Preliminary Exam in Applied Mathematics - Part A. Applied Mathematics

Preliminary Exam in Applied Mathematics - Part A. Applied Mathematics

Two courses are recommended for students preparing for Part A:

  • Math 613, Advanced Applied Mathematics I: Modeling
  • Math 651, Methods of Applied Mathematics I

 

Material Covered

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

  1. Initial-value and boundary-value problems for ODEs
  2. Basic existence and uniqueness results
  3. Solution techniques for homogeneous and inhomogeneous linear ODEs
  4. Local analysis via series solutions for linear ODEs
  5. Phase plane techniques for nonlinear ODEs, including equilibrium points and their stability

PDEs

  1. Examples of linear and nonlinear PDEs
  2. Solution of boundary-value problems and initial boundary-value problems by eigenfunction expansion (separation of variables)
  3. The Sturm-Liouville eigenvalue problem
  4. Nonhomogenous problems
  5. The D'Alembert solution of the wave equation
  6. Examples of solution of linear PDEs by Fourier and Laplace transforms

 

Detailed Outline, Modeling

  1. Units, scales, and non-dimensionalization
  2. Simple models in mechanics, engineering, and population dynamics
  3. Conservation laws, general balance law, flux-density pairs
  4. Traffic flow, the method of characteristics
  5. Diffusion equation, heat and mass flow
  6. Random walk, the diffusion equation as a limit of a difference equation
  7. Elementary linear stability theory
  8. Wave equation, D'Alembert solution
  9. Traveling wave solutions

 

Copies of past qualifying exams are available here.