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The questions on the test are very similar to those in previous years.
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Basically you will be given one PDE w/ BC's and asked to solve it using
separation of variables, explaing all major steps.
Additional questions along the way ask you about different features
of the PDE, the method, and the solution.
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The part of expanding a BC into a Fourier series is formulated as a
separate large part of the problem.
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More detailed items re Fourier analysis
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When can a function be expanded into a Fourier series, sine/cosine series, or be
written as a Fourier integral?
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Construct various odd, even, periodic extensions etc.
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Calculate the Fourier coefficients: General formulas and decimal approximations.
Recall 17.3/4g -- sometimes need to consider some coefficents separately.
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Write out the Fourier expansion as an infinite series, and
write out
an approximation using a small number of terms.
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Check your work graphically.
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Check your work using Parseval's identity, and quantify the mean-square-error.
Compare e.g. test 1, spring 2001 test 1 and final exam, and exercise 17.6/1.
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More detailed items re PDEs
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Basic modeling features: e.g. explain in physical terms the different meaning
of the ut and the utt in the heat and wave equation.
Also explain the signs (+ versus -) in each equation.
Explain the practical meaning of the various BCs.
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Classify PDE as (non)linear, elliptic/parabolic/hyperbolic, and know the
appropriate kinds of boundary conditions for each -- why are these
physically meaningful, and why are they natural from a mathematical
point of view?
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Explain where linearity is used when solving the PDE.
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Explain the main steps in the method of separation of variables, especially
why what is constant, why a constant must be positive or negative or a
multiple of some fixed number, why there might be no sines or no cosines
(in particular, show which BC determines which of these). Explain why you
choose which periodic extension (leading to which half or quarter range
extension).
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Write out the solution as an infinite series with reasonably simplified formulas
for the Fourier coefficients, and write out an approximation with a small number
of terms, using decimal approximations of the coefficients.
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Sketch "cross-sections" for various x held fixed or various $t$ held fixed,
find e.g. the maximum value of each such cross-section ("how hot will it get
before it cools down", "what is the max temp at time t=30") and evaluate
the solution at any given (x,t), e.g. find u(20,12).
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Special features: Calculate and interpret the steady state of the soln of the
heat eqn, and understand d'Alembert's soln of the 1-d wave eqn (e.g. you should
be able to sketch a "movie" of several frames. Pay attention to how the "waves"
are "reflected" at the ends.
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You may bring one (two-sided) page of formulas.
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I will not accept large numbers of computer printout without
detailed explanations of what is going on. If needed at all,
keep the computer printouts to a minimum -- be very selective.
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