User:EGM6341.s11.team1.Chiu/PEA1 F09 Mtg34

Mtg 34: Tue, 10 Nov 09

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[[media: Egm6321.f09.mtg33.djvu| page33-4]] continued


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Consider $$ f( \mu)=T_{0}(1- \mu^{2})^{2} \in \mathbb{P}_{4} \ $$    [[media: Egm6321.f09.mtg33.djvu| page33-2]] (set of polynomial of degree 4 or less)


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[[media: Egm6321.f09.mtg33.djvu| (2) page33-4:]] $$ P_{m} \perp \mathbb{P}_{4}, \ \forall m \geq 5 \ $$


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[[media: Egm6321.f09.mtg33.djvu| (5) page33-2:]] $$ A_{n}=0, \ \forall n \geq 5, \ n=5,6,7, \cdots \ $$


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$$ + \ $$ [[media: Egm6321.f09.mtg33.djvu| (2) page33-3:]] $$ \Rightarrow \ $$ only need to evaluate $$A_{0}, A_{2}, A_{4} \ $$


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$$ A_{0}= \frac{8}{15}{ \color{blue} T_{0}}, \ A_{2}= -\frac{16}{21}{ \color{blue} T_{0}}, \ A_{4}= \frac{8}{35}{ \color{blue} T_{0}} \ $$


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[[media: Egm6321.f09.mtg32.djvu| (2) page32-1:]] $$ \psi (r, \theta)=A_{0} P_{0}+A_{2} r^{2} P_{2}+A_{4} r^{4} P_{4} \ $$


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[[media: Egm6321.f09.mtg34.djvu| page34-2]]

confer K. p.47


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HW:


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boundary condition $$ \psi(1,\theta)=f(\theta)=T_{0} \cos \theta \ $$ $$ f(\mu)=T_{0} \sqrt{1- \mu^{2}} \ $$ even


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[[media: Egm6321.f09.mtg33.djvu| (5) page33-2:]] $$ A_{n}= \frac{2n+1}{2} \ $$


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1) Without calculation, find property of $$ A_{n}\ $$, i.e., $$ A_{2k}=0?, \ A_{2k+1}=0? \ $$

2) Compose 3 non-zero coefficient $$ A_{n}\ $$

2.a.) analytical using either $$ \theta \ $$ or $$ \mu \ $$ as integrate variable.

2.a.) Numerically using Gauss-Legendre accurately within 5%


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The other homogeneous solution of Legendre equation.

(Recall remark on [[media: Egm6321.f09.mtg31.djvu| page31-2]]): $$ Q_{n}(x) \ $$ $$ n=0,1,2, \cdots \ $$ see [[media: Egm6321.f09.mtg18.djvu| page18-1]]