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## Homework Statement

Consider

[tex]\frac{\partial u}{\partial t} = k\frac{\partial^2u}{\partial x^2}[/tex] subject to

[tex]u(0,t) = A(t),\ u(L,t) = 0,\ u(x,0) = g(x).[/tex] 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 [itex]f(x)[/itex]:

[tex]B_n = \frac{2}{L}\int_0^L f(x)\sin(n\pi x/L)\ dx[/tex]

## The Attempt at a Solution

I'm stuck just getting a start. The problem is that [itex]u(x,t)[/itex] 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?

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