m14.mws

4.6.3 1)  heat2-2.mws    1-D Heat Equation      plane steel wall - both sides are insulated (Neumann BC)   L=0.1 m

>    restart;

>    u[0](x):=10000*(x-L/3)^2+100;

u[0](x) := 10000*(x-1/3*L)^2+100

Eigenfuctions:

>    X[n]:=cos(n*Pi*x/L);

X[n] := cos(n*Pi*x/L)

>    T[n]:=exp(-n^2*Pi^2*t/L^2/a^2);

T[n] := exp(-n^2*Pi^2*t/L^2/a^2)

Fourier coefficients:

>    b[0]:=simplify(factor(int(u[0](x),x=0..L)/L));

b[0] := 10000/9*L^2+100

>    b[n]:=2*int(u[0](x)*X[n],x=0..L)/L;

b[n] := 200/9*(-1800*L^2*sin(n*Pi)+9*n^2*Pi^2*sin(n*Pi)+400*n^2*Pi^2*sin(n*Pi)*L^2+1200*n*Pi*cos(n*Pi)*L^2+600*L^2*n*Pi)/n^3/Pi^3

>    b[n]:=simplify(factor(subs({sin(n*Pi)=0,cos(n*Pi)=(-1)^n},b[n])));

b[n] := 40000/3*L^2/n^2/Pi^2*(2*(-1)^n+1)

>    L:=0.1;a:=500;

L := .1

a := 500

>    u(x,t):=b[0]+sum(b[n]*X[n]*T[n],n=1..10):

>    plot3d(u(x,t),x=0..L,t=0..600,axes=boxed,projection=0.85);

[Maple Plot]

>    with(plots): 

Warning, the name changecoords has been redefined

ua = veraged temperature of the wall:

>    ua:=value(int(u[0](x),x=0..L)/L);

ua := 133.3333333-333.3333334*L+1111.111111*L^2

>    animate({ua,u(x,t),u[0](x)},x=0..L,t=0..1200,frames=500,axes=boxed,color=black);

[Maple Plot]

>   

>    u0:=subs(t=0,u(x,t)):

>    u1:=subs(t=60,u(x,t)):

>    u2:=subs(t=300,u(x,t)):

>    u3:=subs(t=600,u(x,t)):

>    plot({ua,u[0](x),u0,u1,u2,u3},x=0..L,axes=boxed,color=black);

[Maple Plot]

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