PDE_lab1 - Problem 1. Classify the follow differential...

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Unformatted text preview: Problem 1. Classify the follow differential equations as ODE’S or PDE’s, linear or nonlinear, and determine their order. For the linear equations, determine whether or not they are homogeneous. (a) The diffusion equation for h(:v, t): ‘ ht 2 DleC (b) The wave equation for w(x, t): wtt = C2wzzx1: (c) The thin film equation for h(a:, t): - ht = _(hhwww)w (d) The forced harmonic oscillator for y(t): ytt + wfiy = Fcos(§2t)_ (e) The Poisson Equation for the electric potential @(x, y, z): em + <I>yy + <11.” = 47rp(rc, y, 2) where p(:r:, y, z) is a known charge density. (f) Burger’s equation for h(x,t): ht + = l/hmm Problem 2. Suppose when deriving the convection equation, we as— sumed the speed of the cars was given by ,Bx for x > 0. (a) Explain why the flux function now is given by q(x,t) : flxp and the associated transport equation is given by pt + (fix/fix = 0- (b) Explain why p(0,t) = 0, p(x, 0) = we” correspond to a boundary condition of no flux of cars in from the origin and an initial condition specifying the distribution of cars at t = 0. (0) Verify that p(x, t) : me—(2fit+me“3‘) is a solution to both the transport equation given in (a) and the initial and boundary conditions given in Problem 3. Show that the helicoid Zen, :1) = tan-1w) satisfies the minimal surface equation, (1 + 232” — 229,;Zyz$y + (1+ 232% MAPLE may be helpful with the algebra. Problem 4. Show that the soliton h(x, t) = 2a2sech (04$ — 4030) - satisfies the the Korteweg—deVries equation, ht + 6hhx = hamc MAPLE may be helpful with the algebra, in particular if you don’t remember your hyperbolic trigonometric identities. ...
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This note was uploaded on 01/04/2011 for the course MATH 372 taught by Professor Ke during the Spring '10 term at SUNY Geneseo.

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PDE_lab1 - Problem 1. Classify the follow differential...

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