Example 8.14 Heat Conduction in a Cylinder
Equation (8.1.30) is solved in Maple below:
| > |
| > | eq:=diff(u(x,t),t)=diff(u(x,t),x$2)+1/x*diff(u(x,t),x); |
| (1) |
| > | u(x,0):=0; |
| (2) |
| > | bc1:=diff(u(x,t),x)=0; |
| (3) |
| > | bc2:=u(x,t)=1; |
| (4) |
| > | eqs:=laplace(eq,t,s): |
| > | eqs:=subs(laplace(u(x,t),t,s)=U(x),eqs); |
| (5) |
| > | bc1:=laplace(bc1,t,s): |
| > | bc1:=subs(laplace(u(x,t),t,s)=U(x),bc1); |
| (6) |
| > | bc2:=laplace(bc2,t,s): |
| > | bc2:=subs(laplace(u(x,t),t,s)=U(x),bc2); |
| (7) |
| > | dsolve(eqs,U(x)); |
| (8) |
Since BesselY becomes infinite x = 0_C2 should be zero and the solution is taken as:
| > | U(x):=c[1]*BesselJ(0,(-s)^(1/2)*x); |
| (9) |
| > | eq0:=eval(subs(x=0,bc1)); |
| (10) |
| > | eq1:=eval(subs(x=1,bc2)); |
| (11) |
| > | con:=solve({eq1},{c[1]}): |
| > | U(x):=subs(con,U(x)): |
| > | U(x):=factor(simplify(U(x))); |
| (12) |
| > | P(s):=numer(U(x)); |
| (13) |
| > | Q(s):=denom(U(x)); |
| (14) |
| > | solve(Q(s),s); |
| (15) |
| > | eig:=BesselJ(0,(-s)^(1/2)); |
| (16) |
| > | convert(eig,BesselI); |
| (17) |
| > | eiglambda:=simplify(subs(s^(1/2)=I*lambda,(-s)^(1/2)=lambda,s=-lambda^2,eig)); |
| (18) |
| > | plot(eiglambda,lambda=0..20,thickness=3,axes=boxed): |
The roots are:
| > | 0,-lambda^2; |
| (19) |
| > | N:=20; |
| (20) |
| > | l[1]:=fsolve(eiglambda,lambda=0..3); |
| (21) |
| > | for i from 2 to N do l[i]:=fsolve(eiglambda,lambda=l[i-1]..l[i-1]+4);od: |
| > | seq(l[i],i=1..N); |
| (22) |
| > | A(s):=P(s)/diff(Q(s),s): |
| > | A[n]:=simplify(subs(s=mu,A(s))); |
| (23) |
| > | A[0]:=limit(A[n],mu=0); |
| (24) |
| > | A[n]:=simplify(subs(mu^(1/2)=I*lambda,(-mu)^(1/2)=lambda,mu^(3/2)=-I*lambda^3,mu=-lambda^2,A[n])); |
| (25) |
| > | u0s:=A[0]*subs(mu=0,1/(s-mu)); |
| (26) |
| > | u0t:=invlaplace(u0s,s,t); |
| (27) |
| > | uns:=A[n]/(s-mu); |
| (28) |
| > | unt:=invlaplace(uns,s,t); |
| (29) |
| > | unt:=subs(mu=-l[n]^2,lambda=l[n],unt); |
| (30) |
The following solution and plots are obtained:
| > | U:=u0t+Sum(unt,n=1..infinity); |
| (31) |
| > | u:=piecewise(t=0,0,t>0,subs(infinity=20,U)): |
| > | u:=evalf(u): |
| > | plot3d(u,x=0..1,t=0..0.5,axes=boxed,title="Figure Exp. 8.26.",labels=[x,t,"u"],orientation=[-145,60]); |
![]() |
| > | plot([subs(t=0,u),subs(t=0.1,u),subs(t=0.2,u),subs(t=0.5,u)],x=0..1,title="Figure Exp. 8.27.",axes=boxed,thickness=5,labels=[x,"u"]); |
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| > |