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:=L/2-abs(x-L/2);

 

>	plot(F, x=0..L);

 

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

 

>	c:=n->(2/L)*int(F*f(n),x=0..L);

 

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

 

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

 

>

h:=g(3);

 

>	plot({F,h},x=0..L);

 

>	s:=seq(g(2*M-1),M=1..6);

 

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

 

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=0..L); evalf(%);

 

>	normg:=N-> int(g(N)^2,x=0..L);

 

>	seq(normg(n),n=1..9); evalf(%);

 

Heat Equation Revisited 

We can now reconstitute our solution to the heat equation, with the initial condition we derived. 

>	u:=M->sum(c(m)*sin(m*Pi*x/L)*exp(-(m*Pi/L)^2*t),m=1..M);

 

>

u6:=u(6);

 

>	plot3d(u6,x=0..L,t=0..0.5,style=patchcontour,shading=zhue,axes=boxed);

 

>	animate(u6,x=0..L,t=0..0.5,frames=50);

 

>