User:EGM6341.s11.team1.Chiu/PEA1 F09 Mtg30

Mtg 30: Thu, 29 Oct 09

[[media: Egm6321.f09.mtg30.djvu| page30-1]]

Find


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$$ \psi(r, \theta) \ $$at $$ \Delta \psi=0 \ $$ and $$ \psi(r=1, \theta)= T_{0} \cos^{4} \theta = T_{0} \sin^{4} \bar{\theta} \ $$ (1)
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End Find

Separation of variables


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$$ \psi(r, \theta) = R(r) \Theta( \theta) \ $$

(2)
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$$ \Delta \psi=0= \frac{\Theta}{r^{2}} \frac{\mathrm{d} }{\mathrm{d} r} (r^{2} \frac{\mathrm{d} R}{\mathrm{d} r})+ \frac{R}{r^{2} \cos \theta} \frac{\mathrm{d} }{\mathrm{d} \theta} (\cos \theta \frac{\mathrm{d} \Theta}{\mathrm{d} \theta}) \ $$


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Simplify $$ \frac{1}{r^{2}} \ $$ ; divide by $$ R \Theta \ $$


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$$ \Rightarrow \underbrace{ \frac{1}{R} \frac{\mathrm{d} }{\mathrm{d} r} (r^{2} \frac{\mathrm{d} R}{\mathrm{d} r})}_{ \color{red} \alpha(r)}+ \frac{1}{ \Theta \cos \theta} \underbrace{ \frac{\mathrm{d} }{\mathrm{d} \theta} (\cos \theta \frac{\mathrm{d} \Theta}{\mathrm{d} \theta})}_{ \color{red} \beta( \theta)}=0 \ $$

(3)
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$$ \Rightarrow \alpha(r)+ \beta( \theta)=0 \ $$

(4)
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[[media: Egm6321.f09.mtg30.djvu| page30-2]]


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$$ \Rightarrow \alpha(r) = -\beta( \theta)=k \ $$  (constant)


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2 Separate equations from [[media: Egm6321.f09.mtg30.djvu| (3)&(4) page30-1]] (L2-ODE-VC 2 equations)


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$$ \frac{\mathrm{d} }{\mathrm{d} r} (r^{2} \frac{\mathrm{d} R}{\mathrm{d} r})=kR \ $$

(1)
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$$ -\frac{1}{ \cos \theta} \frac{\mathrm{d} }{\mathrm{d} \theta} (\cos \theta \frac{\mathrm{d} \Theta}{\mathrm{d} \theta})= k \Theta \ k \ $$ : separation constant

(2)
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(1) by trial solution:


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$$ R(r)=r^{ \lambda} \ \lambda \ $$: undetermined coefficient

(3)
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$$ R'= \frac{\mathrm{d} }{\mathrm{d} r}R= \lambda r^{ \lambda-1} \ $$


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


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$$ { \color{red} (1) \Rightarrow} \ r^{2}R''+2rR'-kR \ \overset{ \overset{ \color{red} (4)}{\downarrow}} = \ 0 \ $$


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(3) $$ \to \ $$ Euler equation [[media: Egm6321.f09.mtg15.djvu| page15-4]]


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$$ r^{2}R'= \lambda r^{ \lambda-1+ { \color{red}2 \cdots}} \ $$


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HW:


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

(5)
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At this stage, no restriction on sign of $$ \lambda \ $$ (see later)

[[media: Egm6321.f09.mtg30.djvu| page30-3]]

Now 2nd separate equation [[media: Egm6321.f09.mtg30.djvu| (2) page30-2]]

Goal: Transform this equation into Legendre differential equation of order n [[media: Egm6321.f09.mtg14.djvu| (1) page14-2]]


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Let $$ \cos^{2} \theta = 1- \underbrace{\sin^{2} \theta}_{ \color{blue} := \mu^{2}} \ $$


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$$ \mu:= \sin \theta \ $$ (think as "x" in [[media: Egm6321.f09.mtg14.djvu| (1) page14-2]])

(1)
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Transform of variables from $$ \theta \ $$ to $$ \mu \ $$:


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$$ \frac{\mathrm{d} }{\mathrm{d} \theta}= \frac{\mathrm{d} }{\mathrm{d} \mu} \underbrace{ \frac{\mathrm{d} \mu}{\mathrm{d} \theta}}_{ { \color{red} (1) \ \Rightarrow} { \color{blue} \cos \theta}} \ $$

(2)
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$$ \Rightarrow \frac{1}{ \cos \theta} \frac{\mathrm{d} }{\mathrm{d} \theta} (.)= \frac{\mathrm{d} }{\mathrm{d} \mu} (.) \ $$

(3)
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[[media: Egm6321.f09.mtg30.djvu| (2) page30-2:]] $$ { \color{red}-} \underbrace{( \frac{1}{C} \frac{\mathrm{d} }{\mathrm{d} \theta})}_{ \color{blue} \frac{\mathrm{d} }{\mathrm{d} \mu}} \left [ \underset{ \color{blue} \underset{(1- \mu^{2})}{\uparrow}} C^{2}\underbrace{( \frac{1}{C} \frac{\mathrm{d} }{\mathrm{d} \theta})}_{ \color{blue} \frac{\mathrm{d} }{\mathrm{d} \mu}} \Theta \right ]=k \Theta \ $$


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$$ \frac{\mathrm{d} }{\mathrm{d} \mu} \left [ (1- \mu^{2}) \frac{\mathrm{d} }{\mathrm{d} \mu} \Theta \right ]+k \Theta=0 \ $$

(4)
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[[media: Egm6321.f09.mtg30.djvu| page30-4]]

$$ \Theta (.) \ $$ is now considered as a function of $$ \mu \ $$, i.e., $$ \Theta ( \mu) \ $$

[[media: Egm6321.f09.mtg30.djvu| (4) page30-3:]]$$ (1- \mu^{2}) \Theta''-2 \mu \Theta'+k \Theta \underset{ \color{red} \underset{(1)}{\uparrow}}=0 \ $$

confer [[media: Egm6321.f09.mtg14.djvu| (1) page14-2]] is Legendre differential equation if $$ k=n(n+1) \ $$(2)