Finding u(x) for the 1-D Heat Equation

[clope023]

Homework Statement

If there is heat radiation in a rod of length L, then the 1-D heat equation might take the form:

u_t = ku_xx + F(x,t)

exercise deals with the steady state condition => temperature u and F are independent of time t and that u_t = 0.

u_t = partial derivative with respect to t
u_xx = 2nd partial derivative with respect to x

Problem: find u(x) if F(x) = sinx, k = 2, u(0) = u'(0), u(L) = 1

Homework Equations

e^(itheta) = cos(theta) + isin(theta)
e^(-itheta) = cos(theta) - isin(theta)
e^(itheta) + e^(-itheta) = 2cos(theta)
e^(itheta) - e^(-itheta) = 2isin(theta)

The Attempt at a Solution

u_t = ku_xx + F(x,t)
u_t = ku_xx + F(x)G(t)
2u_xx + sin(x)G(t) = 0
characteristic equation: m^2+1=0, m^2 = -1, m = +-i
general soltn: u(x) = c1cos(x) + c2sin(x)
u(0) = c1cos(0) + c2sin(0)
u'(x) = -c1sin(x) + c2cos(x)
u'(0) = -c1sin(0) + c2cos(0)
=> c2 = c1
u(x) = c1cos(x) + c1sin(x)
u(L) = c1cos(L) + c1sin(L) = 1

from here I'm actually a little lost, I'm not sure what to do with u(L) = 1 portion; do I perhaps need to find a particular solution? any help would be appreciated.

 

[Dick]

If everything is time independent then it's really just an ODE with boundary conditions, right? Yes, find a particular solution. Then find a homogeneous solution and adjust the constants to fit the boundary conditions.