1. (a) The initial temperature of the finite rod 0 ≤ x ≤ is given by the function (x) = sin(2x). The ends of the rod are kept at temperature zero for all times, and the rod is heated permanently, with the heat source taking the form f(t) = et. Find the temperature distribution in the rod for t> 0 by solving the problem ut = a²uzz + e-t, u(t,0) = u(t, π) = 0, u(0, x) = sin(2x), 0 0, 0≤x≤ π.

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