Maple in Mathematics Education I:
Fourier Series & Wave Equation, Using Partial Sums

by David Canright, Math. Dept., Code MA/Ca, Naval Postgraduate School, Monterey, CA,
dcanright@nps.navy.mil, 2000 David Canright.

Note: This worksheet allows exploration & plotting of Fourier Series (using partial sums ) in an educational setting.

Introduction: Setting up the Worksheet

> restart;

> with(plots): need "plots" package for animation below

Section I: General Definitions for Fourier Series

Note: do this section only once (before L, a0, etc. are ever defined)

term: one term (#n) in series
SN: partial sum (up to N) of series

> term := an*cos(n*Pi*x/L) + bn*sin(n*Pi*x/L);

> SN := a0/2 + 'sum( an*cos(n*Pi*x/L) + bn*sin(n*Pi*x/L), n=1..N )';

Section II: Define Particular Fourier Series

come back and redo this section for each different Fourier Series

>

You have two choices :

If you know the function f , use the first subsection

If you know the coefficients , use the second subsection

If you know f and the coeffiecients, use either subsection

but do not do both !

Section IIa: Either Calculate Coefficients by Integrals

If you know the function f and choose L,

you can calculate the coefficients:

using the integral formulas

define particular function f
(default example is for f(x)= {0 for -1<x<0 and x for 0<x<1})
modify this definition for other series

> f := piecewise(x>0,x);

choose L

> L := 2;

Now evaluate the integrals to find the coefficients

Note: these steps must be done in order

> Int( f, x=-L..L ) / L ;

> value(%);

> a0 := %;

> Int( f*cos(n*Pi*x/L), x=-L..L ) / L ;

> value(%);

> simplify(%,trig,assume=integer);

> an := %;

> Int( f*sin(n*Pi*x/L), x=-L..L ) / L ;

> value(%);

> simplify(%,trig,assume=integer);

> bn := %;

>

At this point, have defined the series, so skip down to the plots

Section 2b: or Specify Coefficients: & L

come back and redo this section for each different Fourier Series

You must first know the formula for each coefficient

define particular series coefficients and L
(default example is from text pg. 24: )
modify these definitions to plot other series

> L := Pi;

> a0 := 0;

> an := 1/n^2;

> bn := 0;

define particular function f (sum of series)
(default example: we don't know f, so set to 0 )
modify this definition for other series

Note: if do not know f, set f := 0;

> f := 0;

>

At this point, have defined the series, so go on to the plots

Section III: Plot Partial Sum and/or Terms of Series

try changing N on any or all of these

choose N to plot the partial sum

> N := 3;
plot( SN, x=-L .. L);

choose N to plot term #N

> N := 3;
plot( subs(n=N,term), x=-L .. L);

choose N to plot a partial sum and all the terms up to N

> N := 3;
plot( { a0/2, seq(subs(n=i,term),i=1..N), SN }, x=-L .. L);

Section IV: Examine Convergence of Fourier Series

> plot( f, x=-L..L ); look at function over one period

choose N to compare the function f to the partial sum

> N := 5;
plot( [SN,f], x=-L .. L);

choose max N to animate the convergence of the partial sum to f

> Nmax := 5;
display([seq(plot( [SN,f], x=-L .. L),N=0..Nmax)], insequence=true);

Conclusion: This worksheet clearly demonstrates Maple's ability to be used as an Educational tool for advanced mathematical concepts.

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.