User:Egm6322.s12.team2.steele.m2/Mtg42

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

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Plan: $$\Rightarrow$$Heat conduction on a sphere (cont’d) Physically meaningful boundary conditions Series solution $$\Rightarrow$$Vectors in :  Projection on basis vectors Representation in arbitrary vector basis Gram matrix, linear independence Computation of components Functions: Fourier series, projection on Fourier basis functions Gram matrix, orthogonality Computation of components Wronskian, linear independence Complex representation of Fourier series Functions: Fourier-Legendre series, projection on Legendre polynomial basis functions Gram matrix, orthogonality Computation of components

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Orthogonality of Legendre polynomials $$\Rightarrow$$Heat conduction on a sphere (cont’d)

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=Heat Conduction on a Sphere (cont’d)= Physically meaningful boundary conditions Consider $$R_n({r})$$ in [[media:Pea1.f11.mtg41.djvu|Eqn(3) p.41-3]]: $$R_n({r}) = A_nr^n + B_nr^{-(n+1)}$$				(3) p.41-3

Since we want $$R_n(0) < +\infty$$, i.e., finite, we set Consider $$\Theta(\theta)$$ in [[media:Pea1.f11.mtg41.djvu|Eqn(1) p.41-6]]: $$\Theta_n(\theta) = C_n P_n(\mu) + D_n Q_n(\mu)$$		(1) p.41-6 $$\mu := \sin \theta$$			[[media:Pea1.f11.mtg40.djvu|Eqn(7) p.40-2]]

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R*7.8: Show that for n = 0: Plot the following figures Fig.1: $$\left\{P_0, P_1, P_2, P_3 \right\}$$ Fig.2: $$\left\{Q_0, Q_1, Q_2, Q_3 \right\}$$ Observe the limits of $$P_n(\mu)$$ and $$Q_n(\mu)$$ as $$\mu \to \pm 1$$ Since we want $$\Theta_n(\theta = \pm \frac{\pi}{2}) < +\infty$$, i.e., finite, we set

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=Series Solution: = Combine (4) p.42-1, (4) p.42-2 and (2) p.41-6:

(1)	Is similar to the Fourier-Legendre series (below), since the basis functions:

Are the term-by-term products of the polynomial basis functions used in Taylor series in: $$\{1, x, x^2, x^3, \cdots\}$$		(4) p.40-1 [[media:Pea1.f11.mtg40.djvu|Eqn(4) p.40-1]] And the Legendre polynomials in: $$\{P_n(x), \ n=0, 1, 2, \ldots \}$$		(1) p.40-1

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Vectors in $$\mathbb{R}^n$$ : Projection on basis vectors Basis vectors in $$\mathbb{R}^n : \{\mathbf b_1, \mathbf b_2 , \cdots , \mathbf b_n \} \in \mathbb R^n$$ See graph of linearly independent vectors $$b_1, b_2, b_3, b_n, \mathbb R$$

.i.e., any vector $$\mathbf v.$$ in $$\mathbb R^n$$ can be expressed as a linear combination (series) of the basis vectors $${\mathbf b_j}$$ Find the components $$\{{\nu_k} \in \mathbb R^{nx1}\}$$ : Form the scalar (dot) product of (1) with $$\{\mathbf b_i, i = 1, \cdots, n\}$$ to have n equations for n unknowns: $$\sum_{j=1}^{n}(\mathbf b_i \cdot \mathbf b_j) \, v_j = \mathbf b_i \cdot \mathbf v, \text{for} i = 1, \cdots , n$$	(2)

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=Matrix Form: =

Note pattern of nxn  nx1 and nx1

Row index column index Gram matrix:

If the Gram matrix is invertible, then the components of $$\mathbf v$$ with respect to $$\{\mathbf b_j \}$$ can be obtained as $$\{v_j \} = \boldsymbol \Gamma^{-1} \{\mathbf b_i \cdot \mathbf v\}$$	(4) $$\boldsymbol \Gamma(\{\mathbf b_1, \cdots , \mathbf b_n\})$$ Gram matrix of $$\{\mathbf b_i , I = 1, \cdots , n\}$$

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Theorem: R7.9 Consider the non-orthonormal basis $$\{ \mathbf b_1, \mathbf b_2, \mathbf b_3 \}$$ Expressed in the orthonormal basis $$\{\mathbf e_1, \mathbf e_2, \mathbf e_3\}$$

Sum

Row index       column index

similarly for $$b_1, b_3$$

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Sum Note: WA code for det $$\mathbf A $$ Determinant {{5,2,3},{4,5,6},{7,8,5}}

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=Fourier Series= Function: Fourier series, projection on Fourier basis functions Expansion of a function into Fourier series:

Fourier basis functions $$\{1, \cos n \omega \theta, \sin n \omega \theta : \, n = 1, 2, \cdots \}$$		(3) p.40-1 Scalar (inner) product: $$g:[a,b] \to \mathbb R$$		$$h:[a,b] \to \mathbb R$$

R8.1: Construct the Gram matrix for the Fourier basis functions (3) p.40-1 [[media:Pea1.f11.mtg40.djvu|Eqn(3) p.40-1]] using the scalar product (2). What is the property of this Gram matrix?

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To evaluate a scalar product in the Euler formulae for the Fourier series coefficients such as

It is necessary to use trigonometric identities to convert a product into a sum so to integrate easily. With the complex representation of Fourier series, the Euler formulae for the Fourier coefficients can be easily obtained, without the need for trigonometric identities. Complex representation of Fourier series Another way to represent Fourier series is to convert (1) p.42-8 into a complex form using Euler’s formula (5) p.27-6 [[media:Pea1.f11.mtg27.djvu|Eqn(5) p.27-6]]:

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Even though the r.h.s. of (2) p.42-10 involve the complex number “i”, the result of the summation is real (not complex) since the l.h.s. is a real-valued function. It is then easy to compute the Fourier coefficients in (2) p.42-10.

R8.2: Verify (4). It follows from (2)-(4) that

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R8.2: cont’d Recover from (1) the Fourier coefficients found with the trigonometric identities above. R8.1: cont’d Show that $${\cos \theta, \sin \theta}$$ are linearly independent using the Wronskian; see [[media:Pea1.f11.mtg35.djvu|Eqn(2) p.35-1]]. R8.3: Show that, equivalent to (2)-(3) p.42-10, the Fourier series can be written as

An advantage of (2)-(3) p.42-10 is a simple and elegant computation of the coefficients as shown in (1).

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Functions: Fourier-Legendre series, projection on Legendre polynomial basis functions Similar to the Fourier series in (1) p.42-8, consider the following series expansion called the Fourier-Legendre series: (1) Legendre polynomial basis functions $$\{ P_j(\mu), \ j=0, 1, 2, \ldots \}$$ [[media:Pea1.f11.mtg40.djvu|Eqn(1) p.40-1]] $$u := \sin \theta$$		(7) p.40-2 Similar to (2) p.42-4, find the components $$\{A_j\}$$ by forming the scalar products of (1) with $$\{P_i(u), j = 0, 1, 2, \ldots \}$$

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Matrix form:

Row index			column index Gram matrix:

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Orthogonality of Legendre polynomials:

$$\delta_{ij}$$  Kronecker delta defined in [[media:Pea1.f11.mtg38.djvu|Eqn(1) p.38-6]]

Thus the Gram matrix $$\mathbf{\Gamma}$$ in (5) p.42-14 is diagonal, and the inverse is easily obtained. Components of $$f(\theta)$$ along basis functions $$\{P_j(\sin\theta)\}$$ (2)	P.42-13, (1)-(2) p.42-14, and (1):