I have a convection-diffusion PDE in the form of:
$$\frac{{\partial x}}{{\partial t}} = w\frac{{{\partial ^2}x}}{{\partial {z^2}}} - \frac{{\partial x}}{{\partial z}}$$
Assuming I know the initial condition and the parameter $w$, is there a way to estimate how much time it would take the system to reach a steady-state? I know I can just solve it numerically, but I wonder if there is another way.
Thank you!
Let us solve the Cauchy problem by using the Fourier method. Spatial Fourier transforms $\hat x= \int x e^{-\text i k z}\text d z$ satisfy $\hat x_t = -(w k + \text i ) k x$, which solution reads $$ \hat x(k,t) = X(k)\, e^{-w k^2 t}e^{-\text i k t} $$ where $X$ is the Fourier transform of the initial condition. Inverse Fourier transformation yields $$ x(z,t) = \frac{1}{2\pi} \int X(k)\, e^{-w k^2 t} e^{- \text i k ( t-z)} \text dk \, . $$ If the initial condition reads $x(z,0) = \cos(\kappa z)$, then its Fourier transform satisfies $X(k) = \pi\big(\delta(k-\kappa) + \delta(k+\kappa)\big)$. Thus, we find $$ x(z,t) = e^{-w \kappa^2 t} \cos\!\big(\kappa (z-t)\big) . $$ One notes that the characteristic time $w^{-1} \kappa^{-2}$ to reach the steady state $x^* = 0$ depends on the initial condition through the parameter $\kappa$.