## 16 Reputation

16 years, 231 days

## Well......

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

## Analytical Solution Using Maple...

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.

## well......

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!

## Using Crank-Nicholson Method...

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

## Thank you very much!...

(:
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