficients in the PDE has different powers I am confused how to solve this problem. Seems like I can not solve this problem using the methods I used to solve 2nd order PDE's. Any help?

$\endgroup$
  • $\begingroup$ What happens when you plug this into the PDE. Can you show that it works, provided there is a certain relation (which you must find) between $w$ and $k$? $\endgroup$ – GEdgar Dec 12 '16 at 18:09
2
$\begingroup$

Since you have a linear PDE you can use the principle of superposition to form solutions with $$ u \sim \exp(−iwt−ikx) $$ then inserting the plane wave solution you obtain $$ -i\omega u + (-ik)^3u = \left(-i\omega-i^3k^3\right)u = -i(\omega -k^3)u = 0 $$ or the dispersion relation is $$ w(k) = k^3 $$ if this relation is true then you can use the plane wave above.

If you want to solve you are missing quite a few pieces of information such as initial conditions etc..

$\endgroup$
  • 1
    $\begingroup$ I get $w=k^3$. Here $i=\sqrt{-1}$ $\endgroup$ – Aleksas Domarkas Jun 4 '18 at 6:39
  • $\begingroup$ @AleksasDomarkas I agree. I am not sure why I did it that way - I must have had a mini brain lapse. Thanks! $\endgroup$ – Chinny84 Jun 4 '18 at 14:30

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.