Subsection 7.3.5 Three Dimensional Equation of Heat Flow
Three dimensional heat equation is given by
\begin{equation}
\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}+\frac{\partial^{2}u}{\partial z^{2}} =\frac{1}{h^{2}}\frac{\partial u}{\partial t}\tag{7.3.87}
\end{equation}
the solution of which is assumed as
\begin{equation}
u=u(x,y,z,t)=X(x)Y(y)Z(z)T(t)\tag{7.3.88}
\end{equation}
By substitution of this in equation (7.3.87), we get -
\begin{equation}
\frac{1}{X}\frac{\partial^{2}X}{\partial x^{2}}+\frac{1}{Y}\frac{\partial^{2}Y}{\partial y^{2}}+\frac{1}{Z}\frac{\partial^{2}Z}{\partial z^{2}} =\frac{1}{h^{2}T}\frac{\partial T}{\partial t} =-\lambda^{2} \text{(say)}\tag{7.3.89}
\end{equation}
Now
\begin{equation*}
\frac{1}{X}\frac{\partial^{2}X}{\partial x^{2}} =-\lambda_{1}^{2};
\end{equation*}
or,
\begin{equation*}
\frac{1}{Y}\frac{\partial^{2}Y}{\partial y^{2}} = -\lambda_{2}^{2};
\end{equation*}
or,
\begin{equation*}
\frac{1}{Z}\frac{\partial^{2}Z}{\partial z^{2}} =-\lambda_{3}^{2},
\end{equation*}
so that
\begin{equation*}
\lambda^{2} = \lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}
\end{equation*}
Yielding
\begin{equation*}
X=A\cos\lambda_{1}x+B\sin\lambda_{1}x = a\cos(\lambda_{1}x+\alpha\lambda_{1});
\end{equation*}
or,
\begin{equation*}
Y=b\cos(\lambda_{2}y+\alpha\lambda_{2}),
\end{equation*}
or,
\begin{equation*}
Z =c\cos(\lambda_{3}z+\alpha\lambda_{3}),
\end{equation*}
and
\begin{equation*}
T=de^{-\lambda^{2}h^{2}t}=de^{-(\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2})h^{2}t}
\end{equation*}
Hence for all values of \(t\text{,}\) the general solution of equation (7.3.87) is
\begin{equation*}
u=\sum\limits_{\lambda_{1}=0}^{\infty}\sum\limits_{\lambda_{2}=0}^{\infty}\sum\limits_{\lambda_{3}=0}^{\infty}A_{\lambda_{1}\lambda_{2}\lambda_{3}}\cos(\lambda_{1}x+\alpha\lambda_{1})\cos(\lambda_{2}y+\alpha\lambda_{2})
\end{equation*}
\begin{equation*}
\cos(\lambda_{3}z+\alpha\lambda_{3})e^{-h^{2}(\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2})t}
\end{equation*}
