1 dimensional heat flow boundary conditionsby hfitzgerald Tags: boundary conditions, heat flow equation, laplace 

#1
May2311, 05:50 AM

P: 2

1. The problem statement, all variables and given/known data
n is given by: ∂^{2}Θ/∂x^{2}=1/α^{2} ∂Θ/∂t , where Θ(x, t) is the temperature as a function of time and position, and α^{2} is a constant characteristic for the material through which the heat is ﬂowing. We have a plate of inﬁnite area and thickness d that has a uniform temperature of 100◦C. Suddenly from t = 0 onwards we put both sides at 0◦C (perhaps by putting the plate between two slabs of ice). Write down the four boundary conditions for this plate. 2. Relevant equations I can't think of any relevant equations to this 3. The attempt at a solution so far I have got Θ(0, t)=0 Θ(d, t)=0 where d is the thickness of the bar. 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution 



#2
May2311, 06:10 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,881

Well, so far, all you have done is write down the problem!
theta_{xx}= (1/a^2)theta_{t} theta(0, t)= theta(d, t)= 0, theta(x, 0)= 100. Now, attempt a solution. What methods have you learned for solving such problems? Most common are "separation of variables" and "Fourier series", both of which will work here, but no one can make any suggestions until we know which methods you know and where you are stuck with this problem. 



#3
May2311, 07:18 AM

HW Helper
P: 1,584

I would try as separation of variables method, so write:
[tex] \theta (t,x)=T(t)X(x) [/tex] 


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