Applied Mathematics Qualifying Exams

Syllabus Topics

The following topics & references will prepare you for the exam.

1. Ordinary Differential Equations

• Initial value problem—existence, uniqueness
• One-step methods for the numerical solution of the initial value problem—explicit and implicit schemes. Numerical solution of nonlinear equations
• Stability and phase plane analysis, bifurcations
• Applications such as population models, epidemiology
• Boundary value problems; finite difference methods—numerical solution of banded linear systems
• Sturm-Liouville systems

2. Fourier Series and Orthogonal Expansions, Discrete Fourier Series and Convolutions

3. The Heat Equation

• Heat flow, Fick’s law
• Separation of variables
• Fundamental solution from Fourier transforms; scale-invariance
• Smoothing effect, maximum principle
• Finite difference methods for heat equation in one dimension

4. Quasilinear First-Order Equations, Characteristics, Burger’s Equation

5. The Wave Equation

• Separation of variables
• 1D - d’Alembert’s formula, initial-boundary value problems
• 2D, 3D: method of spherical means, Hadamard’s method of descent
• Inhomogeneous equations via Duhamel’s principle
• Domain of influence/dependence, Huygen’s principle
• Conservation of energy

6. Distributions and the Delta Function; Green’s Functions and Fundamental Solutions

7. Laplace and Poisson Equations

• Separation of variables
• Green’s representation for solution to Dirichlet problem, Poisson integral
• Mean value inequality, strong and weak maximum principles, uniqueness for Dirichlet problem
• Dirichlet Principle
• Finite difference methods for Poisson equations
• A simple finite element method

8. Elementary Fluid Dynamics

• Derivation of equations of motion
• Vortex dynamics
• Conformal mappings and fluid flow

9. Dimensional Analysis and Scaling

10. Perturbation Theory for ODE’s, Asymptotic Methods

• Regular perturbation
• Asymptotic series
• Multiple scales, secular terms
• Boundary layers, matching
• Asymptotic methods for integrals; Stirling’s formula

References

1. Fritz John, Partial Differential Equations, Fourth edition, Springer, 1982.
2. Walter Strauss, Partial Differential Equations, an Introduction, Wiley, 1992.
3. H. F. Weinberger, A First Course in Partial Differential Equations, Dover, 1995.
4. R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, 2007.
5. A. J. Chorin and J. E. Marsden, A Mathematical Introduction to Fluid Mechanics, hird edition, Springer, 2000.