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Applied Mathematics & PDE syllabus

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Description

  
 

Requirements

  
 

Ph.D requirements

  
  Preliminary exams  
 

Qualifying exams

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

Topics

  1. ODE's: Qualitative aspects
    1. Picard Theorem, continuous dependence of solutions on initial values and parameters, flows
    2. $ 2D$ autonomous systems: flows, phase plane, classification of equilibria, local stability via linear analysis, stable and unstable manifolds, Hopf bifurcation, Lyapunov functions, $ \omega$-limit set (also known as positive limit set), global stability
  2. ODE's: Perturbation theory, asymptotic methods
    1. Regular perturbation
    2. Asymptotic series
    3. Multiple scales, secular terms
    4. Boundary layers, matching
  3. Discrete models
    1. Examples: population models, transition probabilities, examples from economics
    2. Linear difference equations
      1. Existence and uniqueness
      2. Connections with differential equations
      3. Case of constant coefficients: homogeneous and particular solutions, special case of first-order equations: $ y_{k+1} = ay_k + b$, cobwebbing, Markov processes, higher-order equations, transition matrices: $ y_{k+1} = A y_k$ for an $ n\times n$ matrix $ A$, asymptotic behavior, equilibrium and stability
  4. Quasilinear first-order equations, characteristics
    1. Derivation of the equation of continuity of hydrodynamics
    2. Dimensional analysis: dimensionless form of PDE's, dimensionless parameters
    3. How to solve first-order quasilinear equations, Cauchy problem, Cauchy-Kowalesky theorem
    4. Burger's equation: Steepening of profiles, weak solutions, Rankine-Hugoniot jump condition, Riemann problem
    5. Higher-order equations: characteristic surfaces, classification of equations (elliptic, parabolic, hyperbolic).
    6. Well-posedness
  5. Models giving rise to PDE's
    1. Variational principles, Euler-Lagrange equation, examples
    2. Heat flow: Fourier's law
    3. Reaction-diffusion equations: Fick's law
    4. Random walk, Brownian motion informally
    5. Vibrating membranes
  6. Laplace and Poisson equations
    1. Generalized functions, fundamental solutions, Green's representation for solution to Dirichlet problem, Poisson integral
    2. Mean value inequality, strong and weak maximum principles, uniqueness for Dirichlet problem
    3. Dirichlet Principle
  7. Heat equation
    1. Fundamental solution from Fourier transforms; scale-invariance
    2. Cauchy problem for homogeneous and inhomogeneous heat equations, smoothing effect
    3. Weak maximum principle, uniqueness for initial-boundary value problems
  8. Wave equation
    1. $ 1D$: d'Alembert's formula, initial-boundary value problems
    2. $ 3D$ and $ 2D$: method of spherical means, Hadamard's method of descent
    3. Inhomogeneous equations via Duhamel's principle
    4. Domain of influence/dependence, Huygen's principle
    5. Conservation of energy
  9. Classical maximum principles for 2nd-order elliptic and parabolic equations
    1. Weak and strong maximum principles, Hopf boundary point lemma
    2. Comparison principle and application to reaction-diffusion equations
  10. Sobolev spaces
    1. Distributions, weak derivatives
    2. Mollifier, regularization
    3. $ H^k$ spaces: Density Theorem, Trace Theorem, Poincaré inequality, Kondrachov compact imbedding theorem
  11. $ L^2$ theory for second-order elliptic equations
    1. Weak solutions
    2. Existence and uniqueness: Lax-Milgram Theorem, Fredholm alternative
    3. Interior regularity of weak solutions
    4. Eigenvalues and eigenfunctions: minimax characterization of eigenvalues, eigenexpansion in $ L^2$ and $ H^{1}_0$, simplicity of first eigenspace

References

1. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, Second edition, Springer, 1999.

2. David Bleecker and George Csordas, Basic Partial Differential Equations, International Press, 1996.

3. F. Brauer and J. A. Nohel, The Qualitative Theory of Ordinary Differential Equations: An Introduction, Dover, 1989.

4. G. Carrier and C. Pearson, Partial Differential Equations, Second edition, Academic Press, 1988.

5. Lawrence C. Evans, Partial Differential Equations, American Mathematical Society, 1998.

6. Gerald B. Folland, Partial Differential Equations, Princeton University Press, 1995.

7. P. R. Garabedian, Partial Differential Equations, Chelsea, Second revised edition, 1998. (First published 1964.)

8. Samuel Goldberg, Introduction to Difference Equations, Dover, 1986.

9. K. E. Gustafson, Introduction to Partial Differential Equations and Hilbert Space Methods, Third edition, Dover, 1999. (First published in 1980.)

10. E. J. Hinch, Perturbation Methods, Cambridge University Press, 1991.

11. Mark H. Holmes, Introduction to Perturbation Methods, Springer, 1995.

12. Fritz John, Partial Differential Equations, Fourth edition, Springer, 1982.

13. James P. Keener, Principles of Applied Mathematics, Second edition, Perseus, 2000.

14. J. Ockendon, S. Howison, A. Lacey, and A. Movchan, Applied Partial Differential Equations, Oxford University Press, 1999.

15. Mark Pinsky, Partial Differential Equations and Boundary-Value Problems with Applications, Second edition, McGraw-Hill, 1991.

16. M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Springer, 1992.

17. I. Rubinstein and L. Rubinstein, Partial Differential Equations in Classical Mathematical Physics, Cambridge University Press, 1998.

18. Hans Sagan, Boundary and Eigenvalue Problems in Mathematical Physics, Dover, 1989.

19. S. L. Sobolev, Partial Differential Equations of Mathematical Physics, Dover, 1989.

20. Walter Strauss, Partial Differential Equations, an Introduction, Wiley, 1992.

21. I. Stakgold, Green's Functions and Boundary-Value Problems, Wiley, 1979.

22. Michael Taylor, Partial Differential Equations: Basic Theory, Springer, 1996.

23. H. F. Weinberger, A First Course in Partial Differential Equations, Dover, 1995.

24. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, 1990.

25. E. C. Zachmonoglou and D. W. Thoe, Introduction to Partial Differential Equations with Applications, reprinted by Dover, 1986. (First published in 1976.)

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Last Updated: September 21, 2004
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