(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I am solving the heat equation [tex] 1/a^2\theta_t = \theta_x

[/tex]

boundary conditions are [tex] \theta(0,t) = \theta(L,t) = 0 t > 0 [/tex]

initial conditions are [tex] \theta(x,0) = T_0sin(x\pi/L) [/tex]

now I have derived the steady solution to be 0 and I have derived that the general solution will be of the form

[tex] \theta(x,t) = \sum_{n=1}^\infty\B_n sin\frac{n\pix}{L}exp\frac{-a^2n^\pi^2}{L}t [/tex]

I am next required to determine the sequence [tex] {B_n} \\

\mbox{now usually I would proceed in the following manner, using the initial conditions}\\

\theta(x,0) = \sum_{n=1}^\infty B_n sinn\pix/L = T_0sin(\frac{\pix}{L})[/tex]

I would then multiply both sides by [tex] \int_{0}^{L}sin\frac{mx\pi}{L} [/tex]

but instead i decided first to separted term [tex] B_1 [/tex] and cancel term sin(mx\pi/L)

so I proceeded using the cancellation by the orthogonality conditions and derived the following

[tex] B_mL/2 = (T_0 -B_1)\int_{0}^{L}sin\frac{mx\pi}{L}\\

= \frac{(B_1 -T_0)}{m\pix}\left[cosmx\pi/L\right]_{0}^{L}\\

= \frac{B_1-T_0}{m\pi}\left[(-1)^m -1] [/tex]

you see the problem is I end up with [tex] B_1 [/tex] undetermined

[tex] B_1 = \frac{-4T_0}{m\pi -4} [/tex]

where did I go wrong/ what should I have done differently??

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Heat equation

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