- #1
Kizaru
- 45
- 0
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
[tex]u_{t} = k u_{xx} - h (u-T)[/tex]
where h > 0
Homework Equations
Use the function
[tex] v = e^{ht}(u-t) [/tex]
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
[tex]u_{t} = k u_{xx} [/tex]
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.