Hey everyone, I need a little help with this question.

Relevant equations:

$\displaystyle T(x,t)= \sum\limits_{i=1}^{\infty} B_i e^{-(i^2 \pi^2kt)} sin(i \pi x)$

$\displaystyle T(0,t) = T(1,t) = 0$

$\displaystyle T(x,0) = sin(\pi x)$

$\displaystyle \alpha = \frac{k \Delta t}{(\Delta x)^2}$

The question:

Let $\displaystyle N=20$ and $\displaystyle \alpha = \frac{1}{4}$, given the initial condition $\displaystyle T(x,0) = sin(\pi x)$, what is the exact solution at points on the grid?

So I have worked out $\displaystyle \Delta x = \frac{1}{N} = \frac{1}{20} $ and $\displaystyle k \Delta t = \frac{(\Delta x)^2}{4}=0.000625$, and then worked out $\displaystyle B_i = 1$.

I am just unsure what the exact solution is, since I don't know k.

Can someone help me with the exact solution.

Thanks