User:EGM6341.s11.TEAM1.WILKS/Mtg35

=EGM6321 - Principles of Engineering Analysis 1, Fall 2010=

[[media: 2010_11_05_07_46_02.djvu | Mtg 35:]] Fri, 5 Nov 10

= Legendre functions =

[[media: 2010_11_05_07_46_02.djvu | Page 35-1]]
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 [[media: 2010_11_02_14_58_42.djvu | p.32-2]] and [[media: 2010_11_02_14_58_42.djvu | p.32-3]]).

END NOTE

= Heat conduction on a sphere (cont'd) =

[[media: 2010_11_04_14_51_31.djvu | Eq.(2)p.34-4]] : Euler L2-ODE-VC Trial solution: where $$ \lambda\ \ $$ is a constant to be determined, and $$ r \ $$ is an independent variable.

$$ R'=\frac{dR}{dr} = \lambda\ r^{ \lambda\ -1} \ $$ $$ R''= \lambda\ ( \lambda\ -1) r^{ \lambda\ -2} \ $$ [[media: 2010_11_04_14_51_31.djvu | Eq.(2)p.34-4]] $$ \Rightarrow \ r^2R''+2rR'-kR=0 \ $$

[[media: 2010_11_05_07_46_02.djvu | Page 35-2]]
Hence the characteristic equation:

 Goal:  Transform 2nd separation [[media: 2010_11_04_14_51_31.djvu | Eq.(3)p.34-4]] into Legendre equation of order n, i.e. [[media: 2010_09_02_13_55_50.djvu | Eq.(1)p.5-4]].

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

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 $$ C := \cos \theta\ \ $$ [[media: 2010_11_04_14_51_31.djvu | Eq.(3)p.34-4]] : $$ \left ( \frac{1}{C}\frac{d}{d \theta\ } \right ) \left [ C^2 \left ( \frac{1}{C}\frac{d}{d \theta\ } \right ) \Theta\ \right ] = k \Theta\ \ $$

[[media: 2010_11_05_07_46_02.djvu | Page 35-3]]
where $$ (1- \mu^2) = C^2 \ $$.

is the Legendre equation [[media: 2010_09_02_13_55_50.djvu | Eq.(1)p.5-4]] if

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

[[media: 2010_11_05_07_46_02.djvu | Page 35-4]]
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 [[media: 2010_10_14_14_02_27.djvu | 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