Homogenous Heat equation on 2d rectangle, $[0,a]\times[0,b]$, with time independent initial conditions and homogenous Neumann boundaries
$$\frac{\partial{u}}{\partial{t}} = \alpha^2 (\frac{\partial^2{u}}{\partial^2{x}} + \frac{\partial^2{u}}{\partial^2{y}}), \quad0<x<a, \quad 0<y<b, \quad 0<t$$ $$IV: u(x,y,0) = f(x,y) , \quad0<x<a, \quad 0<y<b$$ $$BC: \begin{cases} \frac{\partial{u}}{\partial{x}}(0,y,t) = \frac{\partial{u}}{\partial{x}}(a,y,t) = 0 , \quad 0<y<b, \quad 0<t \\ \frac{\partial{u}}{\partial{y}}(x,0,t) = \frac{\partial{u}}{\partial{y}}(x,b,t) = 0 , \quad 0<x<a, \quad 0<t \end{cases}$$
I tried to find a solution using separation of variables to get Fourier series. I could not find it anywhere so am asking, is this correct?
$$u(x,y,t) = \sum^\infty_{m=1}{\sum^\infty_{n=1}{A_{nm}\cos{(\frac{ n\pi x}{a})}\cos{(\frac{ m\pi y}{b})}e^{-\alpha^2 \pi^2 (\frac{n^2}{a^2}+\frac{m^2}{b^2})t}}}$$ where $$A_{nm} = \frac{4}{ab} \int^a_0 {\int^b_0 {f(x,y) \cos{(\frac{ n\pi x}{a})}\cos{(\frac{ m\pi y}{b})} dy}dx}$$
The separation of variables leads to $$ \left\{ \begin{array}{lcc} X'' + \lambda X & = & 0 \\ Y'' + \mu Y & = & 0 \end{array} \right. $$ The boundary conditions $$ \left. \begin{array}{rcl} u_x(0,y,t) = 0 \ , & u_x(a,y,t)=0 \\ u_y(x,0,t) = 0 \ , & u_y(x,b,t)=0 \end{array} \right\} $$
Imply
$$ \left. \begin{array}{rcl} X'(0)=0 \ , & X'(a)=0 \\ Y'(0) = 0 \ , & Y'(b)=0 \end{array} \right\} $$
These are problem of Sturm Liouville
The eigenvalues are $ \lambda_0 = 0 \ , \lambda_n = \frac{n^2 \pi^2}{a^2} $ and $ \mu_0 = 0 \ , \mu_n = \frac{n^2 \pi^2}{b^2} $
The eigenfunctions are $ X_0 = 1 , \ X_n = cos(\frac{n\pi}{a}x) $ and $ Y_0 = 1 , \ Y_n = cos(\frac{n\pi}{b}y) $
On the other hand
$$ T(t) = e^{ -\alpha^2 (\lambda + \mu)t } $$
By the superposition principle
$$ u(x,y,t) = A_{00} + \sum_{n=1}^{\infty} A_{n0} e^{ -\alpha^2\frac{ n^2 \pi^2 }{a^2}t }cos\left( \frac{ n \pi }{a} x \right ) + \sum_{m=1}^{\infty} A_{0m} e^{ -\alpha^2\frac{ m^2 \pi^2 }{b^2}t }cos\left( \frac{ m \pi }{b} y \right ) $$ $$ + \sum_{n=1}^{\infty}\sum_{m=1}^{\infty} A_{nm} e^{ -\alpha^2\pi^2(\frac{n^2}{a^2} + \frac{m^2}{b^2} )t } cos\left( \frac{n\pi}{a} x \right)cos\left( \frac{m\pi}{b} y \right) $$
I suppose that you can compute the coefficients of the double cosine series.