I was wondering how you would go about using Fourier Transforms to solve a heat equation that has an extra dimension like:
$u_t = k(u_{xx} + u_{yy})$
From what I understand, if you had just the 2-D heat equation $u_t = u_{xx}$ , you would take that:
$\mathfrak{F}(u(x,t) = U(w,t)$, and then
$\mathfrak{F}({\frac{\partial^n u}{\partial t^n}}) = \frac{d^n U}{dt^n}$, and $\mathfrak{F}({\frac{\partial^n u}{\partial x^n}}) = (iw)^nU(w,t)$
And simply substitute. But how do you deal with this $u_{yy}$ term? What is $\mathfrak{F}({\frac{\partial^n u}{\partial y^n}})$ equal to?
Note: This is from a university intro to partial differential equations course. My understanding of PDE's is pretty basic.
In higher dimension, just replace products by scalar products, that is if $x=(x_1,x_2)$ and $w = (w_1,w_2)$, then $x\cdot w = x_1w_1 + x_2w_2$, your equation is $$ \partial_t u = k\left(\partial_{x_1x_1}u + \partial_{x_2x_2}u\right) = k \,\nabla\cdot\nabla u, $$ and the Fourier transform is $$ U(w)={\mathcal F}(u)(w) = \iint_{\Bbb R^2} e^{-ix\cdot w} \, u(x)\,\mathrm d x $$ So that $$ \partial_t U(w) = k\,iw\cdot iw \,U(w) = - k\,|w|^2 \,U(w) $$