User:Egm6341.s10.Team4.yunseok/HW4

= Problem 11: Application of Engineering Orbital Mech. =

Given
Refer Lecture slide 25-2 for problem statement

Polar form relative to focus
 * {| style="width:100%" border="0" align="left"

r(\theta) = \frac{1-e^2}{1-e \, cos(\theta)} $$
 * [[Image:Ellipse Polar.svg|200px]]
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

where, eccentricity $$\displaystyle e = \sin(\frac{\pi}{12}) $$

Find
'''Find Arc length of ellipse$$\displaystyle (I_n)$$ (orbital)using below integration method.

compute $$\displaystyle I_n$$ to the error $$\displaystyle 10^{-10}$$ and

compute time using tic/toc matlab commend

1) Composit Trapozoidal rule

2) Composit Simpson's rule

3) Romberg Table

4) Chebfun, sum commend

5) matlab commend : Trapz, quad, clencurt

Solution
The circumference $$C$$ of an ellipse is $$\displaystyle 4 a E(e)$$, where the function $$E$$ is the complete elliptic integral of the second kind. Here, a=1:

The complete elliptic integral of the second kind E is defined as


 * $$E(e) = \int_0^{\pi/2}\sqrt {1-e^2 \sin^2\theta}\ d\theta\!$$

So,


 * {| style="width:100%" border="0" align="left"

I= 4\int_0^{\pi/2}\sqrt {1-e^2 \sin^2\theta}\ d\theta $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (1)
 * }.
 * }.

using the equation (1), the arc length of ellipse was calculated by several numerical method.

The result is summarized.



 

Ramanujan approximation is 's:
 * $$C \approx \pi \left[3(a+b) - \sqrt{(3a+b)(a+3b)}\right]= \pi(3(a+b)-\sqrt{10ab+3(a^2+b^2)})$$

where, a = longest radius, b= shortest radius in ellipse

'''(1) Composit Trapoziodal rule

 Matlab Code: 

 subfunction ff(x) 

'''(2) Composit Simpson's rule

 Matlab Code: 

'''(3) Romberg Table

 Matlab Code: 

'''(4) Chebfun

 Matlab Code: 

'''(5) Trapz (matlab commend)

 Matlab Code: 

'''(6) quad (matlab commend)

 Matlab Code: 

'''(7) Clencurt (matlab commend)

 Matlab Code: