Applications Oriented Mathematics Preliminary Exam

(11/93)

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.

*Math 578*- Asymptotic Methods*Math 579*- Singular Perturbations

*Keener*- Principles of Applied Mathematics (Math 573)*Bender and Orszag*- Advanced Mathematical Methods for Scientists and Engineers (Math 578-579)

*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`

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