User:EGM6341.s11.team1.Chiu/PEA1 F09 Mtg32

Mtg 32: Thu, 05 Nov 09

[[media: Egm6321.f09.mtg32.djvu| page32-1]]


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HW: Verify that $$ P_{0}, \cdots, P_{4} \ $$ [[media: Egm6321.f09.mtg31.djvu| (1)-(5) page31-3]] are solutions of Legendre equation [[media: Egm6321.f09.mtg14.djvu| (1) page14-2]], [[media: Egm6321.f09.mtg30.djvu| page30-4]]


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HW:(See [[media: Egm6321.f09.mtg31.djvu| page31-1]]) Obtain the separate equations for Laplace equation on circle, cylindrical coordinates and identify the Bessel differential equation.


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K.p.84 $$ x^{2}y''+xy'-(x^{2}- \nu^{2})y \underset{ \color{red} \underset{(1)}{\uparrow}}=0 \ $$


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Heat Problem continued [[media: Egm6321.f09.mtg31.djvu| page31-3]]

Superposition:


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$$ \psi(r, \theta) \underset{ \color{red} \underset{(2)}{\uparrow}}= \sum_{n=0} \left [ A_{n}r^{n}+B_{n}r^{-(n+1)} \right ] P_{n}(\mu) \ { \color{blue} \mu= \sin \theta= \cos \bar{ \theta}}$$


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As $$ r \to 0, r^{-(n+1)} \to \infty ; \ $$ so for $$ \psi \ $$ to be physical meaningful, set


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$$ B_{n}=0, \ \forall n=0,1,2, \cdots \ $$

(3)
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Boundary Condition $$ \psi(1, \theta)= \underbrace{f( \theta)= \sum_{n}A_{n} P_{n}( \sin \theta)}_{ \color{blue} Fourier-Legendre \ series} \ $$

(4)
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[[media: Egm6321.f09.mtg32.djvu| page32-2]]

Recall Fourier series:


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$$ f( \theta)= { \color{red}1 \cdot}a_{0}+ \sum a_{n} \cos n \omega \theta+ \sum b_{n} \sin n \omega \theta \ $$

(1)
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$$ \left \{ 1, \cos n \omega \theta, \sin n \omega \theta \right \}, \ n=1,2, \cdots \ \ $$   linear independent

(2)
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End Recall Fourier series


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Recall: $$ \left \{ \underbrace{ \mathbf{b}_{1}, \mathbf{b}_{2}, \cdots, \mathbf{b}_{n}}_{ \color{blue} linear \ independent} \right \} \in \mathbb{R}^{n} \ $$

(3)
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Consider $$ \mathbf{v} \in \mathbb{R}^{n}, \ \mathbf{v}= \sum_{i=1}^{n}a_{i} \mathbf{b}_{i} \ $$

(4)
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Find $$ a_{i}, \ i=1, \cdots, n \ $$


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$$ \mathbf{v} \cdot \mathbf{b}_{j}= \sum_{i} a_{i} \mathbf{b}_{i} \cdot \mathbf{b}{j}, \ j=1, \cdots, n \ $$


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$$ \Rightarrow \underbrace{ \left [ ( \mathbf{b}_{i} \cdot \mathbf{b}_{j}) \right ]_{ \color{red} n \times n}}_{ \color{blue} Gram \ matrix \ \Gamma} \left \{ a_{i} \right \}_{ \color{red} n \times 1}= \left \{ ( \mathbf{v} \cdot \mathbf{b}_{j}) \right \}_{ \color{red} n \times 1} \ $$


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Theorem: $$ \left \{ \mathbf{b}_{1}, \cdots, \mathbf{b}_{n} \right \} \ $$ linear independent iff det $$ \Gamma \neq 0 \ $$ End Theorem

[[media: Egm6321.f09.mtg32.djvu| page32-3]]

[[media: Egm6321.f09.mtg32.djvu| (4) page32-1:]] functions

Consider inner (scalar) products:


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$$g,h \ $$ 2 functions on $$ \left [ a,b \right ] \ $$


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$$ = \int_{a}^{b}g(x)h(x)dx \ $$

(1)
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$$ f( \theta)= \sum_{n=0}^{ \infty} A_{n} P_{n} ( \sin \theta) \ $$

(2)
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$$ = \sum_{n { \color{red}=0}}^{ \color{red} \infty} A_{n} < P_{n}, P_{m}> \ $$

(3)
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$$ \underbrace{ \left [ < P_{n}, P_{m}> \right ]_{ \color{red} \infty \times \infty}}_{ \color{blue} \Gamma ( P_{0}, \cdots)} \left \{ A_{n} \right \}_{ \color{red} \infty \times 1}= \left \{  \right \}_{ \color{red} \infty \times 1} \ $$

(4)
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