Example 7.4 Heat Conduction in a rectangle with an Initial Profile
| > |
| > | with(plots): |
The gorging equation is entered here:
| > | eq:=diff(u(x,t),t)=diff(u(x,t),x$2); |
| (1) |
The initial condition is entered here:
| > | IC:=u(x,0)=sin(Pi*x); |
| (2) |
The boundary conditions are entered here:
| > | bc1:=u(x,t)=0; |
| (3) |
| > | bc2:=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:=X(x)=0: |
| > | Bc2:=X(x)=0: |
| > | Eq_Bc1:=eval(subs(x=0,Bc1)): |
| > | c[2]:=solve(Eq_Bc1,c[2]): |
| > | Eq_Bc2:=eval(subs(x=1,Bc2)): |
| > | Eq_Eig:=sin(lambda)=0; |
| (5) |
| > | _EnvAllSolutions := true: |
| > | solve(Eq_Eig,lambda): |
| > | U:=eval(X(x)*T(t)): |
| > | Un:=subs(c[1]=A[n]/T0,lambda=lambda[n],U): |
| > | u(x,t):=Sum(Un,n=1..infinity): |
| > | u(x,t):=subs(lambda[n]=n*Pi,u(x,t)); |
| (6) |
| > | eq_An:=eval(subs(t=0,u(x,t)))=rhs(IC); |
| (7) |
| > | phi[n]:=sin(n*Pi*x): |
| > | r(x):=1: |
| > | I1:=int(phi[n]^2*r(x),x=0..1): |
| > | IC; |
| (8) |
| > | I2:=int(rhs(IC)*phi[n]*r(x),x=0..1); |
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(9) |
| > | vars:={sin(n*Pi)=0}: |
| > | I1:=subs(vars,I1); |
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(10) |
The coefficient of An is obtained as:
| > | A[n]:=I2/I1; |
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(11) |
For n = any integer value other than 1, An is zero:
| > | eval(subs(n=2,A[n])); |
| (12) |
| > | eval(subs(n=3,A[n])); |
| (13) |
| > | eval(subs(n=100,A[n])); |
| (14) |
| > | eval(subs(n=1,A[n])); |
| Error, numeric exception: division by zero |
The limit command is applied to n = 1:
| > | A[1]:=limit(A[n],n=1); |
| (15) |
There is only one non zero term in the infinite series:
| > | u(x,t):=A[1]*sin(1*Pi*x)*exp(-Pi^2*t); |
| (16) |
| > | plot3d(u(x,t),x=1..0,t=0.3..0,axes=boxed,title="Figure Exp.7.9.",labels=[x,t,"u"],orientation=[60,60]); |
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| > | plot([subs(t=0,u(x,t)),subs(t=0.05,u(x,t)),subs(t=0.1,u(x,t)),subs(t=0.2,u(x,t)),subs(t=.3,u(x,t))],x=0..1,title="Figure Exp. 7.10.",axes=boxed,thickness=5,labels=[x,"u"],legend=["t=0","t=0.05","t=0.1","t=0.2","t=0.3"]); |
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| > |
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