User:Egm6321.f12.team5/Report 7

= Report 7, Team 5 =

Given
(From lecture notes sec 64)

Find
Use matlab to plot $$F(5,-10;1;x)$$ near $$x=0$$ to display the local maximum (or maxima) in this region. Show that

Solution
On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.

Using Matlab's 'hypergeom' function, plotting $$F(5,-10;1;x)$$ from $$0.05 \leq x \leq 0.9.$$.

Plotting the polynomial in ($$) on the same graph shows that two are identical for $$ x = [0, 0.9]$$.

Matlab code used to generate the plots is shown below.

The problem is solved

Given
(From lecture notes sec 33)

The L2-ODE-CC:

Initial conditions: $$y(t_0)$$,$$y'(t_0)$$

Find
1. Using variation of parameters to show that

2. Compare ($$) and ($$) to Eqs.(2.4)-(2.5) in Dong 2012,A trigonometric integrator pseudospectral discretization for the N-coupled nonlinear Klein–Gordon equations, Numerical Algorithm, May. Which would be

Solution
On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions. 1.First we should find the homogenous solution.

Using trail solution:

We can get the characteristic equation as following:

Then the roots should be $$ \pm a_0 i $$ Then the homogenous solutions of ($$) should be

Since we already know the initial conditions: $$ y(t_0),y'(t_0) $$ So we can get following equation:

Then we can solve $$C_1$$ and $$C_2$$, answer as following:

Then we can get the 1st and 2nd homogenous answers as following:

And we can further simplify the sum of $$u_1(t)$$ and $$u_2(t)$$ as following:

For ($$), using variation of parameters we can get:

Which can be further simplified

So with ($$) and ($$) We can get the complete answer of ($$) should be:

Hence approved 2.Compare ($$) and ($$) to ($$) and ($$).We can find that:

With these substitution,($$) and ($$) would be identical with($$) and ($$)

Given
Given the equation:

Cartesian coord. in terms of spherical coord.:

Find
Show that the infinitesimal length $$\displaystyle ds $$ in (7.3.1) can be written in spherical coord. follows:

Solution
On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.

First, we can calculate the derivative of $$x_1\ x_2\ x_3 $$ separately. And then get $$ds^2$$ by adding them together.

From (7.3.2), we can find:

Similarly, the derivative of $$x_2\ x_3 $$ are:

And

So that:

Thus,

Which is agree with (7.3.3).

Given
(From lecture notes sec 64 with reference to lecture notes sec 63)

Consider the integral in (3) p.63-8 and (1) p.63.9 with particular values of some parameters:

Initial Vertical Velocity

Find
For each value of n, find the vertical velocity $$ z(t) $$ vs. time $$ t$$; plot this function. Also find and plot the altitude $$ y(t)$$ vs. time $$ t$$, then find the time that the projectile returns to the ground.

If an explicit expression for $$ z(t):=v_y(t)$$ cannot be obtained, use a numerical method to find $$ z(t)$$ for each given value of time $$t$$


 * 1) $$ n=2$$
 * 2) $$ n=3$$. Use matlab command "roots" to find the appropriate root z for each given time $$ t$$; verify with WA. Plot $$ z$$ versus $$ t$$. Find $$ y(t)$$ by integrating $$ z(t)$$ using the trapezoidal rule.
 * 3) $$ n=3$$. Use matlab function "hypergeom" to find the time $$ t$$ for each given value of $$ z$$ in the interval [-10,50]. Plot $$ t$$ versus $$ z$$. Find $$ y(t)$$ by integrating $$ z(t)$$ using the trapezoidal rule. Compare to Part 2.
 * 4) Verify the results in Parts 1 and 2 using the matlab in 2 steps:
 * a) use the command ode45 to integrate the L1-ODE-cc (1) p.63-8 to obtain $$ z(t)$$,


 * b) use the trapezoidal rule to integrate $$ z(t)$$ to obtain $$ y(t)$$


 * 1) The eq. of motion can be written as a system 1st-order ODEs to be integrated using matlab ode45

Verify Parts 2.

Solution
On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.

Part 1: n=2
We will plot (7.4.1) using the Matlab function 'hypergeom' where

Part 2: n=3 using 'roots'
We will plot (7.4.1) using the Matlab function 'roots' where

Part 3: n=3 using 'hypergeom'
We will plot (7.4.1) using the Matlab function 'hypergeom' where

Given
(From lecture notes sec 40)

Find
1. Express $$ \{dx_i\} = \{dx_1, dx_2, dx_3\} $$ in terms of $$ \{\xi_j\} $$ and $$ \{d\xi_j\} $$ 2. Express $$ ds^2 = \sum_{i}{d\xi_i}^2 = \sum_{h}{h_k}^2{d\xi_k}^2 $$ and identify $$ \{h_i\} $$ in terms of $$ \{\xi_j\} $$ 3. Find $$ \Delta u $$ in cylindrical co-ordinates 4. Use separation of variables to find the separated equations and compare to the Bessel Equation.

Solution
On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.

Part 1
We know that,

Therefore,

Equations $$, $$, $$, are the respective expressions for $$ \{dx_i\} = \{dx_1, dx_2, dx_3\} $$ in terms of $$ \{\xi_j\} $$ and $$ \{d\xi_j\} $$

Part 2
Substituting the expressions from equations $$, $$, and $$.

Also, given that

Therefore, comparing $$, and $$,

Part 3
The Laplacian operator in general curvilinear co-ordinates is given by,

We know that $$ h_1 h_2 h_3 = \xi_1 $$, Therefore

Therefore,

Part 4
Using separation of variables, Defining $$ u $$ as,

Substituting in the Laplacian expression for cylindrical co-ordinates,

Dividing the whole equation by $$ A(\xi_1) B(\xi_2) C(\xi_3) $$

Multiplying the equation by $$ \xi_1^2 $$

Now setting,$$ \frac{1}{B(\xi_2)} \frac{\partial^2 B(\xi_2) }{\partial \xi_2^2} = -b $$ and $$ \frac{1}{C(\xi_3)} \frac{\partial^2   C(\xi_3)}{\partial \xi_3^2} = a $$

Suppose, $$ \xi_1 = x $$ and $$ A(\xi_1) = y $$

Setting $$ \Delta u = 0 $$

The above equation is similar to the Bessel Equation given in (sec 27)

Given
(From lecture notes sec 40)

Math/Physics convention is

So that

The infinitesimal distance $$dx^2$$ is given by

and the Laplace operator in general curvilinear coordinates is given by

Find
Determine $$\Delta u$$ in spherical coordinates using math/physics convention (as shown in $$).

Solution
On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.

Determine $$dx_1, dx_2, $$ and $$dx_3$$.

Similarly,

and

Determine $$dx_1^2, dx_2^2, $$ and $$dx_3^2$$.

Therefore,

From $$,

Substituting the result into $$,

Substituting the math/physics convention from $$,

Comments
initial solution by --Egm6321.f12.team5.kim (talk) 08:15, 4 December 2012 (UTC)

= Contributors = Report Leader: Yi Zhongwen Report Co-Leaders : Yu Hou