- #1

clope023

- 992

- 132

## 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.