University of Florida/Egm4313/s12.team11.perez.gp/R5.9

R5.9 edit

Problem Statement edit

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

  (5) p.7b-7

  (1) p.7c-28

and the initial conditions

 .

Part 1 edit

Project the excitation   on the polynomial basis

  (1)

i.e., find   such that

  (2)

for x in  , and for n = 3, 6, 9.

Plot   and   to show uniform approximation and convergence.

Note that:

  (3)

Solution edit

Using Matlab, this is the code that was used to produce the results:

Part 2 edit

Find   such that:

  (1) p.7c-27

with the same initial conditions as in (2) p.7c-28.

Plot   for n = 3, 6, 9, for x in  .

In a series of separate plots, compare the results obtained with the projected excitation on polynomial basis to those with truncated Taylor series of the excitation. Plot also the numerical solution as a baseline for comparison.

Solution edit




Using integration by parts, and then with the help of of

General Binomial Theorem

 

Solution edit

For  :

 

For substitution by parts,  

 

 

 

Therefore:

                                      

Using the General Binomial Theorem:

 

Therefore:  

Which we have previously found that answer as:

                                      




For  :

 

Initially we use the following substitutions:  

 

First let us consider the first term:  

Next, we use the integration by parts:  
 

 

 

Next let us consider the second term:  

Again, we will use integration by parts:  
 

 

 

Therefore:

 

 

Re-substituting for t:

 

 

 

Therefore:

                                      



Using the General Binomial Theorem for the integral with t substitution  :

 

Therefore:  

Which we have previously found that answer as: