Suppose I have following PDE given as: $\frac{\partial f(x,y,t)}{\partial t}=-\Big( u \frac{\partial f(x,y,t)}{\partial x} + v \frac{\partial f(x,y,t)}{\partial y}\Big)$. I want to solve it using Fourier transform in two dimensions. \begin{equation} \cal{F}(a,b,t)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} f(x,y,t) ~e^{-i(ax+by)} dx~dy \end{equation} \begin{equation} f(x,y,t)=\frac{1}{(2\pi)^2}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \cal{F}(a,b,t) ~e^{i(ax+by)} da~db \end{equation} Then can I solve it using : $$\frac{\partial \cal{F(a,b,t)}}{\partial t}=-i\Big ( ua + vb \Big) \cal{F}(a,b,t)$$ and then integrating it w.r.t. 't' and taking 2D inverse Fourier transform.
Am I correct, is the approach in 2-D is just an extension of solving PDE using Fourier transform in 1-D?