Applied Math II (spring 2000)
Applied Math II
G63.2702
Spring 2000
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Partial Differential Equations: Models & Phenomena
Many mathematical models are formulated in terms of partial
differential equations (PDEs). A complete applied mathematical study
addresses two issues: connection of the original context with the
model equations (derivation & interpretation), and investigation of
the mathematical properties of the equations themselves (analysis &
solution). Methods for the derivation, solution and computation of
PDE models are discussed within the context of simple examples taken
from the physical sciences.
The linear PDE trilogy (diffusion, Laplace & wave equations) will be
investigated through the development of various solution techniques:
eigenfunction expansions, Greens functions & integral transforms.
Advanced lectures will address special examples of nonlinear PDEs.
Note: this second-term course will be logically distinct from the
first-term. Familiarity with the CIMS computing environment is
expected.
pre-requisites:
ordinary differential equations
elementary complex variables
familiarity with MATLAB
21 April
homework #7 (postscript,
pdf)
19 April
time-harmonic scattering of a plane wave (code13a.m)
This plot shows the phase of the time-harmonic scattering
solution of a 2D plane wave by a conducting cylinder. The plane wave
is incoming from the left, and the boundary condition on the disc is
zero amplitude. The scattered wave satisfies an out-going radiation
condition and decays inversely with the square root of the radial
distance. Note the strong shadow region in the rear of the cylinder,
and the interference pattern of the plane wave with the scattered waves.
19 April
time-dependent scattering by a circular inhomogeneity (code13y.m)
time-dependent scattering by a composite inhomogeneity (code13z.m)
This plot shows the phase of the scattering solution for a
2D plane wavepacket from a circular inhomogeneity. The wavespeed in
the disc is 25% faster than in the far-field. Two times are shown,
where the wavepacket is incoming from the left. In the top figure,
the faster wavespeed in the disc is clearly evident, as is a circular
reflected wave that is spreading from the front of the disc (as a
result of the jump discontinuity in wavespeed). In the bottom figure,
the circularity of the scattered wave can be seen (with a faint
wrap-around effect of periodicity), as well as the interference with the plane
wave.
12 April
lightning rod potential plot (lect12.m)
The solid lines are contours of the solution to the
Laplace equation that is zero on the x-axis as well as on the unit
height vertical segment on the y-axis, at infinity the solution
approaches -y. The dotted & red lines are orthogonal contours
(of the harmonic conjugate). If the solid lines are interpreted as
an electric potential, the dotted & red lines are the corresponding
field lines. This plot gives a 2D answer to the question, "What does a
lightning rod do?"
06 April
homework #6 (postscript,
pdf)
05 April
de-singularized kernel (lect11.m)
The non-singular part of a boundary Greens function for
the Laplace equation on the square. The kernel function is a Fourier
series in x derived by separation of variables. The convergence is
very slow due to the singular nature of the kernel. The above plot
shows only the non-singular part of the kernel after the singular
part has been subtracted out - both the singular and non-singular parts
are solutions of the Laplace equation.
30 March
homework #5 (postscript,
pdf)
22 March
snell's law computation (code9a.m)
Numerical solution of the 2D wave equation with a Snell's
Law interface (y=0) as in homework #3. Three times are shown: initial
time (top), interface interaction (mid), and final time (bot). The
wavespeed above the interface is half of the value below. The domain
is periodic in x, and two periods are shown in each panel. The
initial wavepacket is designed to travel at 45 degrees to the
interface (this is adjustable in the matlab script). Note the
interference pattern formed by the incident and reflected waves below
the interface in the mid-panel. By the final time, the reflected
wavepacket has cleared the interface and is heading (angle of
reflection should be -45 degrees) towards the Dirichlet lower
boundary. Since the upper wavespeed is smaller, the transmitted wave
has shorter wavelength and travels at a larger angle (~69.3 degrees)
from the interface.
08 March
random walk code (walk.m)
homework #4 (postscript,
pdf)
Radial probability histogram for a two-dimensional random
walk. 200,000 walkers each take 200 unit steps in a random direction.
The blue curve is the theoretical distribution obtained using a
diffusion-limit argument to be discussed in a future lecture.
23 February
finite difference wave code (code6a.m)
interface code (code6b.m)
23 February
homework #3 (postscript,
pdf)
09 February
homework #2 (postscript,
pdf)
02 February
philosophy map (postscript)
26 January
homework #1 (postscript,
pdf)
instability code (code2a.m)
19 January
syllabus (postscript,
pdf)
download adobe (reader)
CIMS library reserve books
40 page (279K) matlab primer (pdf)
very short matlab introduction from Applied Math I (pdf)
student version (PC) matlab info (mathworks)
19 January
homework #0 (postscript,
pdf),
bring some plots to the next class
k(x) code (code1a.m)
k(x) plot (code1a.ps)
level curves code (code1b.m)
level curves plot (code1b.ps)
exact solution code (code1c.m)
exact solution plot (code1c.ps) - color
finite difference code (code1d.m)
finite difference figure (code1d.ps) - color
19 January
first lecture 7:00pm
student info form (postscript,
pdf),
please complete again & bring to next class
download adobe (reader)