Example 7.9
| > |
| > | with(plots): |
| > | eq:=diff(u(x,t),t)=diff(u(x,t),x$2); |
| (1) |
| > | IC:=u(x,0)=0; |
| (2) |
| > | bc1:=diff(u(x,t),x)=0; |
| (3) |
| > | bc2:=diff(u(x,t),x)=delta; |
| (4) |
| > | eq1:=eval(subs(u(x,t)=g(x,t)+w(x)+v(t),eq)); |
| (5) |
The governing equations for v(t), w(x), and g(x, t) are:
| > | eqv:=diff(v(t),t)=k; |
| (6) |
| > | eqw:=diff(w(x),x$2)=k; |
| (7) |
| > | eqg:=diff(g(x,t),t)=diff(g(x,t),x$2); |
| (8) |
The boundary conditions for w(x) are:
| > | bc1w:=diff(w(x),x)=0; |
| (9) |
| > | bc2w:=diff(w(x),x)=delta; |
| (10) |
w(x) can be solved with the boundary condition at x = 0 as:
| > | w(x):=rhs(dsolve({eqw,D(w)(0)=0},w(x))); |
| (11) |
The value of constant k is found using the boundary condition at x = 1:
| > | bc2w:=subs(x=1,diff(w(x),x))=delta; |
| (12) |
| > | k:=solve(bc2w,k); |
| (13) |
| > | w(x):=1/2*delta*x^2+B; |
| (14) |
v(t) is solved as:
| > | v(t):=rhs(dsolve({eqv,v(0)=0},v(t))); |
| (15) |
| > | Eq:=subs(g(x,t)=X(x)*T(t),eqg): |
g(x, t) is obtained as:
| > | 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)=0: |
| > | Eq_Bc1:=eval(subs(x=0,Bc1)): |
| > | c[1]:=solve(Eq_Bc1,c[1]): |
| > | Eq_Bc2:=eval(subs(x=1,Bc2)): |
| > | Eq_Eig:=sin(lambda)=0: |
| > | solve(Eq_Eig,lambda): |
| > | _EnvAllSolutions := true: |
| > | solve(Eq_Eig,lambda): |
| > | G1:=eval(X(x)*T(t)): |
| > | Gn:=subs(c[2]=A[n]/T0,lambda=lambda[n],G1): |
| > | g(x,t):=Sum(Gn,n=1..infinity): |
| > | g(x,t):=subs(lambda[n]=n*Pi,g(x,t)); |
| (16) |
| > | u(x,t):=g(x,t)+w(x)+v(t); |
| (17) |
The constant An is found as:
| > | eq_An:=eval(subs(t=0,u(x,t)))=rhs(IC); |
| (18) |
| > | r(x):=1: |
Next, constant B is found as:
| > | eqB:=int(lhs(eq_An*r(x)),x=0..1)=int(rhs(eq_An*r(x)),x=0..1); |
| (19) |
| > | B:=-delta/6; |
| (20) |
| > | phi[n]:=cos(n*Pi*x): |
| > | I1:=int(phi[n]^2*r(x),x=0..1): |
| > | I2:=int((rhs(IC)-w(x))*phi[n]*r(x),x=0..1): |
| > | vars:={sin(n*Pi)=0}: |
| > | I1:=subs(vars,I1): |
| > | I2:=subs(vars,I2): |
| > | A[n]:=I2/I1; |
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(21) |
| > | u(x,t):=eval(u(x,t)); |
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(22) |
| > | u(x,t):=subs(infinity=N,u(x,t)): |
| > | ua:=subs(N=20,u(x,t)): |
| > | uu:=piecewise(t=0,rhs(IC),t>0,ua); |
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(23) |
| > | plot3d(subs(delta=1,uu),x=1..0,t=0.5..0,axes=boxed,title="Figure Exp. 7.21.",labels=[x,t,"u"],orientation=[-135,60]); |
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| > | plot([subs(t=0,delta=1,uu),subs(t=0.2,delta=1,uu),subs(t=0.5,delta=1,uu),subs(t=0.7,delta=1,uu),subs(t=1,delta=1,uu)],x=0..1,title="Figure Exp. 7.22.",axes=boxed,thickness=5,labels=[x,"u"],legend=["t=0","t=0.2","t=0.5","t=0.7","t=1.0"]); |
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| > |