# Solving an PDE with the Fourier Transform

I am currently trying to solve the Wave Equation with friction, i.e.: $$u_{tt}+\alpha u_t=c^2u_{xx}, \>\>\>\>\alpha>0, \>\>\>\> u(x,0)=\phi(x), \>\>u_t(x,0)=\psi(x)$$ Now, I Fourier Transformed the PDE into the form: $$\widehat u_{tt}+\alpha\widehat u_t+c^2k^2 \widehat u=0$$ Which is just a PDE, with the solution of the form: $$\widehat u(k,t) = c_1\exp\left(\frac{(-\alpha+\sqrt{\alpha^2-4c^2k^2})t}{2}\right)+c_2\exp\left(\frac{(-\alpha-\sqrt{\alpha^2-4c^2k^2})t}{2}\right)$$ How do I reverse this using the Fourier Transform? Also, I don't even know if this is the solution to the PDE as I don't know if the roots of the Auxiliary Equation are even real. Can someone please clarify what I'm supposed to do next?

• It's okay if the roots aren't real, you're going to have three cases depending on that square root, each one giving you a different solution (possibly). – DaveNine Dec 16 '17 at 5:16
• Right? But the question is posed such that it leads to one answer... Which I just don;t understand – Felicio Grande Dec 16 '17 at 5:20
• The Fourier Transform isn't probably the best method for this problem, in general it's already pretty difficult to use it on wave-type equations, I might suggest the Laplace Transform method, which I've found does lead to a integral representation, although it is not very nice looking with general $\phi$ and $\psi$. – DaveNine Dec 16 '17 at 10:32