Oral Exam in Applied Mathematics

Oral Exam in Applied Mathematics

  

Oral Exam in Applied Mathematics - Applied Mathematics

  

This topic of the oral exam is based on the courses Math 689, Advanced Applied Mathematics II: ODEs, and Math 690, Advanced Applied Mathematics III: PDEs.

The best reference for both courses is `Boundary Value Problems of Mathematical Physics, Volume I and II' by I. Stakgold.  SIAM Classics in Applied Mathematics, vol 29, ISBN 0-89871-456-7.  Other sources listed below are also useful.

Topics on which questions can be based are as follows: 

I.   Math 689, Advanced Applied Mathematics II: ODEs
     Linear, two-point boundary value problems for ODEs

  1. Definition, construction, and use of a Green's function:  Existence of a Green's function, the Fredholm alternative, and the modified Green's function.
  2. The regular Sturm-Liouville eigenvalue problem:  Representation of the solution of a two-point BVP in terms of eigenfunctions (eigenfunction expansion).  Eigenfunction expansion of a Green's function.  The relation between the Green's function and the spectrum.  Simple examples of singular problems.

Main Reference:

  • `Boundary Value Problems of Mathematical Physics, Volume I and II' by I. Stakgold.  SIAM Classics in Applied Mathematics, vol 29, ISBN 0-89871-456-7.  The material on Green's functions is in chapter 1. Section 1.3, on theory of distributions, will not be emphasized.  The material on eigenfunction expansion is in chapter 4, sections 4.1 to 4.3.

Other references that may be useful are:

  • `Principles and Techniques of Applied Mathematics' by B. Friedman.  Dover,  ISBN 0-48666-444-6.  Material in chapter 3 on Green's functions and chapter 4 on eigenfunction expansion.
  • `Methods of Mathematical Physics, Vol I' by R. Courant and D. Hilbert. Wiley-Interscience, ISBN 0-47150-447-5.  Chapter 5.
  • `Applied Mathematics' by J.D. Logan.  Wiley-Interscience, ISBN 0-471-74662-2. Chapter 4, omitting section 4.3.

 

II.  Math 690, Advanced Applied Mathematics III: PDEs
     Boundary and initial boundary value problems for the classical PDEs of mathematical physics

     Green's function, eigenfunction expansion, and Fourier and Laplace transform methods for the
     solution of:

  1. Boundary value problems for elliptic equations, i.e. the Laplace, Poisson, and Helmholtz equations.
  2. Initial boundary value problems for the diffusion equation (parabolic) and wave equation (hyperbolic). The D'Alembert solution and the method of characteristics for the wave equation.

Main Reference:

  • `Boundary Value Problems of Mathematical Physics, Volume I and II' by I. Stakgold (details above).  Material on fundamental solutions in section 5.8.  Elliptic problems in chapter 6.  Parabolic and hyperbolic problems in chapter 7.

Other references that may be useful are:

  • `Partial Differential Equations:  analytical solution techniques' by J. Kevorkian.  Springer Verlag, texts in applied mathematics, ISBN 1-44193-139-2.  Chapters 1 to 3.  This is a good, systematic reference for the purpose of the course and exam.
  • `Applied Mathematics' by J.D. Logan (details above).  Chapters 6 and 7.

  

Oral Exam in Applied Mathematics - Ordinary Differential Equations

  

This topic of the oral exam is based on the course Math 676, Advanced Ordinary Differential Equations.

Topics on which questions can be based are as follows:

  1. Existence, uniqueness, continuous dependence on parameters and initial conditions.
  2. Linear systems of ODEs:  constant coefficient, matrix exponentials including complex eigenvalues and nontrivial Jordan forms. Periodic coefficients and Floquet theory, fundamental solutions.
  3. Fixed Points, stability, and linearization:  classification of fixed points, stable and unstable manifold theorems.  Lyapunov's method. The Hartman-Grobman theorem (emphasizing its use and interpretation rather than its proof) and normal forms.
  4. Periodic orbits: the Poincare-Bendixson Theorem.  Poincare maps, illustrating the connection between continuous and discrete dynamical systems. 
  5. Hamiltonian systems.  Their definition and basic properties.
  6. Bifurcations:  structural stability as the antithesis of bifurcation. Classification of basic local bifurcations.  Saddle-node, transcritical, pitchfork, and Hopf bifurcation of periodic and homoclinic orbits.
  7. Fundamentals of chaotic dynamics:  definition of chaos.  Lyapunov exponents and rudimentary fractal dimension.

Primary References:

  • `Differential Equations and Dynamical Systems' by Lawrence Perko, 3rd Edition, Springer Verlag, ISBN 0-387-95116-4.  Chapter 1 (covered briefly in Math 676). Chapter 2, all except sections 2.10, 2.11 and 2.13.  Chapter 3, sections 3.2-3.5, 3.7 and 3.9.  Chapter 4, sections 4.2-4.5.
  • `Differential Dynamical Systems' by James Meiss.  SIAM, ISBN 0-898-71635-7.  Similar material with a slightly more applied emphasis and a more readable style.  Chapter 1 sections 1.3-1.5 (considered prerequisite for Math 676).  Chapter 2 and chapter 3, all.  Chapter 4, all except sections 4.3 and 4.11. Chapter 5, all.  Chapter 6, sections 6.5 and 6.6.  Chapter 7, sections 7.1 and 7.2.  Chapter 8, sections 8.1-8.6 and 8.8.

