University of Florida/Egm4313/s12.team11.R6

Report 6


Intermediate Engineering Analysis
Section 7566
Team 11
Due date: April 11, 2012.

R6.1Edit

Solved by Daniel Suh

Problem StatementEdit

  • Find the fundamental period of   and  
  • Show that these functions also have period  .
  • Show that the constants   is also a periodic function with period  .

SolutionEdit

  • Find the fundamental period, and show that these functions have a period of  .

 

 

 

 

 

 

 

                is the fundamental period for  .

 

 

 

 

 

                is the fundamental period for  .


  • Show that the constants   is also a periodic function with period  .

From Fourier Series,
 


 

 

 

 

           


 

 

 

 

           

R6.2Edit

Part AEdit

Problem StatementEdit

Solve the problems 11 and 12 from Kreyszig p.491

SolutionEdit

First, for problem 11:
  from   where   We know that   so that  

We check is the function is even, odd or neither.
Since   satisfies the condition   we know that the function is even.
Given that the function is even and that   for even functions, we exclude the   term from the Fourier Series.

We find  :
 
Plugging in,
 

We find  
 
 
We can use the equivalent expression:
 

Now we evaluate the integral by using integration by parts. Below are the steps:
Integration by parts uses  
 
Then, we have:
 
Repeating for the second term:
 
Then, we have:
 
Bringing all the terms together:
 
Then,
 
Evaluating at the boundaries and simplifying:
 
Further,
 
We can simplify one step further by acknowledging that  .
 

Finally,
 

The Fourier series is:

                                    



Now problem 12:
  from   where   We know that   so that  

We check is the function is even, odd or neither.
Since   satisfies the condition   we know that the function is even.
Given that the function is even and that   for even functions, we exclude the   term from the Fourier Series.

We find  :
 
Plugging in,
 

We find  
 
 
Now we evaluate the integral by using integration by parts. This process is similar to the one shown in the solution for problem 11 above and therefore will be performed in Wolfram Alpha.
The following command was given in Wolfram Alpha: integral (1-(x^2/4))*cos(pi*x) from -2 to 2.
Wolfram Alpha yields the following
 

Finally,
 

The Fourier series is:

                               


Part BEdit

Solved by Francisco Arrieta

Problem StatementEdit

Find the Fourier series expansion for   on p. 9.8 as follows:

Part 1Edit

Develop the Fourier series expansion of  

Plot   and the truncated Fourier series  

 

for n=0,1,2,4,8. Observe the values of   at the points of discontinuities, and the Gibbs phenomenon. Transform the variable so to obtain the Fourier series expansion of  

SolutionEdit

 
 
 


 

 
 


 

 
 

Note:

If k is even  

If k=1, 5, 9...  

If k=3, 7, 11...  


 

Note:

  for n=1, 2...


Then Fourier series becomes a Fourier cosine series:


                                               


For n=0

                                               

For n=1

                                               

 

For n=5

                                               

 

File:Perft.jpg


After transforming the variable of  

                                               

Part 2Edit

Do the same as above, but using   to obtain the Fourier expansion of   ; compare to the result obtained above

SolutionEdit

 
 
 


 

 
 


 

 
 

Note:

  for n=1, 2..


 

 
 

Note:

If k is even  

If k=1, 3, 5...  


Then Fourier series becomes a Fourier sine series:

                                               


For n=0:

                                               

For n=1:

                                               

File:2n=1.jpg

For n=5:

                                               

File:2n=5.jpg

File:2perft.jpg


After transforming the variable of  

                                               

R6.3Edit

Problem StatementEdit

Page 491, #15,17. Plot the truncated Fourier series for n=2,4,8.

SolutionEdit

15.


 

Since  , the function is odd.

 
After simplification,  
. Using Equation (5) on page 490, the exapansion becomes;
 
With   this becomes;
  for n is "odd".
This means when n is even,  

 

17.
 

 

 
 

  

 
Simplifying results in;
 
So,

  

when n is odd.
 
Simplifying results in;

 

 

Since   for even n,   becomes   for even n.

 


R6.4Edit

solved by Luca Imponenti

Problem StatementEdit

Consider the L2-ODE-CC (5) p.7b-7 with the window function f(x) p.9-8 as excitation:

 

and the initial conditions

 

1. Find   such that:

 

with the same initial conditions as above.

