# Heat Equation with insulated endpoints.

1. Oct 25, 2009

### Kizaru

1. The problem statement, all variables and given/known data
Assume that a bar is insulated at the endpoints. If it loses heat through its lateral surface at a rate per unit length proportional to the difference u(x,t) - T, where T is the temperature of the medium surrounding the bar, the equation of heat propagation is now

$$u_{t} = k u_{xx} - h (u-T)$$

where h > 0
2. Relevant equations
Use the function

$$v = e^{ht}(u-t)$$

to reduce this BVP to one already solved.

3. The attempt at a solution

"To one already solved" is referring to heat equation variants in which the PDE is of form

$$u_{t} = k u_{xx}$$

I can solve it from that form, I just need to convert into something of that form.

Some things I noticed, partial derivative of v with respect to t, and equated to the second partial derivative with respect to x yields u_t = u_xx - h(u-t)
This is off by the constant k which is in front of the u_xx in the original PDE. Not sure what I'm missing from here.

2. Oct 26, 2009

### tiny-tim

Hi Kizaru!

(try using the X2 and X2 tags just above the Reply box )
Yes, you have the correct basic technique, I can't see quite how you haven't got there.

If v = eht(u - t),

then vt = … and kvxx = … ?