User:Egm6341.s10.team3.heejun/Meeting 36 transcript

 '''MTG 36: Tue. 30 Mar 10'''


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Note:  Typeset of Transparencies, not lecture transcript  Heejun Chung 20:58, 11 August 2010 (UTC)

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 * (4) [[media: Egm6341.s10.mtg35.djvu | Page 35-2]]:   $$ \displaystyle {Z}' = h \dot{Z} $$
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 * $$ \displaystyle Z(s) = \sum^{4}_{i=1} N_{i}(s) d_{i} $$
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 * $$ \displaystyle Z(t) = \sum^{4}_{i=1} N_{i}(t) d_{i}\,\, since\,\, s=s(t) $$
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 * $$ \displaystyle \underbrace{Z(t_{j})}_{\color{blue}{d_{j}}} = \sum^{4}_{i=1} \underbrace{N_{i}(t_{j})}_{\color{blue}{\delta_{ij}}} d_{i} $$
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HW: Express $$ \displaystyle t(s) $$ in terms of $$ \displaystyle s(t) $$
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Recall collocation at $$ \displaystyle t_{i},\,\,  t_{i+1} $$ Now want to enforce compliance with differential equation at $$ \displaystyle t_{i+ \frac{1}{2}} $$ i.e., collocation

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 * $$ \displaystyle Z_{i+ \frac{1}{2}} = Z(s= \frac{1}{2}) = \frac{1}{2}(Z_{i}+Z_{i+1}) + \frac{h}{8}(f_{i}-f_{i+1}) $$
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HW: Proof that $$ \displaystyle Z(\varsigma=2)= \frac{1}{2}(Z_i-Z_{i+1})+ \frac{h}{8}(f_i-f_{i+1}) $$ using   (2)  [[media: Egm6341.s10.mtg35.djvu | Page 35-2]],     (1)  [[media: Egm6341.s10.mtg35.djvu | Page 35-4]] and     (1)  [[media: Egm6341.s10.mtg36.djvu | Page 36-1]]
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 * Collocation at $$ \displaystyle t_{i+ \frac{1}{2}} : \dot{Z}_{i+ \frac{1}{2}} = \underbrace{f_{i+ \frac{1}{2}}}_{\color{blue}{:= f(Z_{i+ \frac{1}{2}},\,\, t_{i+ \frac{1}{2}})}} $$
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 * $$ \displaystyle \dot{Z}_{i+ \frac{1}{2}} = {Z}'_{i+ \frac{1}{2}} \frac{1}{h} $$
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 * $$ \displaystyle {Z}'_{i+ \frac{1}{2}} \underbrace{=}_{\color{red}{(3)} \color{blue}{p.35-3}} {Z}' (s= \frac{1}{2}) = - \frac{3}{2} (Z_{i} - Z_{i+_1}) - \frac{1}{4} ({Z}'_{i}+{Z}'_{i+1}) $$
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HW: Proof that $$\displaystyle {Z}' (s= \frac{1}{2}) = - \frac{3}{2}(Z_i-Z_{i+1})-\frac{1}{4}(Z_i^\prime-Z_{i+1}^\prime) $$
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 * $$ \displaystyle {Z}' = h \dot{Z} $$
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& \dot{Z}_{i+1} = f_{i+1} \\\end{align} $$
 * Collocation at $$ \displaystyle \begin{align} t_{i},\,\, t_{i+1}:\,\, & \dot{Z}_{i} = f_{i} \\
 * Collocation at $$ \displaystyle \begin{align} t_{i},\,\, t_{i+1}:\,\, & \dot{Z}_{i} = f_{i} \\
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 * $$ \displaystyle \dot{Z}_{i+ \frac{1}{2}} = - \frac{3}{2h}(Z_{i} - Z_{i+1}) - \frac{1}{4} (f_{i}+f_{i+1}) \neq f_{i+ \frac{1}{2}}\,\,\,\, $$   in general
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[[media: Egm6341.s10.mtg36.djvu | Page 36-3]] 

Gap: $$ \displaystyle \bigtriangleup := \dot{Z}_{i+ \frac{1}{2}} - f_{i+ \frac{1}{2}} $$

Collocation at $$ \displaystyle t_{i+ \frac{1}{2}} : $$ Find $$ \displaystyle (Z_{i},\,\, Z_{i+1}) $$ such that $$ \displaystyle \bigtriangleup = 0 $$

$$ \displaystyle \bigtriangleup = 0 \Rightarrow Z_{i+1} = Z_{i} + \underbrace{\frac{(\frac{h}{2})}{3}[f_{i} + 4f_{i+ \frac{1}{2}} + f_{i+1}]}_{\color{blue}{Simpson's\,\, rule}} $$


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HW: Proof that $$ Z_{i+1}=Z_i+\frac{h/2 }{3}[f_i+4f_{i+\frac{1}{2}}+f_{i+1}]$$
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$$ \displaystyle \underbrace{\int^{t_{i+1}}_{t_{i}} \dot{Z} dt}_{\color{blue}{Z_{i+1}-Z_{i}}} = \int^{t_{i+1}}_{t_{i}} f(Z,\,\,t) dt $$
 * $$ \displaystyle \dot{Z} = f(Z,\,\, t) $$
 * $$ \displaystyle \dot{Z} = f(Z,\,\, t) $$
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