Material Covered

The exam  focuses on elements of mathematical modeling and on some of the principal mathematical methods and theories pertinent to model solution. Past exams have usually consisted of six questions, with one or at most two questions on modeling and the remainder on methods. The material is exemplified by the following texts:

Modeling (with some material on methods)

• Mathematics Applied to Deterministic Problems in the Natural Sciences, by C.C. Lin and L.A. Segel.
• Mathematics Applied to Continuum Mechanics, by L.A. Segel.

Methods

• Principles and Techniques of Applied Mathematics, by B. Friedman.
• Partial Differential Equations, Analytical Solution Techniques, by J. Kevorkian.
• Applied Mathematics, a Contemporary Approach, by D. Logan.
• Principles of Applied Mathematics, by J. Keener.
• Green's Functions and Boundary Value Problems, by I. Stakgold.
• Mathematical Physics (Volume I), by R. Courant and D. Hilbert.

Detailed Outline

Modeling

1. Basic physical and mathematical understanding of conservation laws (e.g., density and flux for mass, momentum, energy, and electric charge).
2. Identification of dimensionless parameters.
3. Scaling. Simplification of governing equations.
4. Solution of governing equations and consistency with 3.

Methods

1. Basic elements of distribution theory.
2. Boundary value problems for ODEs.
   • Construction of Green's functions.
   • Spectral theory. Eigenfunction expansion.
   • Rayleigh quotient and variational methods.
3.  Boundary value problems for elliptic PDEs (e.g., Laplace, Poisson, and Helmholtz equations).
   • Construction of Green's functions.
   • Spectral theory. Eigenfunction expansion.
   • Variational methods.
   • Transform techniques (Fourier, Fourier sine, Fourier cosine, Laplace, and complex variable methods).
4. Initial boundary value problems for parabolic and hyperbolic PDE's.
   • Construction of Green's functions.
   • Eigenfunction expansion and transform techniques (Fourier and Laplace).
   • Method of characteristics for hyperbolic problems.
     

Copies of past qualifying exams are available here.