Example 7.6 Heat Conduction in a Slab with Radiation Boundary Conditions
| > |
| > | with(plots): |
The governing equation, initial and boundary conditions are entered here:
| > | eq:=diff(u(x,t),t)=diff(u(x,t),x$2); |
| (1) |
| > | IC:=u(x,0)=1; |
| (2) |
| > | bc1:=diff(u(x,t),x)=0; |
| (3) |
| > | bc2:=diff(u(x,t),x)+u(x,t)=0; |
| (4) |
| > | Eq:=subs(u(x,t)=X(x)*T(t),eq): |
| > | Eq:=expand(Eq/X(x)/T(t)): |
| > | Eq_T:=lhs(Eq)=-lambda^2: |
| > | T(t):=rhs(dsolve({Eq_T,T(0)=T0},T(t))): |
| > | Eq_X:=rhs(Eq)=-lambda^2: |
| > | Eq_X:=expand(Eq_X*X(x)): |
| > | dsolve({Eq_X},X(x)): |
| > | X(x):=c[1]*sin(lambda*x)+c[2]*cos(lambda*x): |
| > | Bc1:=diff(X(x),x)=0: |
| > | Bc2:=diff(X(x),x)+X(x)=0: |
| > | Eq_Bc1:=eval(subs(x=0,Bc1)): |
| > | c[1]:=solve(Eq_Bc1,c[1]): |
| > | Eq_Bc2:=eval(subs(x=1,Bc2)): |
The eigenvalue equation is:
| > | Eq_Eig:=cos(lambda)-lambda*sin(lambda); |
| (5) |
| > | solve(Eq_Eig,lambda); |
| (6) |
Maple predicts negative eigenvalues:
| > | fsolve(Eq_Eig,lambda); |
| (7) |
Positive eigenvalues can be obtained by specifying a positive range in the 'fsolve' command.
| > | fsolve(Eq_Eig,lambda=0..3); |
| (8) |
The eigenvalue equation can be plotted as a function of the eigenvalue as:
| > | plot(Eq_Eig,lambda=0..20,thickness=3,title="Figure Exp. 7.14."); |
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The first 20 eigenvalues are obtained as:
| > | N:=20; |
| (9) |
| > | l[1]:=fsolve(Eq_Eig,lambda=0..3); |
| (10) |
| > | for i from 2 to N do l[i]:=fsolve(Eq_Eig,lambda=l[i-1]..l[i-1]+4);od: |
| > | seq(l[i],i=1..N); |
| (11) |
| > | U:=eval(X(x)*T(t)): |
| > | Un:=subs(c[2]=A[n]/T0,lambda=lambda[n],U): |
| > | u(x,t):=Sum(Un,n=1..infinity); |
| (12) |
| > | eq_An:=eval(subs(t=0,u(x,t)))=rhs(IC); |
| (13) |
| > | phi[n]:=cos(lambda[n]*x): |
| > | r(x):=1: |
| > | I1:=int(phi[n]^2*r(x),x=0..1): |
| > | IC; |
| (14) |
| > | I2:=int(rhs(IC)*phi[n]*r(x),x=0..1): |
| > | A[n]:=I2/I1; |
| (15) |
| > | u(x,t):=eval(u(x,t)); |
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(16) |
| > | u(x,t):=subs(infinity=N,u(x,t)): |
| > | for i to N do lambda[i]:=l[i];od: |
The transient analytical solution is:
| > | Digits:=5:u(x,t):=evalf(u(x,t)); |
| (17) |
| > | uu:=piecewise(t=0,rhs(IC),t>0,u(x,t)): |
The plots are obtained as:
| > | plot3d(uu,x=1..0,t=0.5..0,axes=boxed,title="Figure Exp. 7.15.",labels=[x,t,"u"],orientation=[45,60]); |
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| > | plot([subs(t=0,uu),subs(t=0.05,uu),subs(t=0.1,uu),subs(t=0.2,uu),subs(t=0.5,uu)],x=0..1,axes=boxed,title="Figure Exp. 7.16.",thickness=5,labels=[x,"u"],legend=["t=0","t=0.05","t=0.1","t=0.2","t=0.5"]); |
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| > |