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

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Mtg 36: Thu, 12 Nov 09

page36-1

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

Example

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

 f \in \mathbb{P}_{3} \Rightarrow f(x)= \sum_{j=0}^{3}c_{j}x^{j} \ (1) page33-4  \Rightarrow f^{(4)}(x)=0 \

End Example

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

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

Trapezoidal Rule:

PEA1.F09.Mtg36.pg1.fig1.svg

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

page36-2

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

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

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 page31-3 and Legendre equation (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.

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} \

page36-4

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

 P_{0}(x)=1 \ \ \ \ \ \ \ \ \ \ \ Q_{0}(x)= \frac{1}{2} \log( \frac{1+x}{1-x}) \ HW4 page19-1  = \tanh^{-1}(x) \ HW