EGM6321 - Principles of Engineering Analysis 1, Fall 2010
Mtg 35: Fri, 5 Nov 10
1. Legendre polynomials : Finite series, where is the family of 1st homogeneous solutions of the Legendre equation.
2. Legendre functions:
3. Solving Laplace equation: Use the two families and (infinite series); similar to the use of Fourier series (see also p.32-2 and p.32-3).
Heat conduction on a sphere (cont'd)
Eq.(2)p.34-4 : Euler L2-ODE-VC
where is a constant to be determined, and is an independent variable.
Hence the characteristic equation:
With , define , thus
Think of as "" (independent variable) for the Legendre polynomial (function).
Transform the independent variable from to
, where from Eq.(1)
is the Legendre equation Eq.(1)p.5-4 if
Next step: solve for .
Heat conduction on a cylinder
HW6.5 Circular Cylinder Coordinates (cylindrical)
In terms of and .
2) Find .
Identify in terms of .
3) Find in cylindrical coordinates ( Bessel equation Eq.(2)p.24-1)
Find in spherical coordinates using math/physics convention of