7. Solve the heat equation \( k \frac{\partial^2 u(x,t)}{\partial x^2} = \frac{\partial u(x,t)}{\partial t} \), \( 0 < x < L \), \( t > 0 \) subject to the given conditions \( u(0,t) = 0 \), \( u(L,t) = 0 \), and \( u(x,0) = \begin{cases} 1, & 0 < x < L/2 \\ 0, & L/2 < x < L \end{cases} \). (12%)