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Heat Conduction Equation--Disk

To solve the Heat Conduction Equation on a 2-D disk of radius $R=1$, try to separate the equation using

T(r, \theta, t) = R(r)\Theta (\theta)T(t).
\end{displaymath} (1)

Writing the $\theta$ and $r$ terms of the Laplacian in Spherical Coordinates gives
\nabla^2={d^2R\over dr^2}+ {2\over r} {dR\over dr} + {1\over r^2} {d^2\Theta \over d\theta^2},
\end{displaymath} (2)

so the Heat Conduction Equation becomes
{R\Theta\over \kappa}{d^2T\over dt^2} =
{d^2R\over dr^2} \T...
...ver dr} \Theta T + {1\over r^2} {d^2\Theta\over d\theta^2} RT.
\end{displaymath} (3)

Multiplying through by $r^2/R\Theta T$ gives
{r^2\over \kappa T}{d^2T\over dt^2}
= {r^2\over R} {d^2R\ov...
...over R} {dR\over dr}+{d^2\Theta\over d\theta^2}{1\over\Theta}.
\end{displaymath} (4)

The $\theta$ term can be separated.
{d^2\Theta \over d\theta^2} {1\over \Theta } = -n(n+1),
\end{displaymath} (5)

which has a solution
\Theta(\theta ) = A\cos\left[{\sqrt{n(n+1)}\, \theta}\right]+B\sin\left[{\sqrt{n(n+1)}\,\theta}\right].
\end{displaymath} (6)

The remaining portion becomes
{r^2\over \kappa T}{d^2T\over dt^2}= {r^2\over R} {d^2R\over dr^2} + {2r\over R} {dR\over dr} -n(n+1).
\end{displaymath} (7)

Dividing by $r^2$ gives
{1\over \kappa T}{d^2T\over dt^2}
= {1\over R} {d^2R\over d...
...ver rR} {dR\over dr} -{n(n+1)\over r^2} = -{1\over \lambda^2},
\end{displaymath} (8)

where a Negative separation constant has been chosen so that the $t$ portion remains finite
T(t)=C e^{-\kappa t/\lambda^2}.
\end{displaymath} (9)

The radial portion then becomes
{1\over R} {d^2R\over dr^2} + {2\over rR} {dR\over dr} -{n(n+1)\over r^2} +{1\over \lambda^2}=0
\end{displaymath} (10)

r^2{d^2R\over dr^2} + 2r {dR\over dr} +\left[{{r^2\over \lambda^2}-n(n+1)}\right]R=0,
\end{displaymath} (11)

which is the Spherical Bessel Differential Equation. If the initial temperature is $T(r,0)=0$ and the boundary condition is $T(1,t)=1$, the solution is
T(r,t)=1-2\sum_{n=1}^\infty {J_0(\alpha_n r)\over \alpha_nJ_1(\alpha_n)} e^{{\alpha_n}^2t},
\end{displaymath} (12)

where $\alpha_n$ is the $n$th Positive zero of the Bessel Function of the First Kind $J_0(x)$.

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© 1996-9 Eric W. Weisstein