Heat Equation + 2 Robin Boundary Conditions

[PAR]

Homework Statement

Find the temperature distribution in the long thin bar −a ≤ x ≤ a with a
given initial temperature u(x,0) = f(x).
The side walls of the bar are insulated, while heat radiates from the ends into
the surrounding medium whose temperature is u = 0.
The radiation is taken to obey Newton’s Law.

Homework Equations

$$u_{t} = \alpha^{2}u_{x}_{x}$$
$$u_{x}(-a,t) = (h/k)u(-a,t)$$
$$u_{x}(a,t) = -(h/k)u(a,t)$$
$$u(x,0) = f(x)$$

h and k are constants.

The Attempt at a Solution

My main concern is: I do not know if according to the problem description, the BCs should be as written above OR
$$u_{x}(-a,t) = -(h/k)u(-a,t)$$
$$u_{x}(a,t) = (h/k)u(a,t)$$

I had to construct the BCs myself, they were not given explicitly in equation form.

notice that the only difference is that the negative sign has gone from one equation to the other. I would appreciate it if someone could tell me where the negative sign belongs, and perhaps also explain why, I did not fully grasp the explanation I was given in class. Thank You.

 

[tiny-tim]

Hi PAR!

(have an alpha: α and try using the X² and X₂ tags just above the Reply box )

Find the temperature distribution in the long thin bar −a ≤ x ≤ a with a
given initial temperature u(x,0) = f(x).
The side walls of the bar are insulated, while heat radiates from the ends into
the surrounding medium whose temperature is u = 0.
The radiation is taken to obey Newton’s Law.
…
My main concern is: I do not know if according to the problem description, the BCs should be as written above OR
$$u_{x}(-a,t) = -(h/k)u(-a,t)$$
$$u_{x}(a,t) = (h/k)u(a,t)$$

Physics tells us that if u > 0, then the bar is hottest at x = 0 …

so at the +a end, the temperature is decreasing in the + direction (and at the -a end, the temperature is increasing in the + direction).

In other words, at the +a end, u_x is negative.

Does that help?

 

[PAR]

If u$$_{x}$$ is negative in the +x direction at x=+a that means that there should a negative sign in front of the h/k, otherwise that side of the equation wouldn't be negative, right?

 

[tiny-tim]

(wrong tag! )

If u$$_{x}$$ is negative in the +x direction at x=+a that means that there should a negative sign in front of the h/k, otherwise that side of the equation wouldn't be negative, right?

s'right!