• Introduction
• Section 1: Forced Oscillations
  • The Fourier series of the Triangle wave
  • The particular solution
  • The general solution
• References

Section 10.6-Fourier Series & Forced ODEs

by Alain Goriely, goriely@math.arizona.edu,
( http://www.math.arizona.edu/~goriely)

Abstract: This section illustrates Section 10.6 in Kreyszig 's book (8th ed.)

Application Areas/Subjects: Engineering, Applied Mathematics

Keywords: Fourier series, Oscillations, ODEs
See Also: Other Worksheets in the same package.

Prerequisites: plots

Note: Send me an e-mail (comments-criticisms) if you use this worksheet.

> restart;assume(n,integer):with(plots):
setoptions(thickness=2): #set the tickness of the lines in the plots

Warning, the name changecoords has been redefined

Introduction

In this worksheet, I use Maple to illustrate Section 10.6 of Kreyszig 's book: Advanced Engineering Mathematics.
I show how to compute the solution of second order ODEs periodically forced by computing the Fourier series of the particular solution

We solve the equation

where r(t) is a periodic triangle wave

Section 1: Forced Oscillations

We solve the equation

where r(t) is a periodic triangle wave

The Fourier series of the Triangle wave

> f:=abs(t);g:=abs(t+2*Pi)*(Heaviside(t+3*Pi)-Heaviside(t+Pi))+abs(t)*(Heaviside(t+Pi)-Heaviside(t-Pi))+abs(t-2*Pi)*(Heaviside(t-Pi)-Heaviside(t-3*Pi)):

> plot( g,t=-3*Pi..3*Pi,title="The triangle wave");

The coefficients are all zero

The coefficients

> a[0]:=1/2/Pi*int(f,t=-Pi..Pi);
a[k]:=subs(n=k,2/Pi*int(f*cos(n*t),t=0..Pi));

Let call S(n,t), the Fourier series approximations with n coefficients:

> S:=(m,t)->a[0]+sum(a[k]*cos(k*t),k=1..m):

> plot({g,S(11,t)},t=-3*Pi..3*Pi,title="Approximation (n=11)");

The particular solution

We assume the particular solution is of the form

and we look for the coefficients

> SY:=y(t)=A[k]*cos(k*t);

To compute we substitute y(t) in the differential equation (where r(t) is expressed as a Fourier series)

> S:=simplify(subs(SY,r(t)=a[k]*cos(k*t),diff(y(t),`$`(t,2))+2*y(t) = r(t)));

> A[k]:=solve(simplify(subs(t=0,S)),A[k]);

> S2:=(m,t)->a[0]/2+sum(A[k]*cos(k*t),k=1..m):

> plot({g,S2(21,t)},t=-3*Pi..3*Pi,title="The forcing(green) and The particular solution (red)");

The general solution

The general solution is the sum of the general solution of the homogeneous system and the particular solution

> Y:=C1*cos(sqrt(2)*t)+C2*sin(sqrt(2)*t)+S2(m,t);

For instance, we can choose the intial condition so that .

> Y1:=subs(C2=0,C1=1,m=11,Y);

> plot([Y1,S2(11,t)],t=0..15*Pi,title="The general solution (red) and The particular solution (green)",color=[red,green],numpoints=200);

>

The general solution (with or different from zero) is not periodic. Can you prove it?

References

E. Kreyszig : Advanced Engineering Mathematics (8th Edition) John Wiley New York (1999)

Disclaimer: While every effort has been made to validate the solutions in this worksheet, Waterloo Maple Inc. and the contributors are not responsible for any errors contained and are not liable for any damages resulting from the use of this material.