Plot   for   for x in  

2. Use the matlab command ode45 to integrate the L2-ODE-CC, and plot the numerical solution to compare with the analytical solution.

Level 1:  

Fourier SeriesEdit

One period of the window function p9.8 is described as follows

 

From the above intervals one can see that the period,   and therefore   Applying the Euler formulas from   to   the Fourier coefficients are computed:

 

 

 

 

 

The integral from   to   can be omitted from this point on since it is always zero.

 

 

 

 

and

 

 

 

 

The coefficients give the Fourier series:

 

 

 

Homogeneous SolutionEdit

Considering the homogeneous case of our ODE:

 

The characteristic equation is

 

 

 

Therefore our homogeneous solution is of the form

 

Particular SolutionEdit

Considering the case with f(x) as excitation

 

 

The solution will be of the form

 

Taking the derivatives

 

 

Plugging these back into the ODE:

 

 

 

Setting the two constants equal

 

 

This is valid for all values of n. Since the coefficients of the excitation   and   are zero for all even n, then the coefficients   and   will also be zero, so we must only find these coefficients for odd n's. Now carrying out the sum to   and comparing like terms yields the following sets of equations. Written in matrix form:

 

Assuming   this matrix can be solved to obtain

 

For the remaining coefficients to be solved all sums will be used so a more general equation may be written:

 

Results of these calculations are shown below:

 

The solution to the particular case can be written for all n (assuming A=1):

       

General SolutionEdit

The general solution is

 

where

 

Different coefficients   will be calculated for each n. These coefficients are easily solved for by applying the given initial conditions. Below are the calculations for n=2.

 

Applying the first initial condition  

 

Taking the derivative

 

Applying the second initial condition  

 

Solving the two equations for two unknowns yields:

 

So the general solution for n=2 is:

 

Below is a plot showing the general solutions for n=2,4,8:

 

 

Matlab PlotsEdit

Using ode45 the following graph was generated for n=0:

 

and for n=1

 

R6.5Edit

Solved by: Gonzalo Perez

Problem StatementEdit

Part R4.2, p.7c-26Edit

For each value of n=3,5,9, re-display the expressions for the 3 functions  , and plot these 3 functions separately over the interval  .

R4.2 p.7-26: Consider the L2-ODE-CC (5) p.7b-7 with   as excitation:

  (5)p.7-7

  (1)p.9-15

and the initial conditions:

  (3b)p.3-7

Exact solution:   (2)p.9-15

Re-display the expressions for  .

Superpose each of the above plot with that of the exact solution.

SolutionEdit

The graphs are to be separately graphed as follows.

For n = 3, the code that will generate the graph looks like this:

File:N=3 code.JPG

File:N=3 picture.jpg

For n = 6:

File:N=6 code.JPG

File:N=6 picture.jpg

For n = 9:

File:N=9 code.JPG

File:N=9 picture.jpg

Part R4.3, p.7c-28Edit

Understand and run the TA's code to produce a similar plot, but over a larger interval  . Do zoom-in plots about the points   and comment on the accuracy of different approximations.

R4.3 p.7-28: Consider the L2-ODE-CC (5) p.7b-7 with   as excitation:

  (5)p.7-7

  (1)p.7-28

and the initial conditions:

  (2)p.7-28

SolutionEdit

For n=4 (from x = 0 to x = 10):

File:For n-4 (code).JPG

File:For n-4.jpg


For n=4 (zoom-in plots around x = -0.5):

File:For n-4 zoom 1 (code).JPG

File:For n-4 zoom 1.jpg


For n=4 (zoom-in plots around x = 0):

File:For n-4 zoom 2 (code).JPG

File:For n-4 zoom 2.jpg


For n=4 (zoom-in plots around x = +0.5):

File:For n-4 zoom 3 (code).JPG

File:For n-4 zoom 3.jpg


For n=7 (from x = 0 to x = 10):

File:For n-7 (code).JPG

File:For n-7.jpg


For n=11 (from x = 0 to x = 10):

File:For n-11 (code).JPG

File:For n-11.jpg



Where the black line represents n=11 and the blue line represents log(1+x).

