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

${\frac {1}{r}}={\frac {1}{r_{PQ}}}={\frac {1}{r_{Q}}}\left(1-2\mu \ \rho \ +\rho \ ^{2}\right)^{-{\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}\left(\mu \ ,r_{P},r_{Q}\right)\$ , where $H_{n}\left(\mu \ ,r_{P},r_{Q}\right)=H_{n}\left(x,y,z\right)\$

$\Delta \ {\frac {1}{r}}=0=\sum _{n}\Delta \ H_{n}(x,y,z)\ \Rightarrow \ \Delta \ H_{n}=0$
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 \ \$