Compute the steady state temperature ur and then compute

Question

Q1. Find the deflection u(x, y, t) satisfying the wave equation u_tt = 4(u_xx + u_yy) for a rectangular plate with fixed ends and dimensions: horizontal a = 2π and vertical b = π. The initial velocity is g(x, y) = 0. The initial displacement is f(x, y)  = -3sin(5x)sin(6y) + 11sin(6x)sin(9y).

Q2. Find the deflection u(x, y, t) satisfying the wave equation u_tt = 9(u_xx + u_yy) for a rectangular plate with fixed ends and dimensions: horizontal a = 3 and vertical b = 2. The initial velocity is f(x, y) = 0. The initial velocity is g(x, y) = 2.

Q3. Compute the steady state temperature u(r, θ) and then compute u(2, π/4) of a circular plate of radius R = 4, if the temprature on the boundary is given by f(θ) = 12sin³θ. (Hint: rewrite sin³θ by using a trig identity where the trig functions are in the first power.)

Attachment:- Assignment Notes.rar