The problem asks for n=4, 7, and 11. The TA had up to n=16, but the problem does not ask for this. Manipulating the code and graphs to only account for these n values, n=16 has been disregarded.


Combined plots for n=7 and n=11 (zoom-in plots around x = -0.5):

File:For n-7 and n-11 zoom 1 (code).JPG

File:For n-7 and n-11 zoom 1.jpg


Combined plots for n=7 and n=11 (zoom-in plots around x = 0):

File:For n-7 and n-11 zoom 2 (code).JPG

File:For n-7 and n-11 zoom 2.jpg

File:For n-7 and n-11 zoom 2 ZOOM.JPG



Combined plots for n=7 and n=11 (zoom-in plots around x = 0.5):

File:For n-7 and n-11 zoom 3 (code).JPG

File:For n-7 and n-11 zoom 3.jpg

Part R4.4, p.7c-29Edit

Understand and run the TA's code to produce a similar plot, but over a larger interval  , and for n=4,7. Do zoom-in plots about   and comments on the accuracy of the approximations.

R4.4 p.7-29: Extend the accuracy of the solution beyond  .

SolutionEdit

The general code that can be used to graph the plots of n=4,7,11 is:

File:4.4 general code.JPG

With the above code, we can add the following code to graph n=4.

For n = 4 (from x = 0.9 to x = 10):

File:N=4 code.JPG

File:N=4.jpg

For n = 7 (from x = 0.9 to x = 10):

File:N=7 code.JPG

File:N=7.jpg

For n=11 (from x = 0.9 to x = 10):

File:N=11 code.JPG

File:N=11.jpg

Repeating the same common part of the code, let's graph the other plots around x = 1, 1.5, 2, 2.5:

For n = 4, 7, 11 at x = 1:

File:N=4,7,11 at x=1 code.JPG

File:N=4,7,11 at x=1.jpg

For n = 4, 7, 11 at x = 1.5:

File:N=4,7,11 at x=1.5 code.JPG

File:This might work for x = 1.5.JPG

For n = 4, 7, 11 at x = 2:

File:N=4,7,11 at x=2 code.JPG

File:N=4,7,11 at x=2.jpg

For n = 4, 7, 11 at x = 2.5:

File:N=4,7,11 at x=2.5 code.JPG

File:X=2.5.jpg


For the last part, the first (and unchanged TA's code) is the following:

File:Might work last part code 1.JPG

The second part includes the change that had to be made in order to solve this problem:

File:Might work last part code 2.JPG

File:Might work last part.JPG


R6.6Edit

Problem StatementEdit

Given: For the following differential equation:  

With a particular solution of the form:  

Verify that this solution has a final expression of   as follows:

1) Simplify the term  
2) Simplify the 2nd term   and combine with the simplified first term
3) Finally add the 3rd term  
4) Find the final expression for  

SolutionEdit

1) To find the derivative and simplify   we will use www.wolframalpha.com[1].

The "Simplify" command will be exploited as well as the notation for taking a derivative in mathematica. The following was entered into www.wolframalpha.com:

Simplify[D[D[x*exp(-2x)*(Mcos(3x) + Nsin(3x)),x],x]]

The answer that was yielded is as follows:

 
 



2) In the same fashion,   was evaluated and simplified using www.wolframalpha.com

The following was entered into www.wolframalpha.com:

simplify[4*[D[x*exp(-2x)*(Mcos(3x) + Nsin(3x)),x]]]

The answer that was yielded is as follows:

 

This term can be added to the previous, and then simplified yet again using www.wolframalpha.com

The following was entered into www.wolframalpha.com

Simplify[D[D[x*exp(-2x)*(Mcos(3x) + Nsin(3x)),x],x]+ 4*[D[x*exp(-2x)*(Mcos(3x) + Nsin(3x)),x]]

The answer that was yielded is as follows:

 


3) The final term   is now added to this result, as follows:

 

 

 

 


4) As verified above, the final term is as follows:

 

  1. www.wolframalpha.com

--Egm4313.s12.team11.sheider (talk) 22:30, 10 April 2012 (UTC)

R6.7Edit

Solved by Solved by Daniel Suh

Problem StatementEdit

Find the separated Ordinary Differential Equations for the heat equation.

 

 

SolutionEdit

 

 

Combine   and   into  

 

  (constant)

Separate