Applied Differential Equations

Question

1. Use the method of images to find the solution u(x,t) to the initial-value problem for the wave equation Uₜₜ = Uₓₓ on the half-line x 0 with Dirichlet boundary condition u(0,t) = 0 and initial data uₜ(x, 0) = 0 and u(x,0) = 1 if 4 < x < 6, u(x,0) = 0 otherwise.
Sketch the solution at the times t ∈ (0,2,5,7}.

2. Recall that the general normal-mode solution for the wave equation uₜₜ = c²uₓₓ on a finite domain x ∈ [0, L] with Dirichlet boundary conditions u = 0 at both ends can be written as
u(x,t) = Σₙ ₌ ₁ to infinity [Aₙ cos ωₙ t + Bₙ sin ωₙ t] sin kₙ x
where ωₙ = ckₙ, kₙ = nπ/L, and the coefficients (Aₙ, Bₙ) are arbitrary.
(a) Compute the total energy
ε = ∫₀ᴸ 1/2 (u²ₜ + c² u²ₓ) dx
in terms of the normal modes and show that it is conserved in time (you may assume that summation and integration commute).
(b) How are the normal modes in (2) changed if the boundary condition at x = L is changed to the Neumann form uₓ = 0, which corresponds to a freely sliding string? Is the lowest possible frequency in this case higher or lower than before?
Without repeating the calculation, is ε still conserved?
(c) Revert to Dirichlet boundary conditions, set c = 1 and L = 2, and compute the solution for the "plucked string" initial conditions uₜ(x, 0) = 0 and u(x,0) = x if x ≤ 1 and u(x,0) = 2 - x if x ≥ 1.

3. Use characteristics to find the strong solution u(x,t) to the inviscid Burgers equation uₜ + uuₓ = 0 on the half-line x ≥ 0 with initial data u(x,0) = x². (This involves solving a quadratic equation, you need to pick the root that is consistent with the initial conditions.) Make a sketch that illustrates some of the characteristics in the xt-plane.

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