I have this: $$\frac{\partial c}{\partial t} + p\frac{\partial c}{\partial z}+\lambda p\frac{\partial^{2} c}{\partial z\partial t}-\frac{\partial^{2} c}{\partial z^2}=0\quad(1)$$ $$c(z,0)=\delta(z)$$
I attempted the solution using Fourier Transform and I got: $$\mathcal F\{(1)\}\to \frac{dC}{dt}+p(i\omega)C+\lambda p(i\omega )\frac{dC}{dt}+\omega^2 C=0\quad(2)$$ $$\mathcal F\{c(z,0)\}=1$$ Note that $C(\omega,t)=\mathcal F\{c(z,t)\}$
Rearrange (2) I got: $$C=\exp\left(-\frac{pi\omega+\omega^2}{\lambda pi\omega+1}t\right)$$
Then I don't know how to inverse Fourier Transform to the solution in time domain. I have heard of residue theorem but don't know how to use it. Would anyone verify my work and/or give insight into this? I have been stuck for two days now :(
Thank you in advance!