Use Fourier transform to solve PDE damped wave equation

[math2011]

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Use Fourier transforms to solve the PDE
$$\displaystyle \frac{\partial^2 \phi}{\partial t^2} + \beta \frac{\partial \phi}{\partial t} = c^2 \frac{\partial^2 \phi}{\partial x^2}$$, $$- \infty < x < \infty$$, $$t > 0$$
subject to
$$\phi(x,0) = f(x)$$
$$\displaystyle \left. \frac{\partial \phi}{\partial t} \right\lvert_{t=0} = g(x)$$.

ATTEMPT:

I took Fourier transforms in $$x$$ and got the ODE
$$\Phi_{tt}(k,t) + \beta \Phi_t(k,t) + c^2 k^2 \Phi(k,t) = 0$$.

Trying to solve this ODE I get
$$\lambda = \frac{-\beta \pm \sqrt{\beta^2 - 4 c^2 k^2}}{2}$$.

This results in there are three solutions to the ODE depending on the value of $$\lambda$$. Am I supposed find solve $$\phi$$ for all three solutions?

How can I proceed next?

Should I have taken Fourier transforms in $$t$$ instead in the first step?

 

[jvicens]

I would first like to clarify the problem and the approach that you have taken. The given PDE is a damped wave equation with a damping coefficient of \beta. You have correctly taken Fourier transforms in x to obtain an ODE in k and t. However, I would suggest taking Fourier transforms in t as well, to simplify the ODE further. This will result in an ODE in k and \omega, where \omega is the frequency variable in the Fourier transform of time. This will give you a characteristic equation in terms of \omega, which can then be solved for \Phi(k,\omega).

After obtaining the solution in terms of \Phi(k,\omega), you can then take inverse Fourier transforms in both variables to obtain the solution in terms of x and t, i.e. \phi(x,t). This will give you a general solution in terms of the Fourier transform of the initial conditions f(x) and g(x). You can then use the inverse Fourier transform of the initial conditions to obtain a specific solution for \phi(x,t).

In terms of the three solutions that you have obtained, they may correspond to different boundary conditions or initial conditions. It is important to check if these solutions satisfy the given boundary conditions and initial conditions. If they do, then you can consider all three solutions as part of the general solution. Otherwise, you may need to consider only the solutions that satisfy the given conditions.

I hope this helps in your understanding and approach towards solving the PDE using Fourier transforms. If you need further assistance, please do not hesitate to reach out for help.