I'm trying to solve a PDE subjected to one boundary condition.
$u_t=u_{xx}+u_{yy}+5u_x-6u_y$ Subjected to the boundary condition $u(x, y,0)=f(x,y) \forall f\in L^1(\Bbb{R^2})$ Also assume conditions to apply F.T. to derivatives hold.
Attempt: I've applied F.T. to both sides and obtained an equation. From there, I couldn't proceed.
$\hat u'=\hat u(w,t)(-w^2+5iw)+\hat u(v,t)(-w^2-6iw)$
Since the function depends on more than 1 variable, the two-dimensional Fourier Transform is required. How would I implement it? My idea was to ignore the boundaries and get to where inverse F.T. is required, and then apply the 2D F.T. I'm stuck now as I've never dealt with such an equation before. What do I do with the 2 terms? I thought about treating it as 2 different equations then merging it, however I doubt that will work.
Any help is greatly appreciated.
The Fourier transform will be of the form $\hat u(w,v,t)$. Taking Fourier transform on both sides ion the equation you will get $$ \frac{d\hat u}{dt}=(-w^2-v^2+5\,i\,w-6\,i\,v)\,\hat u $$ and $$ \hat u=C\,e^{(-w^2-v^2+5\,i\,w-6\,i\,v)t}. $$