# Heat equation with nonhomogeneous boundary conditions

1. ### tjackson3

150
1. The problem statement, all variables and given/known data

Consider
$$\frac{\partial u}{\partial t} = k\frac{\partial^2u}{\partial x^2}$$ subject to
$$u(0,t) = A(t),\ u(L,t) = 0,\ u(x,0) = g(x).$$ Assume that u(x,t) has a Fourier sine series. Determine a differential equation for the Fourier coefficients (assume appropriate continuity).

2. Relevant equations

Coefficients of the Fourier sine series of $f(x)$:
$$B_n = \frac{2}{L}\int_0^L f(x)\sin(n\pi x/L)\ dx$$

3. The attempt at a solution

I'm stuck just getting a start. The problem is that $u(x,t)$ will not converge to its Fourier sine series at x = 0 unless A(t) is identically zero. This also kills the ability to perform term-by-term differentiation on the FSS. It seems impossible to impose the boundary condition at x = 0. Thoughts?

Last edited by a moderator: Feb 15, 2012
2. ### Ninty64

46
BUMP.

I also have this same question. I was also stuck at the same point. Since by assumption,
$u(x,t) ≈ \sum B_n(t)sin ( \frac{n \pi x}{L})$
then it follows that you can't differentiate since it is a sine series. Sine series implies it must be continuous and u(0,t)=u(L,t)=0. However, u(0,t)=A(t).

I'm not sure how to approach this problem.