User:Egm6322.s12.team2.steele.m2/Mtg40

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

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Plan: $$\Rightarrow$$Heat conduction on a sphere (cont’d) Legendre functions as basis functions 1st separated eq. Characteristic equation 2nd separated eq. Transform the independent variable from $$\theta$$ to $$\mu$$. Legendre equation $$\Rightarrow$$Heat conduction on a cylinder Bessel equation

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=Heat Conduction on a Sphere (cont’d)= Note: Legendre functions as basis functions See Note p.7-1: Legendre equation [[media:Pea1.f11.mtg7.djvu|Eqn(1) p.7-1]]. See also Example [[media:Pea1.f11.mtg37.djvu|p.37-3]].

$$\{]P_n, Q_n], \ n=0,1,2,\ldots \}$$ Similar to Fourier series w/basis functions

Taylor series w/basis functions

The series solution here is constructed using both families $$\{ P_n(x) \}$$ and $$\{ Q_n(x) \}$$, since for each n both 1st and 2nd homogeneous soln. should be used to obtain the overall solution.

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=1st Separated Eq. (1) p.39-6: Euler L2-ODE-VC= Note [[media:Pea1.f11.mtg30.djvu|Eqn(5) p.30-4]] : Trial solution:

Here is the characteristic equation: Next: Transform the 2nd separated eq. (2) p.39-6 into the Legendre equation of degree n, i.e., [[media:Pea1.f11.mtg7.djvu|Eqn(1) p.7-1]].

Think of u as x (independent variable) in the Legendre polynomials (functions) in (1) p.40-1.

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Transform the independent variable from $$\theta$$ to $$\mu$$.

No. (2) p.39-6 [[media:Pea1.f11.mtg39.djvu|Eqn(2) p.39-6]] :

No. (6)-(7) p.40-2 and (3)-(4):

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Which would be the Legendre equation [[media:Pea1.f11.mtg7.djvu|Eqn(1) p.7-1]] if

Note: King 2003 p.46: “This choice of constant may seem rather ad-hoc at this point, will become clear later.”  (not really) For (2) to be possible, one must have known the form of the Legendre equation (1) p.7-1 and that its solutions are ready to be used. But then where did the Legendre equation come from? How to get its 1st homogeneous solutions? The mystery thickens. It is clear that one cannot begin from the Legendre equation, but needs to explain its origin before one can solve the heat equation on a sphere. The next question would then be why is there a connection between spherical geometry and the Legendre equation?

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R*7.2: Heat conduction on a cylinder See graph of $$x_3 = z = \xi_3$$

See graph of $$x_1 = x$$, $$\theta = \xi_2$$ and $$r = \xi_1$$

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Identify $${h_i}$$ in terms of $${\xi_i}$$ 3)	Find $$\Delta u$$ in cylindrical coordinates 4)	Use separation of variable to find the separated equations and compare to the Bessel equation [[media:Pea1.f11.mtg27.djvu|Eqn(1) p.27-1]]. R*7.3: Find $$\Delta u$$ in spherical coordinates using the math/physics convention [[media:Pea1.f11.mtg38.djvu|Eqn(4) p.38-6]].