First of all, thank you very much!
Unfortunately, I need the exact expression which is
written above, and no other expression.
I understand that the problem may be with the root
but I really have to find another solution for this.
Thanks again!
RedFox

Hi everybody!
I solved my problem analytically by myself, and now
I want to check the solution in maple.
It will be great if someone could look at my code and
say if everything is fine, and if it suppose to work.
comment: the value Lambda[m] is the solution of the
equation tan(Lambda[m]*d) = h/(k*Lambda[m]), that I
developed during solving the heat equation (using
the separation of variables method).
T[amb] is T1.
"up" means the temp' on the surface.(x=d)
"down" means the temp' on the heater.(x=0)
MAPLE:
restart;Digits:=20:
> a:=k/(c*rho):
> lambda[m]:=piecewise(m=0, RootOf(tan(y*d)=h/(k*y), y, 0..Pi/(2*d)), m>0, RootOf(tan(y*d)=h/(k*y), y, Pi*(2*m-1)/(2*d)..Pi*(2*m+1)/(2*d)));
{ Pi
lambda[m] := { RootOf(tan(_Z d) k _Z - h, 0 .. ---) , m = 0
{ 2 d
Pi (2 m - 1) Pi (2 m + 1)
RootOf(tan(_Z d) k _Z - h, ------------ .. ------------) ,
2 d 2 d
0 < m
> beta[m]:= sqrt(h^2+k^2*(lambda[m])^2):
> A[m]:= 2*beta[m]^2/(d*beta[m]^2+h*k)*(h/(beta[m]*lambda[m])*(T[0]-T[amb])-q/(k*(lambda[m])^2));
2 2 2 / h (T[0] - T[amb]) q \
2 (h + k %2 ) |------------------- - -----|
| 2 2 2 1/2 2|
\(h + k %2 ) %2 k %2 /
A[m] := ---------------------------------------------
2 2 2
d (h + k %2 ) + h k
%1 := tan(_Z d) k _Z - h
{ Pi
{ RootOf(%1, 0 .. ---) m = 0
{ 2 d
%2 := {
{ Pi (2 m - 1) Pi (2 m + 1)
{ RootOf(%1, ------------ .. ------------) 0 < m
{ 2 d 2 d
> U:= (x,t,n) -> sum(A[m]*cos(lambda[m]*x)*exp(-a*(lambda[m])^2*t) , m=0..infinity)+T[amb]+q/h+q*(d-x)/k;
U := (x, t, n) ->
/infinity \
| ----- |
| \ 2 |
| ) A[m] cos(lambda[m] x) exp(-a lambda[m] t)|
| / |
| ----- |
\ m = 0 /
q (d - x)
+ T[amb] + q/h + ---------
k
> U(x,t,n):
> d:=0.05:
> T[0]:=6:
> T[amb]:=-100:
> k:=0.7:
> c:=2093:
> rho:=2100:
> q:=400:
> h:=5.6:
> up:=t->simplify(U(d,t,infinity)):
> up(t):
> down:=t->simplify(U(0,t,infinity)):
> down(t):
Thanks A lot !!! (:
RedFox.

this really helps, of course.
But if you can explain how to use this method
exactly in the maple - I mean, with building the
scheme and then use them to find the solution.
Thanks again!

Hi!
Well, I need now to use Crank-Nicholson scheme to solve
the equation above. [in maple!]
I hope someone will help (:
Thanks
RedFox.