User:Egm6341.s10.team3.sa/Mtg37

Mtg 37: Thu, 1 Apr 10

 Typeset of transparencies, not lecture transcript. Subramanian Annamalai 16:19, 12 August 2010 (UTC)

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HW:
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1. Run Kessler's code to reproduce the table of constants and the values of polynomials at t=1 obtained by Kessler. 2. Explain Kessler's code by giving comments for every line. 3. Obtain $$ \displaystyle \color{blue} (p_2,p_3) \; (p_4,p_5) \; (p_6,p_7)$$ by understanding Kessler's code line by line.


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Optimal Control (Continued): Note: 1. Optimal control: $$\displaystyle z_i \;\; i=0,1,2, \cdots $$ are unknows solved bu NLP (Non Linear Programming). 2. Solve non-linear ODE (Eq.4 [[media: Egm6341.s10.mtg36.djvu | Page 36-3]]) and IVP(Initial value problem) $$\displaystyle \rightarrow z(t_0) = z_0 \rightarrow \;known $$

Using (Eq.5) on [[media: Egm6341.s10.mtg36.djvu | Page 36-3]] and Simpson's rule, we obtain (Eq.3) on [[media: Egm6341.s10.mtg36.djvu | Page 36-3]]

Assume $$\displaystyle z_i $$ is known. Hence find $$\displaystyle z_{i+1} $$ using (Eq.3) on [[media: Egm6341.s10.mtg36.djvu | Page 36-3]]

1. $$\displaystyle f_i = f(z_i,t_i) \; \rightarrow $$ Can be computed.

[[media: Egm6341.s10.mtg37.djvu | Page 37-2]] 

2. $$\displaystyle f_{i+1} = f(\underbrace{z_{i+1}}_{unknown},\underbrace{t_{i+1}}_{known}) \; \rightarrow \;Unknown\;\;(t_{i+1} = t_i + h). $$

3. $$\displaystyle f_{i+\frac{1}{2}} = f(\underbrace{z_{i+\frac{1}{2}}}_{unknown},\underbrace{t_{i+\frac{1}{2}}}_{known}) \; \rightarrow \;Unknown\;\;(t_{i+\frac{1}{2}} = t_i + \frac{h}{2}) $$.

Time-stepping algorithm: $$\displaystyle z_{i+\frac{1}{2}} = z(s=\frac{1}{2}) = g(z_i,z_{i+1}) \rightarrow $$ $$\displaystyle (Eq. 1) $$

$$\displaystyle z_{i+1} = z_i + \underbrace{\frac{h/2}{3} \bigg[f_i + f(g(z_i,z_{i+1}),t_{i+\frac{1}{2}}) + f_{i+1} \bigg]}_{Non-linear function in z_{i+1}}$$ $$\displaystyle (Eq. 2) $$

$$\displaystyle \Rightarrow F(z_{i+1}) = 0 \rightarrow $$ Non-linear algebraic equation in $$\displaystyle z_{i+1} $$ $$\displaystyle (Eq. 3) $$