[itex]\imath\frac{\partial u}{\partial t} + \frac{\partial^2 u}{\partial x^2}=0[/itex](adsbygoogle = window.adsbygoogle || []).push({});

[itex]\left(x,t\right) = \int^{\infty}_{-\infty}A\left(k\right)e^{\imath\left(kx-wt\right)}dk[/itex]

[itex]u\left(x,0\right)=\delta\left(x\right) [/itex]

This is what I am working with. I am supposed to find the dispersion relation. So far I have gotten

[itex]A\left(k\right) = \frac{1}{2\pi}\int^{\infty}_{-\infty}\delta\left(x\right)e^{-\imath\left(kx\right)}dx = \frac{1}{2\pi}[/itex]

plugging this in to u(x,t) do I work with

[itex] u\left(x,t\right) =\frac{1}{2\pi}\int^{\infty}_{-\infty}e^{\imath\left(kx-wt\right)}dk[/itex]

This is where I am stuck. I know w(k) is the dispersion relation. If I put in the pde do I just deal with

[itex] \imath \left(-\imath w\right) + \frac{d^{2}u}{dt^{2}} = w +\frac{d^{2}u}{dt^{2}}=0 [/itex]

not sure how to pull out the dispersion equation or if I am even going the right route. Any clues on how to proceed would be most appreciated. Solving this equation does not seem to get me to where I want to be. Thanks!

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# Finding Dispersion relation

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