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In this section we consider separation of variables for multidimentional equations in the simplest cases. More complicated cases will be considered in Chapter 7.
We consider only wave equation \begin{equation} u_{tt}-c^2\Delta u =0 \label{eq-6.6.1} \end{equation} in the rectangular box $\Omega=\{\mathbf{x}=(x_1,x_2,\ldots ,x_n)\colon 0 < x_1 < a_1, \ldots, 0<x_n< a_n\}$.
For $n=2$ we have a rectangle $\Omega=\{(x,y)\colon 0< x <a,0<y< b\}$.
Then, plugging $u(x,y,t)=X(x)Y(y)T(t)$ we get \begin{equation*} T''(t)X(x)Y(y)- c^2 T(t)X''(x)Y(y)- c^2 T(t)X(x)Y''(y), \end{equation*} which can be rewritten as \begin{equation} \underbracket{\frac{T''(t)}{T(t)}}- c^2 \underbracket{\frac{X''(x)}{X(x)}}- c^2 \underbracket{\frac{Y''(y)}{Y(y)}}=0. \label{eq-6.6.2} \end{equation} Observe that each of selected terms depends only on it's own variable, and repeating arguments of 1D-case we conclude that it must be a constant: \begin{equation*} \frac{X''}{X}=-\lambda, \frac{Y''}{Y}=-\mu, \quad \frac{T''}{T}=-\nu = -c^2(\lambda+\mu). \end{equation*} In other words \begin{equation} X''+\lambda X=0,\qquad Y''+\mu Y=0,\qquad T''+\nu T=0.\label{eq-6.6.3} \end{equation}
Next, we assume that $u$ must satisfy Dirichlet boundary conditions: \begin{equation} u|_{\Gamma}=0, \label{eq-6.6.4} \end{equation} where $\Gamma=\partial\Omega$ is the boundary of $\Omega$, which in our case consists of four segments $\{(x,y)\colon x=0,\ 0<y<b\}$, $\{(x,y)\colon x=a,\ 0<y<b\}$, $\{(x,y)\colon y=0,\ 0<x<a\}$, $\{(x,y)\colon y=b,\ 0<x<a\}$.
In $n$-dimensional case $\Gamma$ consists of $2n$ rectangular boxes of dimension $(n-1)$ each.
Then, repeating arguments of 1D-case we conclude that
\begin{equation}
X(0)=X(a)=0,\qquad
Y(0)=Y(b)=0.
\label{eq-6.6.5}
\end{equation}
Thus we got the same problems for $X$ and for $Y$ as in 1D-case, and therefore
\begin{align}
&X_m(x)=\sin (\frac{\pi m x}{a}),
&&Y_n (y)=\sin (\frac{\pi n y}{b}),
\label{eq-6.6.6}\\
&\lambda_m= (\frac{\pi m }{a})^2,
&&\mu_n =(\frac{\pi n }{b})^2
\label{eq-6.6.7}
\end{align}
with $m,n=1,2,\ldots$, and therefore
\begin{gather}
T_{mn}(t)= A_{mn}\cos (\omega_{mn}t)+
B_{mn}\sin (\omega_{mn}t),\label{eq-6.6.8}\\
\omega_{mn}= c \pi \bigl( \frac{m^2}{a^2}+\frac{n^2}{b^2}\bigr)^{\frac{1}{2}}.
\label{eq-6.6.9}
\end{gather}
Finally,
\begin{multline}
u_{mn}(x,y,t)=
\Bigl(A_{mn}\cos (\omega_{mn}t)+
B_{mn}\sin (\omega_{mn}t)\Bigr)
\sin (\frac{\pi m x}{a})
\sin (\frac{\pi n y}{b})
\label{eq-6.6.10}
\end{multline}
and the general solution is derived in the form of multidimensional Fourier series
\begin{multline}
u(x,y,t)=
\sum_{m=1}^\infty \sum_{n=1}^\infty
\Bigl(A_{mn}\cos (\omega_{mn}t)+
B_{mn}\sin (\omega_{mn}t)\Bigr)
\sin (\frac{\pi m x}{a})
\sin (\frac{\pi n y}{b}).
\label{eq-6.6.11}
\end{multline}
One can find coefficients from initial conditions $u|_{t=0}=g(x,y)$ and $u_t|_{t=0}=h(x,y)$.
Remark 1. While solution (\ref{eq-6.6.10}) is periodic with respect to $t$, solution (\ref{eq-6.6.11}) is not (in the generic case).
Remark 2.
The same arguments work in higher dimensions.
The same arguments work for Neumann and Robin boundary conditions and on each face of $\Gamma$ could be its own boundary condition.
The same arguments work for many other equations, like heat equation, or Schrödinger equation.
Consider Laplace equation in the rectangular box $\Omega$, with the Dirichlet boundary conditions on $\Gamma=\partial\Omega$: \begin{align} &\Delta u=0, \label{eq-6.6.12}\\ &u|_\gamma =g. \label{eq-6.6.13} \end{align} We consider $n=3$, so $\Omega=\{(x,y,z)\colon 0<x< a, \ 0<y< b, 0<z< c\}$. Without any loss of the generality we can assume that $g=0$ on all faces, except two opposite $z=0$ and $z=c$.
Again we are looking for solution $u(x,y,z)=X(x)Y(y)Z(z)$ and separating variables we get \begin{equation} \underbracket{\frac{X''(x)}{X(x)}} + \underbracket{\frac{Y''(y)}{Y(y)}}+ \underbracket{\frac{Z''(z)}{Z(z)}}=0\label{eq-6.6.14} \end{equation} and therefore \begin{equation} X''+\lambda X=0,\qquad Y''+\mu Y=0,\qquad Z''-\nu Z=0\label{eq-6.6.15} \end{equation} with $\nu =\lambda+\mu$.
We also get (\ref{eq-6.6.5}) and then (\ref{eq-6.6.6})-(\ref{eq-6.6.7}). Therefore \begin{gather} Z_{mn}(z)= A_{mn}\cosh (\omega_{mn}z)+ B_{mn}\sinh (\omega_{mn}z),\label{eq-6.6.16}\\ \omega_{mn}= \pi \bigl( \frac{m^2}{a^2}+\frac{n^2}{b^2}\bigr)^{\frac{1}{2}}. \label{eq-6.6.17} \end{gather} Finally, \begin{multline} u_{mn}(x,y,t)=\\ \Bigl(A_{mn}\cosh (\omega_{mn}z)+ B_{mn}\sinh (\omega_{mn}z)\Bigr) \sin (\frac{\pi m x}{a}) \sin (\frac{\pi n y}{b}) \label{eq-6.6.18} \end{multline} and the general solution is derived in the form of multidimensional Fourier series \begin{multline} u(x,y,t)=\\ \sum_{m=1}^\infty \sum_{n=1}^\infty \Bigl(A_{mn}\cosh (\omega_{mn}z)+ B_{mn}\sinh (\omega_{mn}z)\Bigr) \sin (\frac{\pi m x}{a}) \sin (\frac{\pi n y}{b}). \label{eq-6.6.19} \end{multline} One can find coefficients from boundary conditions $u|_{z=0}=g(x,y)$ and $u|_{z=c}=h(x,y)$.
Remark 3.
The same arguments work in higher dimensions.
The same arguments work for Neumann and Robin boundary conditions and on each face of $\Gamma$ could be its own boundary condition.
Later we consider separation of variables for 2D-wave equation in the polar coordinates.
Later we consider separation of variables for 3D-wave and Laplace equations in the cylindrical and spherical coordinates.