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Heat equation with nonhomogeneous boundary conditions

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Feb26-11, 05:34 PM
P: 150
1. The problem statement, all variables and given/known data

[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).

2. Relevant 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]

3. 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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Feb14-12, 08:38 PM
P: 46

I also have this same question. I was also stuck at the same point. Since by assumption,
[itex]u(x,t) ≈ \sum B_n(t)sin ( \frac{n \pi x}{L}) [/itex]
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.

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