Tutorial 10 2021 PDF

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Tutorial 10 2021...

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Lecturer: Johanna Knapp School of Mathematics and Statistics

Second Semester, 2021 University of Melbourne

MAST20030: Differential Equations Tutorial 10

Question 1. In this question, we’re going to solve the 1D wave equation with an an additional damping term: ∂u ∂ 2u ∂ 2u +µ = c2 2 . 2 ∂t ∂t ∂x Here, µ > 0 is the coefficient of friction. (a) Show that we can use separation of variables to reduce this PDE to two ODEs. (b) Assuming homogeneous Neumann boundary conditions at x = 0 and x = 1, solve the ODE in x . (c) Assuming that 0 < µ < 2πc, carefully solve the ODE in t. You should get two linearly independent real solutions for each eigenvalue (and be careful to consider all eigenvalues). (d) Now, write down a series solution that formally satisfies the PDE and the boundary conditions. What happens to this solution (formally) as t → ∞? (e) If the initial conditions are u(x, 0) = cos(2πx) + 1 and ∂t u(x, 0) = 2 cos(πx) − 1, find the solution to the IBVP. You should see a “resonance” (pole) in the limit as µ → 2πc from below....