University of Florida/Egm6321/F10.TEAM1.WILKS/Mtg33

EGM6321 - Principles of Engineering Analysis 1, Fall 2009

Mtg 33: Thurs, 5Nov09

Page 33-1

$\displaystyle {\begin{aligned}F:=\left\{P_{0},P_{1},P_{2},...\right\}\end{aligned}}$

(1)

$\displaystyle {\begin{aligned}\left\langle f,P_{m}\right\rangle =\int _{\theta \ =-{\frac {\pi \ }{2}}}^{\frac {\pi \ }{2}}f(\theta \ )P_{m}(\sin \theta \ )\,d\theta \ \end{aligned}}$

(2)

Where: $\sin \theta \ =\mu \ \$ and $d\theta \ \$ becomes $d\mu \ \$

Similarly for $\left\langle P_{n},P_{m}\right\rangle$

Orthogonality of Legendre polynomial

$\displaystyle {\begin{aligned}\left\langle P_{n},P_{m}\right\rangle =\int _{-1}^{1}P_{n}(x)P_{m}(x)\,dx={\frac {2}{2n+1}}\delta \ _{mn}\end{aligned}}$

(3)

Where $\delta \ _{mn}=\$ kronecker delta

$\displaystyle {\begin{aligned}\delta \ _{mn}={\begin{cases}0&m\neq n\\1&m=n\end{cases}}\end{aligned}}$

(4)

$\displaystyle {\begin{aligned}\left\langle f,P_{m}\right\rangle =\int _{-1}^{1}f(\mu \ )P_{m}(\mu \ )\,d\mu \ \end{aligned}}$

(5)

Page 33-2

Orthogonality of $\displaystyle {\begin{aligned}\left\{P_{n}\right\}=:F\end{aligned}}$

(1)

$\Rightarrow \ {\underline {\Gamma \ }}(F)\$ is diagonal with diagonal coefficient:

$\displaystyle {\begin{aligned}\left\langle P_{n},P_{m}\right\rangle ={\frac {2}{2n+1}}\end{aligned}}$

(2)

$\displaystyle {\begin{aligned}\Rightarrow \ A_{n}={\frac {1}{\left\langle P_{n},P_{m}\right\rangle }}\left\langle f,P_{n}\right\rangle \end{aligned}}$

(3)

F is complete, i.e. any continuous function, f, can be expressed as an infinite series of function in F:

$\displaystyle {\begin{aligned}f(\mu \ )=\sum _{u=0}^{\infty }A_{n}P_{n}(\mu \ )\end{aligned}}$

(4)

Eq(4) is an equality due to the completeness of F

p29-5: $f(\theta \ )=T_{0}\cos ^{4}\theta \ =T_{0}(1-\mu \ ^{2})^{2}\sum _{u=0}^{\infty }A_{n}P_{n}(\mu \ )\$

Where $\mu \ =\sin |theta\ \$

$\displaystyle {\begin{aligned}A_{n}={\frac {2n+1}{2}}\int _{-1}^{1}T_{0}(1-\mu \ ^{2})^{2}P_{n}(\mu \ )\,d\mu \ \end{aligned}}$

(5)

Where n=0,1,2...n

Page 33-3

HW: $f=\sum _{i}g_{i}\$

Show that if $\left\{g_{i}\right\}\$ is odd, then f is odd

Show that if $\left\{g_{i}\right\}\$ is even, then f is even

HW: Show that $P_{2k}\$ is even for k=0,1,2... and $P_{2k+1}\$ is odd

Eq.(5) P.33-2 $A_{n}={\frac {\left\langle f,P_{n}\right\rangle }{\left\langle P_{n},P_{n}\right\rangle }}\$, f even

$\Rightarrow \ A_{n}=0\$ for $n=2k+1\$, since $P_{2k+1}(x)\$ is odd

$A_{1}=A_{3}=A_{5}=...=0\$

It turns out that $A_{n}=0\$ for all $n\geq 5\$ due to linear independance of $F=\left\{P_{n}\right\}\$ and the orthogonality of $F\$

Page 33-4

Linear independance of $F\$

$P_{n}(x)\$ is a polynomial of order n

$P_{n}\in \mathrm {P} \ _{n}\$ set of all polynomials of degree (order) $\leq \ n\$

$\displaystyle {\begin{aligned}\forall q\in \mathrm {P} \ _{n}\Rightarrow \ q(x)=\sum _{i=0}^{n}a_{i}P_{i}(x)\end{aligned}}$

(1)

HW: $q\in \mathrm {P} \ _{4}\$

$q(x)=\sum _{i=0}^{4}c_{i}x^{i}\$

Given $c_{0}=3,c_{1}=10,c_{2}=15,c_{3}=-1,c_{4}=5\$

Find $\left\{a_{i}\right\}\$ such that $q(x)=\sum _{i=0}^{4}a_{i}P_{i}\$

Plot $q=\sum _{i}c_{i}x^{i}=\sum _{i}a_{i}P_{i}\$

Where $\sum _{i}c_{i}x^{i}=\$ figure 1 and $\sum _{i}a_{i}P_{i}=\$ figure 2

Othogonality of $F=\left\{P_{k}\right\}\$ Eq.(3) P.33-1

$\displaystyle {\begin{aligned}P_{m}\perp \mathrm {P} \ _{n}\ \forall m\geq \ n+1\end{aligned}}$

(2)