Word Count : 221

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We are given the heat equation: \[ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} \] with the initial and boundary conditions: \[ u(x,0) = \begin{cases} 2x, & 0 \leq x \leq \frac{1}{2} \\ 2(1-x), & \frac{1}{2} \leq x \leq 1 \end{cases} \] \[ u(0,t) = 0, \quad u(1,t) = 0 \] We need to solve ___ _______ ____ _____ ________ ________ _____ ______.
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