MATH4PD — Assignment 2 due on 4 September at 12pm (a) Heat conduction in bar with two radiating ends. Investigate the behaviour of the solu- tion to the initial boundary value problem (IBVP) ut − uxx = 0, for t > 0, 0 < x < 1, ux − u = 0, for t > 0, x = 0, ux + αu = 0, for t > 0, x = 1, u = x sin pix, for t = 0, 0 < x < 1, where α ≥ 0 is a parameter. (b) Damped vibrations of a string with fixed ends. Solve the following IBVP utt + 2kut − uxx = 0, for t > 0, −1 < x < 1, u = 0, for t > 0, x = ±1, u = (x2 − 1)(x− c), for t = 0, −1 < x < 1, ut = 0, for t = 0, −1 < x < 1, where k ≥ 0 and c ∈ [0, 1] are parameters, and discuss the effect of the parameters on the solution. (c) Liquid sloshing in a tank. The periodic motion of an non-viscous fluid in a tank of width a and depth h, and with angular frequency ω, can be described by the following BVP φxx + φzz = 0, for 0 < x < a, −h < z < 0, φx = U, for x = 0, −h < z < 0, φx = 0, for x = a, −h < z < 0, φz = 0, for 0 < x < a, z = −h, φz − αφ = 0, for 0 < x < a, z = 0, where α = 0.1ω2 and U > 0 is the forcing term. (i) Find a formula for the natural frequencies, defined as the values of ω for which the homogeneous BVP (i.e. with U = 0) has non-trivial solutions, in terms of a and h. Plot the first six eigenfunctions corresponding the six lowest natural frequencies, for several combinations of a and h. 1 (ii) Fixing a and h, find and plot the general solution to the non-homogeneous BVP (taking, e.g., U = 1). Investigate the response for different values of α (in particular at the natural frequencies found in (i)). Along with your report (preferably written in LATEX), hand in all your Matlab code and video files. Also, please reference any published resource you may use to complete this assignment. 2
