Example 4.5 Heat Conduction with a Source Term
The technique illustrated in the previous examples can be applied for the case where there is a source term (this source term can be a function of x and t). Consider the transient heat conduction problem in a slab[5]
=α
u(x,0) =0 (4.1.5)
(0,t) = 0 and u(∞,t) is defined
Equation (4.1.5) is solved in Maple below. The programs given for the previous example can be modified to solve equation (4.1.5).
| > | restart:with(linalg):with(inttrans):with(plots): |
Only the governing equation has to be changed since:
| > | eq:=diff(u(x,t),t)=alpha*diff(u(x,t),x$2)+sin(x)*exp(-t); |
| (1) |
| > | u(x,0):=0; |
| (2) |
| > | bc1:=u(0,t)=0; |
| (3) |
| > | bc2:=u(infinity,t)=defined; |
| (4) |
The following solution and plots are obtained:
| > | eqs:=laplace(eq,t,s); |
| (5) |
| > | eqs:=subs(laplace(u(x,t),t,s)=U(x),eqs); |
| (6) |
| > | bc1:=laplace(bc1,t,s); |
| (7) |
| > | bc1:=subs(laplace(u(0,t),t,s)=U(0),bc1); |
| (8) |
| > | U(x):=rhs(dsolve({eqs,bc1},U(x))); |
| (9) |
| > | eqc:=coeff(U(x),exp(1/alpha^(1/2)*s^(1/2)*x)); |
| (10) |
| > | _C2:=solve(eqc,_C2); |
| (11) |
| > | U(x):=eval(U(x)); |
| (12) |
| > | u:=invlaplace(U(x),s,t); |
| (13) |
| > | pars:={alpha=1/5}; |
| (14) |
| > | plot3d(subs(pars,u),t=5..0,x=10..0,axes=boxed,title="Figure Exp. 4.1.11.",labels=[x,t,"u"]); |
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| > | plot([subs(pars,t=0,u),subs(pars,t=0.5,u),subs(pars,t=1,u),subs(pars,t=2,u)],x=0..10,axes=boxed,title="Figure Exp. 4.1.12.",thickness=5,labels=[x,"u"]); |
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| > | animate(subs(pars,u),x=0..10,t=0..5,thickness=5,axes=boxed,labels=[x,"u"]);
|
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| > |