User:EGM6341.s11.team1.Chiu/PEA1 F09 Mtg36

Mtg 36: Thu, 12 Nov 09

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Let $$ f \in \mathbb{P}_{2n-1}, \ $$ i.e. $$ f \ $$ is a polynomial of degree $$ \leq 2n-1 \Rightarrow f^{(2n)}(x)=0 \ $$

Example


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$$ 2n-1=3 \Rightarrow n=2 \Rightarrow 2n=4 \ $$


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$$ f \in \mathbb{P}_{3} \Rightarrow f(x)= \sum_{j=0}^{3}c_{j}x^{j} \ $$ [[media: Egm6321.f09.mtg33.djvu| (1) page33-4]] $$ \Rightarrow f^{(4)}(x)=0 \ $$


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End Example


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$$ f^{(2n)}(x)=0 \Rightarrow E_{n}(f)=0 \ $$


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i.e., we can integrate exactly any polynomial of degree $$ \leq 2n-1 \ $$ using only $$ n \ $$ integral points (almost half).

Trapezoidal Rule:


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$$ I(f)= \int_{a}^{b}f(x)dx \ $$


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[[media: Egm6321.f09.mtg36.djvu| page36-2]]


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$$ h= \frac{b-a}{n}, \ n= \ $$ number of panels (trapezoidal)


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$$ E_{n}(f)=- \frac{(b-a)h^{2}}{12} f^{(2)}( \eta), \ \eta \in \left [ a,b \right ] \ $$


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Trapezoidal Rule can only integrate exact a straight line. Otherwise, $$ E_{n}(f) \to 0 \ $$ as $$ n \to \infty, \ $$ even for a simple polynomial of degree 3. (not even degree 2)

End Trapezoidal Rule

Question 1: Origin (2) of Legendre polynomial [[media: Egm6321.f09.mtg31.djvu| page31-3]] and Legendre equation [[media: Egm6321.f09.mtg14.djvu| (1) page14-2]].

End Question 1

Question 2: Why solving Laplace equation (heat, fluid,...) in a sphere gave rise to Legendre equation (2) ?

End Question 2

Answer 1: Legendre's idea: Expand Newtonian potential $$ \frac{1}{r} \ $$ into power series in his study of attraction of spheres $$ \Rightarrow \ $$ Legendre polynomial.

[[media: Egm6321.f09.mtg36.djvu| page36-3]]

Legendre found the differential equation that admits Legendre polynomial $$ P_{n}(x) \ $$ as solutions $$ \Rightarrow \ $$ Legendre differential equation.

End Answer 1

Answer 2: Newtonian potentail $$ \frac{1}{r} \ $$ is a solution of Laplace equation, and thus each term in power series of $$ \frac{1}{r} \ $$ and thus $$ P_{n}(x) \ $$ is also solution $$ \Rightarrow \ $$ spherical harmonics (solution of Laplace equation in a sphere)

End Answer 2

$$ \color{blue} \left \{ Q_{n}(x) \right \}: \ $$ 2nd set of homogeneous solutions (non-polynomial) to Legendre equation.

$$ \left \{ P_{n}(x) \right \}: \ $$ 1st set of homogeneous solutions (polynomial) to Legendre equation.

Legendre functions= $$ \underbrace{ \left \{ P_{n}(x) \right \}}_{ \color{red} polynomial} + \underbrace{ \left \{ Q_{n}(x) \right \}}_{ \color{red} non-polynomial} \ $$

[[media: Egm6321.f09.mtg36.djvu| page36-4]]


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$$ P_{n} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Q_{n} \ $$


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$$ P_{0}(x)=1 \ \ \ \ \ \ \ \ \ \ \ Q_{0}(x)= \frac{1}{2} \log( \frac{1+x}{1-x}) \ $$ [[media: Egm6321.f09.mtg19.djvu| HW4 page19-1]] $$ = \tanh^{-1}(x) \ $$ HW


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