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

EGM6321 - Principles of Engineering Analysis 1, Fall 2009

Mtg 33: Thurs, 5Nov09

Page 33-1

$\begin{array}{c}F:=\{P_0,P_1,P_2,...\}\end{array}$

$\begin{array}{c}⟨f,P_m⟩=\underset{\theta =-\frac{\pi }{2}}{\overset{\frac{\pi }{2}}{\int }}f(\theta )P_m(\sin \theta )d\theta \end{array}$

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

Similarly for $⟨P_n,P_m⟩$

Orthogonality of Legendre polynomial

$\begin{array}{c}⟨P_n,P_m⟩=\underset{-1}{\overset{1}{\int }}{P}_n(x)P_m(x)dx=\frac{2}{2n+1}\delta {}_{mn}\end{array}$

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

$\begin{array}{c}\delta {}_{mn}=\{\begin{array}{cc}0& m\ne n\\ 1& m=n\end{array}\end{array}$

$\begin{array}{c}⟨f,P_m⟩=\underset{-1}{\overset{1}{\int }}f(\mu )P_m(\mu )d\mu \end{array}$

Page 33-2

Orthogonality of $\begin{array}{c}\left\{P_n\right\}=:F\end{array}$

$\Rightarrow \underset{\_}{\mathrm{\Gamma }}(F)$ is diagonal with diagonal coefficient:

$\begin{array}{c}⟨P_n,P_m⟩=\frac{2}{2n+1}\end{array}$

$\begin{array}{c}\Rightarrow A_n=\frac{1}{⟨P_n,P_m⟩}⟨f,P_n⟩\end{array}$

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

$\begin{array}{c}f(\mu )=\sum _{u=0}^{\mathrm{\infty }}{A}_n{P}_n(\mu )\end{array}$

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

p29-5: $f(\theta )=T_0{\cos }^4\theta =T_0(1-\mu {}^2{)}^2\sum _{u=0}^{\mathrm{\infty }}{A}_n{P}_n(\mu )$

Where $\mu =\sin |theta$

$\begin{array}{c}{A}_n=\frac{2n+1}{2}\underset{-1}{\overset{1}{\int }}{T}_0(1-\mu {}^2{)}^2{P}_n(\mu )d\mu \end{array}$

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 $A_n=\frac{⟨f,P_n⟩}{⟨P_n,P_n⟩}$, 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 $n\ge 5$ 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) $\le n$

$\begin{array}{c}\mathrm{\forall }q\in \mathrm{P}{}_n\Rightarrow q(x)=\sum _{i=0}^n{a}_i{P}_i(x)\end{array}$

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

$\begin{array}{c}{P}_m\perp \mathrm{P}{}_n\mathrm{\forall }m\ge n+1\end{array}$

References

Template:CourseCat