# EGM6321 - Principles of Engineering Analysis 1, Fall 2010

Mtg 35: Fri, 5 Nov 10

# Legendre functions

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NOTE:

1. Legendre polynomials $\{ P_i(x), i=0,1,2,... \} \$ : Finite series, where $\{ P_i(x) \} \$ is the family of 1st homogeneous solutions of the Legendre equation.

2. Legendre functions: $\begin{cases} {P_i(x)} & \mbox{ }\mbox{ polynomial} \\ {Q_i(x)} & \mbox{ } \mbox{ functions} \end{cases} \$

3. Solving Laplace equation: Use the two families $\{ P_i \} \$ and $\{ Q_i \} \$ (infinite series); similar to the use of Fourier series (see also p.32-2 and p.32-3).

END NOTE

# Heat conduction on a sphere (cont'd)

Eq.(2)p.34-4 : Euler L2-ODE-VC

Trial solution:

 $\displaystyle R(r)=r^{ \lambda\ }$ (1)

where $\lambda\ \$ is a constant to be determined, and $r \$ is an independent variable.

$R'=\frac{dR}{dr} = \lambda\ r^{ \lambda\ -1} \$

$R''= \lambda\ ( \lambda\ -1) r^{ \lambda\ -2} \$

Eq.(2)p.34-4 $\Rightarrow \ r^2R''+2rR'-kR=0 \$

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Hence the characteristic equation:

 $\displaystyle \lambda\ ( \lambda\ +1)=k$ (0)

Goal: Transform 2nd separation Eq.(3)p.34-4 into Legendre equation of order n, i.e. Eq.(1)p.5-4.

With $\cos^2 \theta = 1 - \sin^2 \theta \$ , define $\mu^2 := \sin ^2 \theta \$, thus

 \displaystyle \begin{align} \mu\ := \sin \theta\ \end{align} (1)

Think of $\mu\ \$ as "$\displaystyle x$" (independent variable) for the Legendre polynomial (function).

Transform the independent variable from $\theta\ \$ to $\mu\ \$

$\frac{d}{d \theta\ } = \frac{d}{d \mu\ }\frac{d \mu\ }{d \theta\ } \$ , where $\frac{d \mu\ }{d \theta\ } =\cos \theta\ \$ from Eq.(1)

$\Rightarrow \ \frac{1}{\cos \theta\ }\frac{d(.)}{d \theta\ } = \frac{d(.)}{d \mu\ } \$ , where $C := \cos \theta\ \$

Eq.(3)p.34-4 : $\left ( \frac{1}{C}\frac{d}{d \theta\ } \right ) \left [ C^2 \left ( \frac{1}{C}\frac{d}{d \theta\ } \right ) \Theta\ \right ] = k \Theta\ \$

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 $\displaystyle \Rightarrow \ -\frac{d}{d \mu} \left[ (1- \mu^2) \frac{d \Theta}{d \mu} \right] = k \Theta$ (1)

where $(1- \mu^2) = C^2 \$.

 $\displaystyle \Rightarrow \ (1- \mu^2) \Theta''-2 \mu \Theta'+ k \Theta = 0$ (2)

is the Legendre equation Eq.(1)p.5-4 if

 $\displaystyle k = n(n+1)$ (3)

Next step: solve for $\lambda , \ R(r), \ \Theta\ (\theta) \$.

# Heat conduction on a cylinder

HW6.5 Circular Cylinder Coordinates (cylindrical)

Given:

$x=r\cos \theta\ = \xi_1 \cos \xi_2 \$

$y=r\sin \theta\ = \xi_1 \sin \xi_2 \$

$z= \xi_3 \$

1) Find $\left \{ dx_i \right \} = \left \{ dx_1, dx_2, dx_3 \right \} \$

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In terms of $\left \{ \xi_j \right \} = \left \{ \xi_1, \xi_2, \xi_3 \right \} \$ and $\left \{ d \xi_k \right \} \$.

2) Find $ds^2= \sum_{i} (dx_i)^2 = \sum_{k} (h_k)^2 (d \xi_k)^2 \$.

Identify $\{ h_i \} \$ in terms of $\{ \xi_j \} \$.

3) Find $\Delta \Psi \$ in cylindrical coordinates ($\Rightarrow \ \$ Bessel equation Eq.(2)p.24-1)

END HW6.5

HW6.6

Find $\Delta \Psi\$ in spherical coordinates using math/physics convention of $(r , \bar \theta , \varphi ) = ( \xi_1, \xi_2, \xi_3 ) \$

END HW6.6