Applied Differential Equations

1. Find and sketch the Fourier transforms of (a) f1(x) = cos 3x (b) f2(x) = sin 5x 􏰄1 −L<x<L forsomelengthL>0 (c) f3(x) = 0 otherwise 2. The Green's function G(x,x0,t) for the heat equation ut = kuxx on the real line is de􏰀ned for t ≥ 0 by Gt =kGxx, lim G=0, and G(x,x0,0)=δ(x−x0) |x|→∞ Find the equivalent equations for its Fourier transform Gˆ(k,x0,t) and solve them. Com- pare with the known expression for G(x, x0, t). You may use that 􏰂−x2􏰃 ˆ √ 􏰂 k2σ2􏰃 f(x)=exp 2σ2 ⇔f(k)=σ 2πexp − 2 . 3. Write down the de􏰀nition of the Green's function G for ut = uxxx, 􏰀nd its Fourier transform Gˆ, and hence show that G can be written as 1􏰅∞ exp i[k(x − x0) − k3t]dk. Do not attempt to do this integral, which cannot be computed in closed form. G(x, x0, t) = 2π 4. It is shown in 􏰁uid dynamics that ocean waves in deep water that are propagating ∞ in the positive x-direction are governed by the dispersion relation ω(k) = 􏰆gk where k > 0 is the wavenumber and g is gravity. Sketch ω(k), compute the phase velocity cp and the group velocity cg, and indicate them in the sketch. What is the wavelength of a wave with period of ten seconds and what is its group velocity? 5. More generally, water waves in a 􏰁uid of depth H propagating in the positive x- direction are governed the dispersion relation ω(k) = 􏰆gk tanh(kH) for k > 0. Sketch the last expression and show that it reduces to previously mentioned expression in the deep-water limit where kH ≫ 1. What is the limiting form of the last equation in the opposite, shallow-water limit kH ≪ 1 ? Are the waves in this limit dispersive? Consider a Tsunami that has a wavelength of 100 kilometers in an ocean of depth 4 kilometres, does it behave like a shallow-water wave or a deep-water wave? What is its group velocity? 1