We are adding to the equation found in the 2-D heat equation in cylindrical coordinates, starting with the following definition:
![{\displaystyle D:=(0,a)\times (0,b)\times (0,L)~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04a79abc99e7906fcd34ed876b30a8478a4d4142)
By changing the coordinate system, we arrive at the following nonhomogeneous PDE for the heat equation:
![{\displaystyle u_{t}=k\left[{\frac {1}{r}}\left(u_{r}+ru_{rr}\right)+{\frac {1}{r^{2}}}u_{\theta \theta }+u_{zz}\right]+h(r,\theta ,z,t),{\text{ where }}(r,\theta ,z)\in D,t\in (0,\infty )~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07e15b84158ea89eedd76a115d2182fb5d58a51b)
We choose for the example the Robin boundary conditions and initial conditions as follows:
All of the boundary conditions are homogeneous, so we don't have to partition the solution into a "steady-state" portion and a "variable" portion. Otherwise, that would be the way to solve this problem.
![{\displaystyle u(r,\theta ,z,t)=R(r)\Theta (\theta )Z(z)T(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5c6e17e9f4572bd91e660f8585e7159b0da18f5)
![{\displaystyle \Rightarrow R\Theta ZT'=k\left(R''\Theta ZT+{\frac {1}{r}}R'\Theta ZT+{\frac {1}{r^{2}}}R\Theta ''ZT+R\Theta Z''T\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d49086cb172e2a2a33b38b61c2059d0077773db3)
![{\displaystyle {\frac {T'}{kT}}={\frac {R''}{R}}+{\frac {1}{r}}{\frac {R'}{R}}+{\frac {1}{r^{2}}}{\frac {\Theta ''}{\Theta }}+{\frac {Z''}{Z}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c1d324c39359b1eae8cacfc836ea07e057e6d6f)
There is a separation constant
that both sides of the equation are equivalent to. This yields:
![{\displaystyle {\frac {T'}{kT}}=-\gamma ^{2}\Rightarrow {\color {Blue}T'+k\gamma ^{2}T=0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aad93cdd7033ac7ac1b800cc4c3ac520657e8a66)
![{\displaystyle {\frac {R''}{R}}+{\frac {1}{r}}{\frac {R'}{R}}+{\frac {1}{r^{2}}}{\frac {\Theta ''}{\Theta }}+\gamma ^{2}=-{\frac {Z''}{Z}}=\mu ^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53006b7c993eb0b0d4ffe0240bea50bacad11299)
The second equation yields the equations:
![{\displaystyle {\color {Blue}Z''+\mu ^{2}Z=0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/424c288756f8231151f45b4a7aa7112f64e82aa4)
![{\displaystyle {\frac {R''}{R}}+{\frac {1}{r}}{\frac {R'}{R}}+\gamma ^{2}-\mu ^{2}=-{\frac {1}{r^{2}}}{\frac {\Theta ''}{\Theta }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/607ca569f4121ef12066ef4a3707e43c5f14c426)
![{\displaystyle \Rightarrow r^{2}{\frac {R''}{R}}+r{\frac {R'}{R}}+\left(\gamma ^{2}-\mu ^{2}\right)r^{2}=-{\frac {\Theta ''}{\Theta }}=\rho ^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31f59d0f25e60200a79799c69478a2b01d2f5954)
This yields the following equations:
![{\displaystyle {\color {Blue}\Theta ''+\rho ^{2}\Theta =0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1623e927e8975c3f3bcb363bf232fe786c0e586)
![{\displaystyle {\color {Blue}r^{2}R''+rR'+\left[\left(\gamma ^{2}-\mu ^{2}\right)r^{2}-\rho ^{2}\right]R=0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b573afb5f5f93ebbb11ff2e6b9094640dd0bf60a)
Just like in the 2-D heat equation, the boundary conditions yield:
![{\displaystyle {\begin{cases}\left\vert R(0)\right\vert <\infty \\\alpha _{1}R(a)+\beta _{1}R'(a)=0\\\alpha _{2}\Theta (0)-\beta _{2}\Theta '(0)=0\\\alpha _{3}\Theta (b)+\beta _{3}\Theta '(b)=0\\\alpha _{4}Z(0)-\beta _{4}Z'(0)=0\\\alpha _{5}Z(L)+\beta _{5}Z'(L)=0\\\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1960492d6796475aef634d0c386eb95d13793828)
![{\displaystyle \left.{\begin{aligned}&Z''+\mu ^{2}Z=0\\&\alpha _{4}Z(0)-\beta _{4}Z'(0)=0\\&\alpha _{5}Z(L)+\beta _{5}Z'(L)=0\end{aligned}}\right\}{\begin{aligned}&{\text{Eigenvalues }}\mu _{k}{\text{: solutions to equation }}(\alpha _{4}\alpha _{5}-\beta _{4}\beta _{5}\mu ^{2})\sin(\mu L)+(\alpha _{4}\beta _{5}+\alpha _{5}\beta _{4})\mu \cos(\mu L)=0\\&Z_{k}(z)=\beta _{4}\mu _{k}\cos(\mu _{k}z)+\alpha _{4}\sin(\mu _{k}z),\;k=0,1,2,\cdots \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/778a606e006e1f6f26ef6e4f57f1f26f730cf66d)
![{\displaystyle \left.{\begin{aligned}&\Theta ''+\rho ^{2}\Theta =0\\&\alpha _{2}\Theta (0)-\beta _{2}\Theta '(0)=0\\&\alpha _{3}\Theta (b)+\beta _{3}\Theta '(b)=0\end{aligned}}\right\}{\begin{aligned}&{\text{Eigenvalues }}\rho _{m}{\text{: solutions to equation }}(\alpha _{2}\alpha _{3}-\beta _{2}\beta _{3}\rho ^{2})\sin(\rho b)+(\alpha _{2}\beta _{3}+\alpha _{3}\beta _{2})\rho \cos(\rho b)=0\\&\Theta _{m}(\theta )=\beta _{2}\rho _{m}\cos(\rho _{m}\theta )+\alpha _{2}\sin(\rho _{m}\theta ),\;m=0,1,2,\cdots \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2862a5757db3f067f3b34ae6e316ce5a7865250)
![{\displaystyle \left.{\begin{aligned}&r^{2}R''+rR'+\left[\left(\gamma ^{2}-\mu ^{2}\right)r^{2}-\rho ^{2}\right]R=0\\&\left\vert R(0)\right\vert <\infty \\&\alpha _{1}R(a)+\beta _{1}R'(a)=0\end{aligned}}\right\}{\begin{aligned}&{\text{Substitute }}\lambda ^{2}=\gamma ^{2}-\mu _{k}^{2}{\text{ and }}\nu ^{2}=\rho _{m},\;k,m=0,1,2,\cdots \\&{\text{Eigenvals }}\lambda _{kmn}{\text{: solns to eqn }}(\alpha _{1}\lambda a+\beta _{1}\rho _{m})J_{\rho _{m}}(\lambda a)-\beta _{1}a\lambda J_{\rho _{m}+1}(\lambda a)=0\\&R_{kmn}(r)=J_{\rho _{m}}(\lambda _{kmn}r),\;k,m,n=0,1,2,\cdots \\&\gamma _{kmn}^{2}=\lambda _{kmn}^{2}+\mu _{k}^{2}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ada783da032d643506b636a9d5cb1d57520cee61)
![{\displaystyle T'+k\gamma _{kmn}^{2}T=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af97665b5e54a4fd809e07ae57f09d4dbef85f22)
The solution to the equation is:
![{\displaystyle T_{kmn}(t)=C_{kmn}e^{-k\left(\lambda _{kmn}^{2}+\mu _{k}^{2}\right)t},\;k,m,n=0,1,2,\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9baa9262fde4640db9c8d24eb502168d4264d948)
Define:
![{\displaystyle u(r,\theta ,z,t)=\sum _{k,m,n=0}^{\infty }T_{kmn}(t)R_{kmn}(r)\Theta _{m}(\theta )Z_{k}(z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c495029067d185f0bde83a662c6400435417724)
Applying the initial condition:
![{\displaystyle {\begin{aligned}u(r,\theta ,z,0)&=f(r,\theta ,z)\\&=\sum _{k,m,n=0}^{\infty }T_{kmn}(0)R_{kmn}(r)\Theta _{m}(\theta )Z_{k}(z)\\&=\sum _{k,m,n=0}^{\infty }C_{kmn}R_{kmn}(r)\Theta _{m}(\theta )Z_{k}(z)\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b37be25843d6294a95f89ec3f097b001ae157080)
This is the orthogonal expansion of
in terms of
Hence,
![{\displaystyle C_{kmn}={\frac {\int \limits _{0}^{L}\int \limits _{0}^{b}\int \limits _{0}^{a}f(r,\theta ,z)R_{kmn}(r)\Theta _{m}(\theta )Z_{k}(z)rdrd\theta dz}{\int \limits _{0}^{a}R_{kmn}^{2}(r)rdr\int \limits _{0}^{b}\Theta _{m}^{2}(\theta )d\theta \int \limits _{0}^{L}Z_{k}^{2}(z)dz}},\;k,m,n=0,1,2,\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/63f24a830b1213d8f888df037b7735ad0838137d)
Let:
![{\displaystyle u(r,\theta ,z,t)=\sum _{k,m,n=0}^{\infty }T_{kmn}(t)R_{kmn}(r)\Theta _{m}(\theta )Z_{k}(z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c495029067d185f0bde83a662c6400435417724)
![{\displaystyle h(r,\theta ,z,t)=\sum _{k,m,n=0}^{\infty }H_{kmn}(t)R_{kmn}(r)\Theta _{m}(\theta )Z_{k}(z),\;H_{kmn}(t)={\frac {\int \limits _{0}^{L}\int \limits _{0}^{b}\int \limits _{0}^{a}h(r,\theta ,z,t)R_{kmn}(r)\Theta _{m}(\theta )Z_{k}(z)rdrd\theta dz}{\int \limits _{0}^{a}R_{kmn}^{2}(r)rdr\int \limits _{0}^{b}\Theta _{m}^{2}(\theta )d\theta \int \limits _{0}^{L}Z_{k}^{2}(z)dz}},\;k,m,n=0,1,2,\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b4412525c23af271a58d7fb4a2e59976fb6c6d9)
Substitute the expansions for u and h into the non-homogeneous equation:
![{\displaystyle \sum T_{kmn}'R_{kmn}\Theta _{m}Z_{k}=k\left\{\sum T_{kmn}\left[\underbrace {\left(R_{kmn}''+{\frac {1}{r}}R_{kmn}'\right)} _{-\left[\left(\lambda _{kmn}^{2}-\mu _{k}^{2}\right)-{\frac {1}{r^{2}}}\rho _{m}^{2}\right]R_{kmn}}\Theta _{m}Z_{k}+{\frac {1}{r^{2}}}R_{kmn}\underbrace {\Theta _{m}''} _{-\rho _{m}^{2}\Theta _{m}}Z_{k}+R_{kmn}\Theta _{m}\underbrace {Z_{k}''} _{-\mu _{k}^{2}Z_{k}}\right]\right\}+\sum H_{kmn}R_{kmn}\Theta _{m}Z_{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1472f2f9a5be6eb2cd9b16a7fbcfb56ea8c921ed)
![{\displaystyle \Leftrightarrow \sum T_{kmn}'\left[R_{kmn}\Theta _{m}Z_{k}\right]=\sum \left[(-k\gamma _{kmn}^{2})T_{kmn}+H_{kmn}\right]\left[R_{kmn}\Theta _{m}Z_{k}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b856a111f8616c7ec034db0904389ed59c6806bc)
From the linear independence of
:
![{\displaystyle T_{kmn}'(t)=k\gamma _{kmn}^{2}T_{kmn}(t)=H_{kmn}(t),\;k,m,n=0,1,2,\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/64233cde11578821f1715371bf39c7e0f0f08d96)
![{\displaystyle T_{kmn}(t)=e^{-k\gamma _{kmn}^{2}t}\int \limits _{0}^{t}e^{k\gamma _{kmn}^{2}s}H_{kmn}(s)ds+C_{kmn}e^{-k\gamma _{kmn}^{2}t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3decd6f13ebf7bc9edd606805ca4fd835a08f92)
The undetermined coefficient satisfies the initial condition: