User:Eml5526.s11.team01.roark/Mtg21

[[media: Fe1.s11.mtg21.djvu| Mtg 21]]: Fri, 18 Feb 11 [[media: Fe1.s11.mtg21.djvu| Page 21-1]]

HW 4.2:

FB, p.73, Problem 3.5.

End HW 4.2

HW 4.3:

FB, p.73, Problem 3.6.

End HW 4.3

HW 4.4:

FB, p.73, Problem 3.7. Use G1DM1.0/D1b Use [[media: Fe1.s11.mtg26.djvu| p. 26-2]]

End HW 4.4

Possible Basis Functions: Poly
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$$\displaystyle {{\mathcal{F}}_{p}}:=\left\{ {{x}^{j}},j=0,1,2,... \right\}$$ Cosine:
 *  (1)
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$$\displaystyle {{\mathcal{F}}_{c}}:=\left\{ \cos jx,j=0,1,2,... \right\}$$ Sine:
 *  (2)
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$$\displaystyle {{\mathcal{F}}_{s}}:=\left\{ 1,\sin jx,j=0,1,2,... \right\}$$ Fourier:
 *  (3)
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$$\displaystyle {{\mathcal{F}}_{F}}:=\left\{ \begin{matrix} \cos jx,j=0,1,2,... \\  \sin kx,k=0,1,2,... \\ \end{matrix} \right\}$$ Exponential:
 *  (4)
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$$\displaystyle {{\mathcal{F}}_{e}}:=\left\{ {{e}^{jx}},j=0,1,2,... \right\}$$ HW 4.5: For $$\displaystyle {{\Gamma }_{g}}=\left\{ \beta \right\}$$ and each of the above families of basis functions, i.e.,
 *  (5)
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$$\displaystyle {{\mathcal{F}}_{I}}=\left\{ {{b}_{j}},j=0,1,... \right\},I\in \left\{ p,c,s,F,e \right\}$$ [[media: Fe1.s11.mtg21.djvu| Page 21-2]]
 *  (6)
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find corresponding family $$\displaystyle {{\bar{\mathcal{F}}}_{I}}=\left\{ {{b}_{j}} \right\}$$ satisfying CBS, i.e., [[media: Fe1.s11.mtg20.djvu|(1) p. 20-2]]. Let $$\displaystyle \Omega =\left] \alpha ,\beta \right[=\left] -2,4 \right[$$.
 * 1) $$\displaystyle {{\bar{b}}_{0}}(x)=const$$; plot $$\displaystyle {{b}_{j}}$$ and $$\displaystyle {{\bar{b}}_{j}}$$ for j=1,2,3.
 * 2) Show $$\displaystyle \left\{ {{e}^{jx}},j=0,1,2 \right\}$$ linear independence on $$\displaystyle \Omega $$

Hint:



End Hint Section

Summary: Pros and cons on boundary conditions

(see also summation on [[media: Fe1.s11.mtg14.djvu| p. 14-1]], [[media: Fe1.s11.mtg15.djvu| p. 15-2]])

WRF:
 * 1) no constraint on w (and thus $$\displaystyle {{w}^{h}}$$ and $$\displaystyle \left\{ {{c}_{i}} \right\}$$) -> select $$\displaystyle \left\{ {{c}_{i}} \right\}$$ arbitrarily
 * [[media: Fe1.s11.mtg21.djvu| Page 21-3]]
 * 1) $$\displaystyle u$$ (and thus $$\displaystyle {{u}^{h}}$$ and $$\displaystyle \left\{ {{d}_{j}} \right\}$$) must satisfy 2 constraints.: essential boundary conditions and natural boundary conditions.

WF:
 * 1) $$\displaystyle w$$ and thus $$\displaystyle {{w}^{h}}$$ and $$\displaystyle \left\{ {{c}_{i}} \right\}$$) must satisfy homogenous essential boundary conditions -> cannot select $$\displaystyle \left\{ {{c}_{i}} \right\}$$ arbitrarily -> need CBS [[media: Fe1.s11.mtg20.djvu|(1) p. 20-2]].
 * 2) $$\displaystyle u$$ (and thus $$\displaystyle {{u}^{h}}$$ and $$\displaystyle \left\{ {{d}_{j}} \right\}$$) must satisfy essential boundary conditions at same point $$\displaystyle x=\beta $$ => Also use CBS [[media: Fe1.s11.mtg20.djvu|(1) p. 20-2]]

End Summary Section