This is one of the two cluster examinations required for the Applied
Mathematics Option. Students are required to answer 5 of 8 questions. The
questions emphasize the derivation of analytical relations and the
applications of mathematical techniques. Solutions of mathematical problems
in terms of series, integrals, or in closed form, as well as approximations
based on convergent or asymptotic expansions are called for. Below is a list
of topics, courses, and several references that cover the required material.
Questions are drawn from the relevant courses and their prerequisites. Sample
exams indicate the intent and level.
TOPICS:
asymptotic analysis - asymptotic sequences; Laplace's method;
applications of Watson's lemma; saddle point and stationary phase
approximations; asymptotic expansion solutions of differential equations;
WKB method; singular perturbation; boundary layer analysis; matched
asymptotic expansions.
ordinary differential equations - exact and approximate method
for initial and boundary problems; Green's functions, classification of
spectra.
stability theory - almost linear theory and Liapunov
functions; resonances; bifurcation; limit cycles and nonlinear oscillations;
multiple scales and other perturbation theories.
partial differential equations - exact and approximate methods
for elliptic, hyperbolic, and parabolic equations; separation of variables;
Green's function methods; transform methods; variational procedures;
maximum princples; energy methods; characteristics.
RELEVANT COURSES
Math 578 - Asymptotic Methods
Math 579 - Singular Perturbations
COURSE REFERENCES
Keener - Principles of Applied Mathematics (Math 573)
Bender and Orszag - Advanced Mathematical Methods for Scientists
and Engineers (Math 578-579)
GENERAL REFERENCES
Kevorkian and Cole - Perturbation Methods
Jeffreys & Jeffreys - Methods of Mathematical Physics
Courant & Hilbert - Methods of Mathematical Physics, Vols. I
and II
John - Partial Differential Equations
Kevorkian - Partial Differential Equations
Web Source: http://www.math.uic.edu/~hanson/prelaomsyl.html
