Example 7.12 Diffusion in a Cylinder
| > | restart: |
| > | with(plots): |
| > | eq:=diff(u(x,y),y$2)=-diff(u(x,y),x$2)-1/x*diff(u(x,y),x); |
| (1) |
| > | bc1:=diff(u(x,y),x)=0; |
| (2) |
| > | bc2:=u(x,y)=0; |
| (3) |
| > | bc3:=u(x,y)=1; |
| (4) |
| > | bc4:=diff(u(x,y),x)=0; |
| (5) |
| > | Eq:=subs(u(x,y)=X(x)*Y(y),eq); |
| (6) |
| > | Eq:=expand(Eq/X(x)/Y(y)); |
| (7) |
| > | Eq_Y:=lhs(Eq)=lambda^2: |
| > | Eq_Y:=eval(Eq_Y*Y(y)); |
| (8) |
| > | dsolve(Eq_Y,Y(y)); |
| (9) |
| > | Y(y):=C[1]*sinh(lambda*y)+C[2]*cosh(lambda*y); |
| (10) |
| > | Bc4:=diff(Y(y),y)=0; |
| (11) |
| > | Eq_Bc4:=eval(subs(y=1,Bc4)); |
| (12) |
| > | C[2]:=solve(Eq_Bc4,C[2]); |
| (13) |
| > | Y(y):=eval(Y(y)); |
| (14) |
| > | combine(Y(y)); |
| (15) |
| > | Y(y):=C[1]*cosh(lambda*(1-y))/sinh(lambda); |
| (16) |
| > | Eq_X:=rhs(Eq)=lambda^2: |
| > | Eq_X:=expand(Eq_X*X(x)); |
| (17) |
| > | dsolve({Eq_X},X(x)); |
| (18) |
| > | X(x):=c[1]*BesselJ(0,lambda*x)+c[2]*BesselY(0,lambda*x); |
| (19) |
| > | Bc1:=diff(X(x),x)=0; |
| (20) |
| > | Bc2:=X(x)=0; |
| (21) |
| > | BesselJ(0,0); |
| (22) |
| > | limit(BesselY(0,lambda*x),x=0); |
| (23) |
| > | c[2]:=0; |
| (24) |
| > | Eq_Bc2:=eval(subs(x=1,Bc2)); |
| (25) |
| > | Eq_Eig:=BesselJ(0,lambda)=0; |
| (26) |
| > | l[1]:=fsolve(Eq_Eig,lambda=0..3); |
| (27) |
| > | N:=20; |
| (28) |
| > | 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); |
| (29) |
| > | U:=eval(X(x)*Y(y)): |
| > | Un:=subs(c[1]=A[n]/C[1],lambda=lambda[n],U); |
| (30) |
| > | u(x,y):=Sum(Un,n=1..N): |
| > | eq_An:=eval(subs(y=0,u(x,y)))=rhs(bc3); |
| (31) |
| > | phi[n]:=BesselJ(0,lambda[n]*x); |
| (32) |
| > | r(x):=x; |
| (33) |
| > | I1:=int(phi[n]^2*r(x),x=0..1): |
| > | I2:=int(1*phi[n]*r(x),x=0..1): |
| > | A[n]:=I2/I1/cosh(lambda[n])*sinh(lambda[n]); |
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(34) |
| > | u(x,y):=eval(u(x,y)); |
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(35) |
| > | for i to N do lambda[i]:=l[i];od: |
| > | ua:=evalf(u(x,y)): |
| > | uu:=piecewise(y=0,1,y>0,ua): |
| > | plot3d(evalf(uu),x=0..1,y=1..0,axes=boxed,title="Figure Exp. 7.28.",labels=[x,y,"u"],orientation=[30,60]); |
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| > | plot([subs(y=0,uu),subs(y=0.1,uu),subs(y=0.2,uu),subs(y=0.3,uu),subs(y=0.4,uu),subs(y=1,uu)],x=0..1,axes=boxed,title="Figure Exp. 7.29.",thickness=5,labels=[x,"u"],legend=["y=0","y=0.1","y=0.2","y=0.3","y=0.4","y=1.0"]); |
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| > |