Example 8.11 Diffusion in a Slab with Nonhomogeneous Flux Boundary Condtions during the Charging of a Battery
Equation (8.1.27) is solved in Maple and the results are given below:
| > | restart:with(inttrans):with(plots): |
| > | eq:=diff(u(x,t),t)=diff(u(x,t),x$2); |
| (1) |
| > | u(x,0):=0; |
| (2) |
| > | bc1:=diff(u(x,t),x)=0; |
| (3) |
| > | bc2:=diff(u(x,t),x)=delta; |
| (4) |
| > | eqs:=laplace(eq,t,s): |
| > | eqs:=subs(laplace(u(x,t),t,s)=U(x),eqs); |
| (5) |
| > | bc1:=laplace(bc1,t,s): |
| > | bc1:=subs(laplace(u(x,t),t,s)=U(x),bc1); |
| (6) |
| > | bc2:=laplace(bc2,t,s): |
| > | bc2:=subs(laplace(u(x,t),t,s)=U(x),bc2); |
| (7) |
| > | dsolve(eqs,U(x)); |
| (8) |
| > | U(x):=c[1]*cosh(s^(1/2)*x)+c[2]*sinh(s^(1/2)*x); |
| (9) |
| > | eq0:=eval(subs(x=0,bc1)): |
| > | eq1:=eval(subs(x=1,bc2)): |
| > | con:=solve({eq0,eq1},{c[1],c[2]}): |
| > | U(x):=subs(con,U(x)); |
| (10) |
| > | U(x):=factor(combine(simplify(U(x)))); |
| (11) |
| > | P(s):=numer(U(x)); |
| (12) |
| > | Q(s):=denom(U(x)); |
| (13) |
| > | solve(Q(s),s); |
| (14) |
| > | _EnvAllSolutions := true; |
| (15) |
| > | solve(Q(s),s); |
| (16) |
| > | 0,0,-n^2*Pi^2; |
| (17) |
| > | mu0:=0; |
| (18) |
| > | b[2]:=(s-mu0)^2*P(s)/Q(s); |
| (19) |
| > | B[2]:=limit(b[2],s=0); |
| (20) |
| > | b[1]:=diff(b[2],s): |
| > | B[1]:=limit(b[1],s=0); |
| (21) |
| > | A(s):=P(s)/diff(Q(s),s): |
| > | A[n]:=simplify(subs(s=mu,A(s))); |
| (22) |
| > | A[n]:=simplify(subs(mu^(1/2)=I*n*Pi,mu=-n^2*Pi^2,A[n])): |
| > | vars:={cos(n*Pi)=(-1)^n,sin(n*Pi)=0}; |
| (23) |
| > | A[n]:=simplify(subs(vars,A[n])): |
| > | A[n]:=simplify(subs(vars,expand(A[n]))); |
| (24) |
| > | b1s:=B[1]*subs(mu0=0,1/(s-mu0)); |
| (25) |
| > | b1t:=invlaplace(b1s,s,t); |
| (26) |
| > | b2s:=B[2]*subs(mu0=0,1/(s-mu0)^2); |
| (27) |
| > | b2t:=invlaplace(b2s,s,t); |
| (28) |
| > | uns:=A[n]/(s-mu); |
| (29) |
| > | unt:=invlaplace(uns,s,t); |
| (30) |
| > | unt:=subs(mu=-n^2*Pi^2,unt); |
| (31) |
| > | U:=b1t+b2t+Sum(unt,n=1..infinity); |
| (32) |
| > | u:=piecewise(t=0,0,t>0,subs(infinity=20,U)): |
| > | plot3d(subs(delta=1,u),x=0..1,t=0..0.5,axes=boxed,title="Figure Exp. 8.20.",labels=[x,t,"u"],orientation=[-135,60]); |
![]() |
| > |