4. Consider heat conduction in a bar (length of \( L \)) with constant temperature ends. The boundary value problem modeling the temperature distribution is \[ \frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}, \quad \text{for} \quad 0 < x < L, \quad t > 0, \] \[ u(0, t) = u(L, t) = 0 \quad \text{for} \quad t \geq 0, \quad \text{and} \]\[ u(x, 0) = f(x) \quad \text{for} \quad 0 \leq x \leq L. \] The partial differential equation can be solved using separation of variables (Fourier method) consists of attempting a solution of the from \( u(x, t) = X(x)T(t) \).