The mathematical methods part of the exam focuses on material in the course Math 651, Methods of Applied Mathematics I. This consists of basic solution techniques for ODEs and PDEs.

The material on ODEs can be found in chapter 1 and sections 3.1. to 3.3 of 'Advanced Mathematical Methods for Scientists and Engineers' by C.M. Bender and S.A. Orszag, together with material on phase plane techniques for ODEs in chapters 5 and 6 of 'Nonlinear Dynamics and Chaos' by S.H. Strogatz.

The material on solution of linear PDEs by eigenfunction expansion is in chapters 2, 5, 7, and 8 of 'Applied Partial Differential Equations' by R. Haberman, and the material on Fourier and Laplace transform methods is in chapters 10 and 13.

The modeling part of the exam focuses on material in the course Math 613, Advanced Applied Mathematics I: Modeling. This consists of developing mathematical models of physical phenomena. Emphasis is on models involving ODEs and PDEs.

Much of the material can be found in 'Mathematics Applied to Deterministic Problems in the Natural Sciences' by C.C. Lin and L.A. Segel. The following sections are especially relevant: 1.1, 2.1-2.2, 3.3, 4.1, 6.1-6.3, 7.1-7.2, 11.1-11.2.

Modeling involving 1st and 2nd order ODEs may be found in most elementary ODE texts. Chapters 2-6 of 'Nonlinear Dynamics and Chaos' by S.H. Strogatz provide numerous applications.

The material on traffic flow may be found in 'Mathematical Models' by R. Haberman.

Detailed Outline, Methods

ODEs

Initial-value and boundary-value problems for ODEs

Basic existence and uniqueness results

Solution techniques for homogeneous and inhomogeneous linear ODEs

Local analysis via series solutions for linear ODEs

Phase plane techniques for nonlinear ODEs, including equilibrium points and their stability

PDEs

Examples of linear and nonlinear PDEs

Solution of boundary-value problems and initial boundary-value problems by eigenfunction expansion (separation of variables)

The Sturm-Liouville eigenvalue problem

Nonhomogenous problems

The D'Alembert solution of the wave equation

Examples of solution of linear PDEs by Fourier and Laplace transforms

Detailed Outline, Modeling

Units, scales, and non-dimensionalization

Simple models in mechanics, engineering, and population dynamics

Conservation laws, general balance law, flux-density pairs

Traffic flow, the method of characteristics

Diffusion equation, heat and mass flow

Random walk, the diffusion equation as a limit of a difference equation

Elementary linear stability theory

Wave equation, D'Alembert solution

Traveling wave solutions

Copies of past qualifying exams are available here.