Least Squares Approximation of The Initial Condition 

Let's approximate the initial condition f(x) with the sum of sines that came from separation of variables. 

>	F:=piecewise(0<x and x<Pi,1, -Pi<x and x<0, -1,0);

 

>	plot(F, x=-Pi..Pi,thickness=2, discont=true, thickness=2);

 

>	f:=n-> sin (n*x);

 

>	c:=n->(1/Pi)*int(F*f(n),x=-Pi..Pi);

 

>	seq(c[n]=c(n),n=1..10);

 

>	g:= M->sum(c(m)*sin(m*x),m=1..M);

 

>

h:=g(10);

 

>	plot({F,h},x=-Pi..Pi);

 

>	s:=seq(g(2*M),M=1..21):

 

>	plot({s,F},x=-Pi..Pi);

 

>	plot({s,F},x=0..1);

 

Bessel's Inequality and Parseval's Theorem 

Let's compute the (norm)^2 of F(x) and the n-term approximation of F 

>	normF:=int(F^2,x=-Pi..Pi); evalf(%);

 

>	S:=N->(16/Pi)*sum(1/(2*j-1)^2,j=1..N);

 

>	seq(evalf(S(n)),n=1..20);

 

Gibb's Phenomena and the Sine Integral 

We can now compare the partial sums to the Sine Integral 

>	h:=x->sin(x)/x;

 

>	plot(h(x),x=0..3*Pi,thickness=2);

 

>	SI:=z->int(h(x),x=0..z);

 

>	SI(infinity);SI(Pi)/(Pi/2);evalf(%);

 

>	G:=M->(2/Pi)*SI(x*(2*M));

 

>	M:=11;G(M);

 

>	plot({g(2*M),G(M)},x=0..Pi,thickness=2);