Apply Fourier Transform to Solve t2u_t-u_x=g(x)

[Markov2]

I need to apply Fourier transform to solve the following: $t^2u_t-u_x=g(x),$ $x\in\mathbb R,$ $t>0$ and $u(x,1)=0,$ $x\in\mathbb R.$
How do I apply the Fourier transform for $t^2u_t$ ?

Thanks!

 

[ThePerfectHacker]

I need to apply Fourier transform to solve the following: $t^2u_t-u_x=g(x),$ $x\in\mathbb R,$ $t>0$ and $u(x,1)=0,$ $x\in\mathbb R.$
How do I apply the Fourier transform for $t^2u_t$ ?

Thanks!

Take the Fourier transform with respect to $x$ then you simply treat $t$ as a constant.

$$ \int \limits_{-\infty}^{\infty} t^2 \frac{\partial u}{\partial t} e^{-i\omega x} dx = t^2 \frac{\partial }{\partial t}\int \limits_{-\infty}^{\infty} u(x,t) e^{-i\omega x} dx = t^2 \frac{\partial U}{\partial t}$$

Here we differenciated under the integral sign, and $U(\omega , t)$ is the notation for the Fourier transform of $u(x,t)$ with respect to $x$.

 

[Markov2]

Ah, then I have ${{t}^{2}}\dfrac{\partial U}{\partial t}-iwU=F(g)$ and I need to solve that ODE, first I need a particular solution.

Does this look right?

 

[Markov2]

Can anyone confirm this please?