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1. For solutions to the heat equation ut = νuxx derive the following energy evolution equation between any two points x = 0 and x = L : d dt Z L 0 u 2 2 dx − νuux| L 0 = −ν Z L 0 (ux) 2 dx. This shows that the energy fiux is F = −v nuuux and that energy is always extracted from any non-constant solution via the dissipation density ν(ux) 2 . (Note that this is very different from the energy for the wave equation, which had no destruction term unless we added damping.) Show that previous equation can be used to prove uniqueness of the initial-value problem for the u = 0 and x = 0 and x = L. Hint: apply the mentioned equation to the difference w(x, t) = u1 − u2 between two assumed solutions u1 and u2 and show that w = 0 at all times. Does the same uniqueness result hold for homogeneous Neumann conditions ux = 0 at x = 0 and x = L? 2. Find the Green's function G(x, x0, t) for the heat equation in a gap x ∈ [0, L] with homogeneous Dirichlet conditions u = 0 at x = 0 and x = L by combining the general solution in terms of Fourier sine series, which is G(x, x0, t) = X∞ n=1 An exp − n 2π 2 L2 νt! sin nπx L ! , with the definition of the Green's function as a solution to the initial-value problem with initial data G(x, x0, 0) = δ(x − x0). Use the Green's function to find the solution to the initial-value problem with u(x, 0) = ( 1 L/4 ≤ x ≤ 3L/4 0 otherwise

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Applied Differential Equations
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