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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) = tan1w) 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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 Spring '10
 Ke
 Partial Differential Equations, Elementary algebra, Boundary value problem, Partial differential equation, Green's function

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