Heat equation with nonhomogeneous boundary conditions

[tjackson3]

Homework Statement

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).

Homework 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

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?

 

[Ninty64]

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.