How can I transform the following PDE into an ODE? I tried using three different functions $H(x),G(y)$ and $F(t)$ but that didn't help hence I did not post it here. I really hope someone can help me out, I did not find any book which shows how to turn a PDE into an ODE in a 3 variable case. Thanks.
$a\frac{\partial^2 V(x,y,t)}{\partial^2 x}+a\frac{\partial^2 V(x,y,t)}{\partial^2 y}+W(x,y,t) - b\frac{\partial V(x,y,t)}{\partial t} =0$
which has boundaries:
$\frac{\partial V(x,0,t)}{\partial y} =0$
$\frac{\partial V(0,y,t)}{\partial x}=0$
$\frac{\partial V(x,7,t)}{\partial y} =0$
$\frac{\partial V(5,y,t)}{\partial x} =0$
for all $t\geq0$, $0\leq x\leq5,0\leq y\leq7$
The functions $X_n(x) = \sin(n\pi x/5)$, $Y_m(y)=\sin(m\pi x/7)$ can be used to expand functions in $x$, $y$ on $[0,5]\times[0,7]$. The set of functions $\{ X_n(x)Y_m(y) \}$ is an orthogonal basis for $L^2([0,5]\times[0,7])$.
The solution $V$ can be written in terms of these $$ V(x,y,t) = \sum_{n=1}^{\infty}\sum_{m=1}^{\infty}C_{n,m}(t)X_{n}(x)Y_{m}(y). \;\;\; (\dagger) $$ The coefficients are determined by multiplying both sides of $(\dagger)$ by $X_n(x)Y_m(y)$ and integrating over $[0,5]\times[0,7]$ in order to obtain \begin{align} C_{n,m}(t) &= \frac{\int_{0}^{7}\int_{0}^{5}V(x,y,t)X_n(x)Y_m(y)dxdy} {\int_{0}^{7}\int_{0}^{5}X_{n}(x)^2Y_{m}(y)^2dxdy} \\ &= \frac{4}{35}\int_{0}^{7}\int_{0}^{5}V(x,y,t)X_n(x)Y_m(y)dxdy. \end{align} Your equation is $$ a\frac{\partial^2 V}{\partial x^2}+a\frac{\partial^2 V}{\partial y^2} + W = b\frac{\partial V}{\partial t}. $$ Multiplying both sides of this equation by $\frac{4}{35}X_n(x)Y_m(y)$, integrating over $[0,5]\times[0,7]$, and performing integration by parts on the spatial derivative terms gives $$ -a\left[\frac{n^2\pi^2}{5^2}+\frac{m^2\pi^2}{7^2}\right]C_{n,m}(t) + \frac{4}{35}\int_{0}^{7}\int_{0}^{5}W(x,y,t)X_{n}(x)Y_{m}(y)dxdy = bC_{n,m}'(t). $$ Assuming you know the function $W$, the above is a first order ODE in $t$ that can be solved, provided $C_{n,m}(0)$ is known, which are the coefficients associated with the initial function $V(x,y,t=0)$.