University of Florida/Egm4313/s12.team11.imponenti/R6.4

R6.4

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solved by Luca Imponenti

Problem Statement

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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 Series

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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 Solution

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Considering the homogeneous case of our ODE:

 

The characteristic equation is

 

 

 

Therefore our homogeneous solution is of the form

 

Particular Solution

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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 Solution

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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 Plots

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

 

and for n=1