91Ó°ÊÓ

An expression has been given as $\cos \mathrm{θ}−3{\sin }^2\mathrm{θ}$.

The total of the second-order partial derivatives of , the unknown function, with respect to the Cartesian coordinates equals , according to Laplace's equation.

Laplace’s equation in cylindrical coordinates is${\mathbf{∇}}^{\mathbf{2}}\mathit{u}\mathbf{=}\frac{\mathbf{1}}{\mathbf{r}}\frac{\mathbf{∂}}{\mathbf{∂}\mathbf{r}}\mathbf{\left(}\mathbf{r}\frac{\mathbf{∂}\mathbf{u}}{\mathbf{∂}\mathbf{r}}\mathbf{\right)}\mathbf{+}\frac{\mathbf{1}}{{\mathbf{r}}^{\mathbf{2}}}\frac{{\mathbf{∂}}^{\mathbf{2}}\mathbf{u}}{\mathbf{∂}{\mathbf{θ}}^{\mathbf{2}}}\mathbf{+}\frac{{\mathbf{∂}}^{\mathbf{2}}\mathbf{u}}{\mathbf{∂}{\mathbf{z}}^{\mathbf{2}}}\mathbf{=}\mathbf{0}$

And to separate the variable the solution assumed is of the form$\mathit{u}\mathbf{=}\mathit{R}\mathbf{\left(}\mathbf{r}\mathbf{\right)}\mathit{Θ}\mathbf{\left(}\mathbf{θ}\mathbf{\right)}\mathit{Z}\mathbf{\left(}\mathbf{z}\mathbf{\right)}$.

Rewrite the given question.

$\begin{array}{c}A\left(\mathrm{θ}\right)=\cos \left(\mathrm{θ}\right)−3{\sin }^2\left(\mathrm{θ}\right)\\ =\cos \left(\mathrm{θ}\right)+3{\cos }^2\left(\mathrm{θ}\right)−3\end{array}$

Due to the fact that $A\left(\mathrm{θ}\right)$is independent of $\mathrm{ϕ},$the associated Legendre polynomials $P_l^m\left(\cos \left(\mathrm{θ}\right)\right)$reduce to the standard Legendre polynomials $P_l\left(\cos \left(\mathrm{θ}\right)\right)$.

In order to find the steady-state temperature distribution function. Express $A\left(\mathrm{θ}\right)$in terms of the standard Legendre polynomials which in turn leads us to determine the corresponding coefficients $c_I.$

Substitute$x=\cos \left(\mathrm{θ}\right)$

$u_{r=1}\left(x\right)=\underset{l=0}{\overset{\mathrm{∞}}{∑}}{c}_l{1}^l{P}_l\left(x\right)$

$u_{r=1}\left(x\right)=3x^2+x−3$ ….. (1)

Multiply the above equation by $P_{\mathrm{ν}}$and integrate.

$\underset{l=0}{\overset{\mathrm{∞}}{∑}}{c}_l\underset{−1}{\overset{1}{∫}}{P}_l\left(x\right)P_{\mathrm{ν}}\left(x\right)dx=\underset{−1}{\overset{1}{∫}}(3x^2+x−3)P_{\mathrm{ν}}\left(x\right)dx$ ….. (2)

Legendre orthogonally relation:

$\underset{−1}{\overset{1}{∫}}{P}_l\left(x\right)P_{\mathrm{ν}}\left(x\right)dx=\frac{2}{(2l+1)}{\mathrm{δ}}_{l,\mathrm{ν}}$ ….. (3)

Insert equation (3) into equation (2):

$\underset{l=0}{\overset{\mathrm{∞}}{∑}}{c}_l\frac{2}{(2l+1)}{\mathrm{δ}}_{l,\mathrm{ν}}=\underset{−1}{\overset{1}{∫}}(3x^2+x)P_{\mathrm{ν}}\left(x\right)dx−3\underset{−1}{\overset{1}{∫}}{P}_{\mathrm{ν}}\left(x\right)$ ….. (4)

Legendre identity:

$\underset{a}{\overset{1}{∫}}{P}_n\left(x\right)dx=\frac{1}{(2n+1)}[P_{n−1}\left(a\right)−P_{n+1}\left(a\right)]$ ….. (5)

Combine equation (4) and (5).

$c_{\mathrm{ν}}=\frac{2\mathrm{ν}+1}{2}\underset{−1}{\overset{1}{∫}}(3x^2+x)P_{\mathrm{ν}}\left(x\right)dx−\frac{3}{2}[P_{\mathrm{ν}−1}(−1)−P_{\mathrm{ν}+1}(−1)]$

Hence the final answer is as given below.

$u(r,\mathrm{θ})=\underset{\mathrm{ν}=0}{\overset{\mathrm{∞}}{∑}}\{\frac{2\mathrm{ν}+1}{2}\underset{−1}{\overset{1}{∫}}(3x^2+x)P_{\mathrm{ν}}\left(x\right)dx−\frac{3}{2}[P_{\mathrm{ν}−1}(−1)−P_{\mathrm{ν}+1}(−1)]\}{r}^{\mathrm{ν}}{P}_{\mathrm{ν}}\left(x\right)$