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

 

(1.1)

 

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

 

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

 

(1.2)

 

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

 

(1.3)

 

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

 

(1.4)

 

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

 

(1.5)

 

>

h:=g(3);

 

(1.6)

 

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

 

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

 

(1.7)

 

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

 

Wave Equation via Fourier Sine Series 

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

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

 

(2.1)

 

>	up:=u(10);

 

(2.2)

 

>	plot3d(up,x=0..L,t=0..2.0,style=patchcontour,shading=zhue,axes=boxed);

 

>	animate(up,x=0..L,t=0..2*L,frames=51);

 

d'Alembert Solution 

We can construct a d'Alembert solution for the same initial condition by constructing the odd  

periodic extension of the initial condition F(x) 

>	p:=x->x-2*L*floor((x-L)/(2*L))-2*L;

 

(3.1)

 

>	plot(p(x),x=-3*L..3*L);

 

>	F1:=x->signum(x)*F(abs(x));

 

(3.2)

 

>	plot(F1(x),x=-L..L);

 

>	F2:=x->F1(p(x));

 

(3.3)

 

>	plot(F2(x),x=-3*L..3*L);

 

>	U:=(F2(x-c*t)+F2(x+c*t))/2:

 

>	plot3d(U,x=0..L,t=0..2.0,style=patchcontour,shading=zhue,axes=boxed);

 

>	animate(U,x=0..L,t=0..2*L,frames=51);

 

>	animate({U,-up},x=0..L,t=0..2*L,frames=51,thickness=3);

 

>