- #1
Hart
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Homework Statement
A differential equation [*] is given by:
[tex] \frac{\partial^{3}u}{\partial x^{3}} + 2 \left( \frac{\partial u}{\partial x} \right) = \frac{\partial u}{\partial t} [/tex]
By first Fourier transforming the equation (*) with respect to x, show by substitution that:
[tex] u(k,t) = A(k)e^{-i \left( k^{3}-2k \right) t} [/tex]
is the Fourier transform of the solution of [*] , where A(k) is an unknown function of k.
Homework Equations
Derivative property of Fourier transforms (with respect to x):
[tex] F\left[ \frac{\partial f}{\partial x} \right] = ikF[f] [/tex]
Also know:
[tex] F(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x)e^{ikx}dx [/tex]
[tex] F(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} F(k)e^{ikx}dk [/tex]
The Attempt at a Solution
.. Fourier transform [*] with respect to x, treating t as a parameter. k used as a variable.
Firstly rearrange [*] to get just [tex] \frac{\partial^{3}u}{\partial x^{3}} [/tex] on the LHS
Then the LHS:
[tex] F\left[ \frac{\partial^{3}u}{\partial x^{3}} \right] = F\left[ \frac{\partial^{2}}{\partial x^{2}}\left( \frac{\partial u}{\partial x} \right)\ right] = -k^{2}\left( \frac{\partial^{2}}{\partial x^{2}} \right)F [/tex]
Then the RHS:
RHS = [tex] \frac{\partial u}{\partial t} - 2 \left( \frac{\partial u}{\partial x} \right) [/tex]
can take the t term outside the integral as it is just a constant parameter, therefore:
[tex] F\left[ \left(\frac{\partial u}{\partial t}\right)-\left(\frac{2\partial u}{\partial x}\right) \right] = \frac{1}{\sqrt{2\pi}}\left(\frac{\partial}{\partial t} \right)\int_{-\infty}^{\infty}\frac{-2\partial u}{\partial x}e^{-ikx} dx [/tex]
.. then obviously some more steps, not sure really where to go next though.
Not sure if I'm even going about this in vaguely the right way .. so help will definitely be appreciated!