m16.mws    018  4.6.3 3)  heat4-1a.mws    Heat Equation (Dirichlet-Robin)

>

restart;

>	with(plots):

Warning, the name changecoords has been redefined

>	L:=2;H:=3;A:=4;

Characteristic equation:

>	w(x):=x*cos(x*L)+H*sin(x*L);

>	plot(w(x),x=0..10);

Eigenvalues:

>	lambda:=array(1..50);

>	n:=1: for m from 1 to 500 do z:=fsolve(w(x)=0,x=m/10..(m+1)/10): if type(z,float) then lambda[n]:=z: n:=n+1 fi od:

>	for i to 6 do lambda[i] od;

>

N:=n-1;

>	n:='n':i:='i':

Eigenfunctions:

>	X[n]:=sin(lambda[n]*x);

Squared-norm:

>	NX[n]:=int(X[n]^2,x=0..L);

GENERALIZED FOURIER SERIES

Function (initial condition):

>	u0(x):=x*(L-x)+1;

Fourier coefficients:

>	a[n]:=simplify(int(u0(x)*X[n],x=0..L)/NX[n]);

Generalized Fourier series:

>	u(x,t):=sum(a[n]*X[n]*exp(-lambda[n]^2*t/A^2),n=1..N):

>	plot3d(u(x,t),x=0..L,t=0..30,axes=boxed,projection=0.9,color=black,style=wireframe);

>	animate({u0(x),u(x,t)},x=0..L,t=0..50,frames=200,axes=boxed);

>	u(x,0):=subs(t=0,u(x,t)):

>	u(x,1):=subs(t=1,u(x,t)):

>	u(x,5):=subs(t=5,u(x,t)):

>	u(x,10):=subs(t=10,u(x,t)):

>	u(x,20):=subs(t=20,u(x,t)):

>	plot({u0(x),u(x,0),u(x,1),u(x,5),u(x,10),u(x,20)},x=0..L,axes=boxed,color=black);

>