Additional references:

  • `Nonlinear Dynamics and Chaos' by Steven Strogatz.  Westview Press, ISBN 0-738-20453-6.  Useful, especially to review more basic material.
  • `Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields' by J. Guckenheimer and P. Holmes.  Springer Verlag, ISBN 0-387-90819-6.  May be useful for more advanced material.

  

Oral Exam in Applied Mathematics - Asymptotic Methods

  

This topic of the oral exam is based on the course Math 671, Asymptotic Methods I.

Topics on which questions can be based are as follows: 

  1. The Landau symbols or order relations o, O, and ~.  Asymptotic series; definitions, properties, and operations.  Asymptotic versus convergent. 
  2. Asymptotic expansion of integrals:  the method of stationary phase and the Riemann-Lebesgue lemma.  Integration by parts.  Watson's lemma, Laplace's method, and the method of steepest descent.
  3. Matched asymptotic expansions for two point boundary value problems for ODEs:  the location of a boundary layer.  Inner and outer expansions, asymptotic matching with an intermediate variable.
  4. The method of multiple time scales for weakly nonlinear oscillators: examples.
  5. The WKB method:  exponential dispersive and dissipative behavior.  The WKB method applied to multiple scale and boundary layer phenomena.  The single turning point problem.  Examples.

References:  There is no single best reference for material on this exam. Two good references are:

  • `Introduction to Perturbation Methods' by M.H. Holmes.  Springer Verlag, ISBN 0-387-94203-3.  Material for (2) above is not discussed in this text (formulas only are given in Appendix A2) but the other material appears in the same order in chapters 1 to 4.
  • `Advanced Mathematical Methods for Scientists and Engineers' by C.M. Bender and S.A. Orszag.  Springer Verlag, ISBN 0-387-98931-5. Some material for (1) above is in section 3.8.  Material for (2) is in chapter 6, omitting section 6.7.  Material for (3) is in chapter 9, omitting sections 9.6 and 9.7.  Material for (5) is in chapter 10, sections 10.1, 10.2 and 10.4. 

   

Oral Exam in Applied Mathematics - Numerical Methods

  

This topic of the oral exam is based on the courses Math 614, Numerical Methods I, and Math 712, Numerical Methods II.

The Main References are:

(a)  `An Introduction to Numerical Analysis' by K. Atkinson.  Wiley, ISBN 0-471-62489-6.

(b)  `Numerical Partial Differential Equations - Finite Difference Methods' by J. W. Thomas.  Springer Verlag, ISBN 0-387-97999-9.

(c)  `Finite Difference Schemes and Partial Differential Equations' by John C. Strikwerda. 2nd Edition, SIAM, ISBN 0-898-71639-X.

Topics on which questions can be based are as follows:

  1. Rootfinding: Newton's method, secant method, fixed point theorem. 
  2. Numerical interpolation: Lagrange interpolation, Newton divided differences, spline interpolation. 
  3. Numerical approximation: least squares approximation, orthogonal polynomials, minimax approximation, Chebyshev equioscillation theorem. 
  4. Numerical integration:  Newton-Cotes formula (trapezoidal rule, Simpson's rule, etc.), Gaussian quadrature, Euler-MacLaurin formula. 
  5. Numerical methods for ODEs:  multistep methods, one-step methods (e.g. Heun's method, Runge-Kutta methods, etc.), order, consistency, convergence, stability, absolute stability, Dalquist equivalence theorem. 
  6. Numerical methods for PDEs:  Parabolic and hyperbolic equations.
    • Order, consistency, convergence, stability, and the Lax equivalence theorem.
    • Fourier analysis, von Neumann condition, the Courant-Friedrichs-Lewy condition.
    • Dispersion and dissipation.
    • Various finite difference schemes for parabolic and hyperbolic equations, such as one-sided scheme, centered scheme, Lax-Wendroff scheme, Crank-Nicolson scheme, leap frog scheme. Alternating direction implicit schemes.
  7. Numerical linear algebra and numerical elliptic PDEs.
    • Classical iterative solvers - Jacobi, Gauss-Seidel, and SOR schemes.
    • Krylov-subspace based iterative solvers, such as conjugate gradient.

Topics (1)-(5) can be found in reference (a) chapters 2-6, respectively. Topic (6) can be found in reference (b) chapters 2-5 and 7, and reference (c).

Other useful references are:

(d) `Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations' by Lloyd N. Trefethen.

(e) `A First Course in the Numerical Analysis of Differential Equations' by Arieh Iserles.

(f)  `Numerical Linear Algebra' by Lloyd N. Trefethen and David Bau.

Topic (7) can be found in references (a), (d), (e), and (f).