University of Florida/Egm6321/F10.TEAM1.WILKS/Mtg41

EGM6321 - Principles of Engineering Analysis 1, Fall 2009

Mtg 41: Tues, 1Dec09

Page 41-1

Review for exam 2

- Historical development - Legendre functions
Question: How to obtain $P_n$ based on known $P_{n-1},P_{n-2},...$ ? - 2 recurring relationships. Same technique in power series.
Solution: Frobenius method
Question: Find a differential equation governing all $\left\{P_n\right\}$ ? - Legendre differential equations
2 families of homogeneous solutions:
- Legendre functions= $\left\{P_n\right\}$ + $\left\{Q_n\right\}$

$L_n=P_n$ or $L_n=Q_n$

Newtonian potential is solution of Laplace equation

i.e., $\mathrm{\Delta }\left(\frac{1}{r}\right)=0$

$\frac{1}{r}=\frac{1}{r_{PQ}}=\frac{1}{r_Q}{(1-2\mu \rho +\rho {}^2)}^{-\frac{1}{2}}$

Page 41-2

$=\frac{1}{r_Q}\sum _n{P}_n(\mu )\rho {}^n$ , where $=\rho :=\frac{r_P}{r_Q}$

$\Rightarrow \frac{1}{r}=\sum _n{P}_n(\mu )\frac{r_P^n}{r_Q^{n+1}}=\sum _n{H}_n(\mu ,r_P,r_Q)$ , where $H_n(\mu ,r_P,r_Q)=H_n(x,y,z)$

$\mathrm{\Delta }\frac{1}{r}=0=\sum _n\mathrm{\Delta }{H}_n(x,y,z)\Rightarrow \mathrm{\Delta }{H}_n=0$
Where this argument is based on the power series
Laplace equations in a sphere
axisymmetrical case P.29-1
separation of variables P.30-1
General solution of axisymmetrical Laplace equations in a sphere

$\psi (r,\theta )=\sum _n(A_n{r}^n+B_n{r}^{n+1})(C_n{P}_n+D_n{Q}_n)$

Where $A_n{r}^n+B_n{r}^{n+1}$ can be found on P.31-2

and $C_n{P}_n+D_n{Q}_n$ can be found on P.32-1

and $\mu =\sin \theta$

References

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