I have a very specific problem concerning the solution of the two-dimensional diffusion equation in a rectangular domain. The problem I have is that I don't understand the way to get to the final solution by applying Neumann boundary conditions on all four domain boundaries. The solution is from the literature and would be helpful for me.
To the problem:
The domain is defined as a rectangle, with its boundaries parallel to the coordinate axis, and $0<x<a$ and $0<y<b$. Given the time-independent diffusion equation $\nabla F=\gamma^2F$, where $F=f(x,y)$. By finding two solutions, both satisfying the equation, and adding them, one can find a general solution of the form $$ F = \sum_{k=0}^K \left(v_{1,k}\cos p_kx + v_{2,k}\sin p_kx\right)\left(v_{3,k}\cosh\xi_ky+v_{4,k}\sinh\xi_ky\right) \\ \hspace{75pt} + \left(v_{5,k}\cosh\zeta_kx + v_{6,k}\sinh\zeta_kx\right)\left(v_{7,k}\cos q_ky+v_{8,k}\sin q_ky\right) $$ when $p_k=k\pi/a$, $q_k=k\pi/b$, $\xi_k=\sqrt{\gamma^2+p_k^2}$ and $\zeta_k=\sqrt{\gamma^2+q_k^2}$. Here, as far as I understand, the cosine and sine terms, with $p_k$ and $q_k$, are supposed to make the solution satisfy arbitrary boundary conditions, by Fourier series expansion.
From here it gets complicated for me:
By applying Neumann boundary conditions to the boundary of the domain, three homogenous and one inhomogeneous, as follows $$ \frac{\partial F}{\partial y}\bigg\vert_{y=0} \neq 0 \\ \frac{\partial F}{\partial y}\bigg\vert_{y=b} = 0 \\ \frac{\partial F}{\partial x}\bigg\vert_{x=0} = 0 \\ \frac{\partial F}{\partial x}\bigg\vert_{x=a} = 0 $$ one can find the much simpler solution from the original general solution, for the case of one boundary condition not being zero as $$ F_1 = \sum_{k=0}^K w_k\cos p_kx\cosh\xi_ky. $$ Repeating this four times, setting every individual boundary $\neq0$, when the other three are zero, and adding the four solutions yields $$ F = \sum_{k=0}^K \left(c_{1,k}\cosh\xi_k(y-b) + c_{3,k}\cosh\xi_ky\right)\frac{\cos p_kx}{\cosh\xi_kb} \\ \hspace{75pt} + \left(c_{4,k}\cosh\zeta(x-a)+c_{2,k}\cosh\zeta_kx\right)\frac{\cos q_ky}{\cosh\zeta_ka}. $$ These final two steps are what I don't understand. In the literature, this is said to be the final solution for the original diffusion equation, and it is used there for further calculations. I want to understand the steps that lead to this solution with four unknowns $c_{i,k}$, coming from the first solution with eight unknowns $v_{j,k}$. Especially tricky for me is the part, where the values on the boundaries are set $\neq0$, but not further specified.
Can someone enlighten me, or give me literature hints, that explain this in particular?
Best regards and a very nice evening! Thomas