Applied Mathematics 5480
Methods in Applied Mathematics 3: Approximation Methods
Click here for
information specific to spring term of 2013
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Course text:
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Perturbation Methods
(E.J. Hinch; Cambridge Texts in Applied Mathematics 1991)
&
Advanced Mathematical Methods for Scientists and Engineers
(Bender and Orszag; McGraw-Hill 1978,
reprinted by Springer 1999).
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Topics
covered:
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Some introductory examples from applications
Expansion methods in case of algebraic equations
Asymptotic Series and techniques for convergence improvement of truncated expansions
Asymptotic expansion of integrals
Approximate solutions of linear and nonlinear ODEs
Perturbation series for ODEs and PDEs
Boundary layer theory - matched asymptotic expansions
Multiple scale analysis
WKB theory
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| Prerequisites: |
Some fimiliarilty with ODE's, basic PDE's, complex variables will be assumed
(e.g. see previous three courses in the APPM Methods-series). |
Course Motivation:
The governing equations for most phenomena in nature and in the sciences
can be formulated in terms of PDEs, ODEs, integral equations, or in combinations
of these. The main approaches to obtain insights from such equations are
Analytic Solutions
This approach alone is virtually never successful. For all but the
most trivial cases, realistic governing equations simply do not admit exact
solutions in terms of elementary functions. This fundamental obstacle is
not due to any limitations in our ability to perform analytic manipulations
- the use of symbolic algebra packages (like Mathematica) help only very
little.
Numerical Solutions
This general approach is immensely powerful - large-scale computer simulations
are now often considered as the third fundamental investigative technique
(besides the long established ones of theory and experiment). No other
approach can come even close to solving systems of hundreds or thousands
of coupled nonlinear differential equations that arise in many applications.
However, 'single-minded number-crunching' suffers from notable limitations
and difficulties, e.g.
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Coding can be very complex,
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We are often interested in situations where some parameter is very small
(or very large). In such limits, computer costs often become prohibitive,
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Numerics is not well suited to get 'leading behaviors' in precise analytic
form, suitable for further analysis.
- Identification of a theoretical foundation can be obscured.
Perturbation / Asymptotic Analysis
By replacing an analytically unsolvable problem with a sequence of
solvable ones, one can often avoid the fundamental barrier that is encountered
when searching for exact solutions (Now, a package like Mathematica becomes
extremely useful in carrying out the difficult and lengthy - but feasible
- manipulations required). Asymptotic methods are usually most powerful
precisely when numerical approaches encounter their most serious difficulties,
such as in cases of small parameters, phenomena on vastly different scales
etc. Perturbation / asymptotic analysis can then provide accurate information
in analytic forms which are very well suited for both understanding and
for further analysis.
The three general approaches above all complement each other. In most applications,
all three are required. In particular, perturbation / asymptotic analysis
is often required in both the problem formulation and again later as one
tool in the subsequent analysis or numerical verification. Several courses
in APPM are devoted to analytical- and numerical techniques; APPM 5480
is the only one fully focussed on these approximate analytical techniques.
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