How do I solve this wave equation with friction using Fourier transform? $$\left\{ \begin{array}{l l} u_{tt} - \alpha u_{t} = c^2u_{xx} & \quad -\infty< x< \infty , t>0, \\ \quad u(x,0) = \phi(x), \\ \quad u_t(x,0) = \psi(x). \end{array} \right. $$
where $\alpha, \beta, c \in \mathbb{R}$ are constants?
I think I understand that for this problem, I need to let $F(k,t) = \int_{-\infty}^{\infty} u(x,t)e^{-ikx}\,dx$
Then, by transforming the equation I think it becomes... $F(k,t)= A(k)\cos(kct)+B(k)\sin(kct)$