Example 7.7.mw

Example 7.7 Heat Conduction in a slab with Nonhomogeneous Boundary Conditions 

> restart:
 

> with(plots):
 

> eq:=diff(u(x,t),t)=diff(u(x,t),x$2);
 

diff(u(x, t), t) = diff(diff(u(x, t), x), x) (1)
 

> IC:=u(x,0)=0;
 

u(x, 0) = 0 (2)
 

> bc1:=u(x,t)=0;
 

u(x, t) = 0 (3)
 

> bc2:=u(x,t)=1;
 

u(x, t) = 1 (4)
 

> eq1:=eval(subs(u(x,t)=g(x,t)+w(x),eq));
 

diff(g(x, t), t) = `+`(diff(diff(g(x, t), x), x), diff(diff(w(x), x), x)) (5)
 

> eqw:=diff(w(x),x$2);
 

diff(diff(w(x), x), x) (6)
 

> eqg:=diff(g(x,t),t)=diff(g(x,t),x$2);
 

diff(g(x, t), t) = diff(diff(g(x, t), x), x) (7)
 

> bc1w:=w(x)=0;
 

w(x) = 0 (8)
 

> bc2w:=w(x)=1;
 

w(x) = 1 (9)
 

> w(x):=rhs(dsolve({eqw,w(0)=0,w(1)=1},w(x)));
 

x (10)
 

> Eq:=subs(g(x,t)=X(x)*T(t),eqg):
 

> 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);
 

`+`(`*`(c[1], `*`(sin(`*`(lambda, `*`(x))))), `*`(c[2], `*`(cos(`*`(lambda, `*`(x)))))) (11)
 

> Bc1:=X(x)=0;
 

`+`(`*`(c[1], `*`(sin(`*`(lambda, `*`(x))))), `*`(c[2], `*`(cos(`*`(lambda, `*`(x)))))) = 0 (12)
 

> Bc2:=X(x)=0;
 

`+`(`*`(c[1], `*`(sin(`*`(lambda, `*`(x))))), `*`(c[2], `*`(cos(`*`(lambda, `*`(x)))))) = 0 (13)
 

> 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:
 

> solve(Eq_Eig,lambda):
 

> _EnvAllSolutions := true:
 

> solve(Eq_Eig,lambda):
 

> G1:=eval(X(x)*T(t)):
 

> Gn:=subs(c[1]=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));
 

Sum(`*`(A[n], `*`(sin(`*`(n, `*`(Pi, `*`(x)))), `*`(exp(`+`(`-`(`*`(`^`(n, 2), `*`(`^`(Pi, 2), `*`(t))))))))), n = 1 .. infinity) (14)
 

> u(x,t):=g(x,t)+w(x);
 

`+`(Sum(`*`(A[n], `*`(sin(`*`(n, `*`(Pi, `*`(x)))), `*`(exp(`+`(`-`(`*`(`^`(n, 2), `*`(`^`(Pi, 2), `*`(t))))))))), n = 1 .. infinity), x) (15)
 

> eq_An:=eval(subs(t=0,u(x,t)))=rhs(IC);
 

`+`(Sum(`*`(A[n], `*`(sin(`*`(n, `*`(Pi, `*`(x)))))), n = 1 .. infinity), x) = 0 (16)
 

> phi[n]:=sin(n*Pi*x):
 

> r(x):=1:
 

> 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:
 

> A[n]:=simplify(A[n]):
 

> u(x,t):=eval(u(x,t)):
 

> 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);
 

piecewise(t = 0, 0, `<`(0, t), `+`(Sum(`*`(piecewise(n = 0, undefined, `+`(`/`(`*`(2, `*`(cos(`*`(n, `*`(Pi))))), `*`(n, `*`(Pi))))), `*`(sin(`*`(n, `*`(Pi, `*`(x)))), `*`(exp(`+`(`-`(`*`(`^`(n, 2), `... (17)
 

> plot3d(uu,x=1..0,t=0.5..0,axes=boxed,title="Figure Exp. 7.17.",labels=[x,t,"u"],orientation=[-135,60]);
 

Plot
 

> plot([subs(t=0,uu),subs(t=0.05,uu),subs(t=0.1,uu),subs(t=0.2,uu),subs(t=0.5,uu)],x=0..1,axes=boxed,title="Figure Exp. 7.18.",thickness=5,labels=[x,"u"],legend=["t=0","t=0.05","t=0.1","t=0.2","t=0.5"]);
 

Plot_2d
 

>