Heat Equation with insulated endpoints.

[Kizaru]

Homework Statement

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

Homework Equations

Use the function

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

to reduce this BVP to one already solved.

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.

 

[tiny-tim]

Hi Kizaru!

(try using the X² and X₂ tags just above the Reply box )

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)

Yes, you have the correct basic technique, I can't see quite how you haven't got there.

If v = e^ht(u - t),

then v_t = … and kv_xx = … ? ​