Problem 4 (damping, Haberman 4.4.3-5) Analyze the initial/boundary value prob- lem (x. t) E (0, L) x (0,00) u(z, 0) = uo(z), FE (0, L) u, (x. 0) = to(I). IE (0, L) u(0, t) = 0 = u(L, t), 130 where p, c, and B are positive constants, and up and to are given functions. Here are some suggestions for your analysis: (a) Solve the problem in general using separation of variables and superposition. (b) Solve the problem in general using eigenfunction expansion. Note: In parts (a) and (b) there should be multiple qualitative cases (under- damped, critically damped, and overdamped) depending on the magnitude of the damping coefficient B. (c) Choose some specific values of the constants (including L) and initial position and velocity to see some simple separated variable solutions illustrating each qualitative case. Animations of the standard (Haberman) "string" model could be good. (d) For at least one choice of "more interesting" initial conditions that require a superposition write down and illustrate the solution