User:EGM6341.s11.TEAM1.WILKS/Mtg37

=EGM6321 - Principles of Engineering Analysis 1, Fall 2010=

[[media: 2010_11_10_14_40_12.djvu | Mtg 37:]] Wed, 10 Nov 10

= Heat conduction on a sphere (cont'd) =

[[media: 2010_11_10_14_40_12.djvu | Page 37-1]]
Heat Problem continued [[media: 2010_11_09_15_00_14.djvu | P.36-3]]

Physically meaningful boundary conditions
[[media: 2010_11_09_15_00_14.djvu | Eq.(1)P.36-2]] : $$ R_n(r) \ $$

$$ r \rightarrow \ 0, r^{-(n+1)} \rightarrow \ + \infty \ $$

$$ n \ge 0 \Rightarrow n+1 > 0 \ $$

$$ \Rightarrow \ $$ set:

[[media: 2010_11_09_15_00_14.djvu | Eq.(1)P.36-3]] : $$ \Theta_n ( \theta\ ) \ $$

$$ \theta \rightarrow \pm \frac{ \pi\ }{2} \Rightarrow \mu = \sin \theta \rightarrow \pm 1 \ $$

$$ \left | Q_n( \mu ) \right| \rightarrow + \infty \ $$

$$ \Rightarrow \ \ $$ set:

HW7.1

Plot $$ \left \{ P_0, P_1, P_2, P_3 \right \} \ $$ in Figure 1

and $$ \left \{ Q_0, Q_1, Q_2, Q_3 \right \} \ $$ in Figure 2.

Observe $$ P_n( \mu ) \ $$ and $$ Q_n( \mu ) \ $$ as $$ \mu \rightarrow \pm 1 \ $$.

END HW7.1

Series solution for homogeneous heat equation
Combine Eq.(1), Eq.(2) and [[media: 2010_11_09_15_00_14.djvu | Eq.(2)P.36-3]] :

$$ \Psi (r, \theta ) = \sum_{n=0}^\infty (A_n r^n) [ C_n P_n( \sin \theta )] =\sum_{n=0}^\infty \bar A_n r^n P_n ( \sin \theta ) \ $$ ,

where $$\bar A_n = A_n C_n \ $$ and $$ \sin \theta = \mu \ $$.

[[media: 2010_11_10_14_40_12.djvu | Page 37-2]]
which is similar to the Fourier-Legendre series (see below), since the basis functions involve the Legendre polynomials $$\displaystyle P_n(\mu)$$, with $$\displaystyle \mu = \sin \theta$$.

Fourier series: Projection on Fourier basis
NOTE: Fourier series $$ f( \theta) = 1 \cdot a_0 + \sum_{n=1}^\infty a_n \cos n \omega \theta + \sum_{n=1}^\infty b_n \sin n \omega \theta \ $$ Basis functions $$ = \left \{ 1, \cos n \omega \theta, \sin n \omega \theta \right \} \ $$ , where $$ n=1,2,3,... \ $$ linearly independent. END NOTE NOTE: Vector space $$\displaystyle \mathbb R^n$$

$$ \left \{ \mathbf b _1, \mathbf b _2 , ... , \mathbf b _n \right \} \in \mathbb R^n \ $$ basis vectors.

$$ \left \{ \mathbf{b}_1, \mathbf{b}_2 , ... , \mathbf{b}_n \right \} \ $$ are linearly independent. Consider: $$ \mathbf{v} \in \mathbb R^n \Rightarrow \ \mathbf v = \sum_{i=1}^n a_i \mathbf b  _i = a_i b_i  \ $$ Find $$ a_i \ $$ for $$ i=1,2,...,n \ $$ $$ \mathbf{v} \cdot \mathbf{b} _j = \sum_{i} a_i \mathbf{b} _i \cdot \mathbf{b} _j \ $$, for $$ j=1,2,...,n \ $$

[[media: 2010_11_10_14_40_12.djvu | Page 37-3]]
$$ \Rightarrow \left[ (\mathbf b _i \cdot \mathbf b  _j) \right] \left\{ a_i \right\} = \left\{ (\mathbf v  \cdot \mathbf b  _j) \right\} \ $$ where $$ [ (\mathbf b _i \cdot \mathbf b _j)] \in \mathbb R^{n \times n} \ $$ is the Gram Matrix $$ \mathbf \Gamma \ $$ and is n rows by n columns, $$ \left \{ a_i \right \} \ $$ is a n row by 1 column matrix and $$ \left\{ (\mathbf v \cdot \mathbf b _j) \right\} \ $$ is a n row by 1 column matrix. Theorem: $$ \left \{ (\mathbf{b}_1, \cdots, \mathbf{b}_n) \right \} \ $$ is linearly independent iff the determinant of $$ \mathbf { \Gamma\ } \ne 0 \ $$ , where iff is defined as "if and only if" $$ ( \iff ) \ $$ END NOTE Generalize to functions: Inner (scalar) product of two functions: $$ g: \left [ a,b \right ] \rightarrow \ \mathbb R \ $$ $$ h: \left [ a,b \right ] \rightarrow \ \mathbb R \ $$ where $$ \left [ a,b \right ] \ $$ is the domain and $$ \mathbb R \ $$ is the range. $$ \left \langle g,h \right \rangle = \int_{a}^{b} g(x) h(x) \, dx \ $$

Fourier-Legendre series: Projection on Legendre polynomial basis
Consider: $$ f( \theta ) = \sum_{n=0}^\infty A_n P_n(\sin \theta ) \ $$ Question: Find $$ A_n, n=0,1,...,\infty \ $$

[[media: 2010_11_10_14_40_12.djvu | Page 37-4]]
where $$ \left [ \left \langle P_n,P_m  \right \rangle \right ] \ $$ is the Gram Matrix $$ \mathbf { \Gamma\ } (P_0, P_1, P_2, \cdots ) \ $$ that has infinite rows and columns.

$$ \left \{ A_n \right \} \ $$ is an infinite row by 1 column matrix and $$ \left \{ \left \langle f, P_m \ \right \rangle \right \} \ $$ is an infinite row by 1 column matrix.

where $$ \sin \theta\ = \mu\ \ $$ Similarly for $$ \left \langle P_n,P_m \right \rangle \ $$

Orthogonality of Legendre Polynomials
where $$ \delta_{mn} \ $$ is the Kronecker delta [[media: 2010_11_04_13_57_39.djvu| Ref Eq.(2)P.33-2]] $$ \Rightarrow \ \mathbf{ \Gamma\ }(P_0, P_1,...) \ $$ is diagonal and $$ \Gamma\ ^{-1} \ $$ can be found easily

Computation of projection coefficients
Use $$ \mu\ = \sin \theta\ \ $$ as integrating variable instead fo $$ \theta\ \ $$ due to [[media:2010_11_04_14_51_31.djvu | Eq.(4)P.34-4]].

Rewrite [[media:2010_11_04_14_51_31.djvu | Eq.(3)P.34-4]] as:

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

Similar to Fourier series, Eq.(3) is an equality, not an approximation, due to competeness of $$\displaystyle \mathcal F$$.

Example: