Differential Equations Comprehensive Exam

Topics:

Ordinary Differential Equations

• Gronwall’s inequality and applications to ODEs.
• Existence and uniqueness: the Cauchy problem, Picard’s theorem, Peano’s theorem, maximum interval of solution.
• Linear equations: constant coefficients, fundamental matrices, variation of constants, stability, asymptotic stability.
• Autonomous equations: phase space, orbits, critical points, linearization, periodic solutions, two dimensional linear systems, phase portraits.
• Stability and asymptotic stability of nonlinear ODE: linearization; Lyapunov functions.
• Boundary value problems: Sturm-Liouville problems, Green’s functions.

Partial Differential Equations

• First-Order Equations
• Linear equations, Quasilinear equations, Nonlinear equations, Characteristics, The Cauchy problem, Hamilton-Jacobi equations.
• Second-Order Equations
• Characteristics for linear and quasilinear equations.
• Classification of linear equations; Canonical (Normal) forms.
• The wave equation; Initial and initial-boundary value problems; D’Alembert’s formula; Duhamel’s principle.
• General formulation of a Cauchy problem; Multi-index notation; The Lagrange-Green identity; Distribution (weak) solutions; Well-posedness in general.
• The Laplace Equation
• Green’s identity; Fundamental solutions; Poisson’s equation.
• The maximum principle.
• The Dirichlet problem; Green’s function; Poisson’s formula.
• Hyperbolic Equations in Higher Dimensions
• The wave equation in n-dimensional space; Spherical means; Hadamard’s method of descent; Duhamel’s principle; Huygen’s principle; Initial-boundary value problems.
• Higher order equations with constant coefficients. 
• Parabolic Equations
• The heat equation; Initial value problems; Maximum principle; Uniqueness; Regularity.
• The initial value problem for general second order linear parabolic equations.

Selected Bibliography:

Ordinary Differential Equations

1. V. I. Arnold, Ordinary Differential Equations
2. E. Coddington and N. Levinson, Theory of Ordinary Differential Equations
3. F. Brauer and J. Nohel, The Qualitative Theory of Ordinary Differential Equations
4. M. Braun, Differential Equations and Their Applications
5. J. K. Hale, Ordinary Differential Equations
6. P. Hartman, Ordinary Differential Equations
7. J. Cronin, Ordinary Differential Equations
8. D. Sanchez, Ordinary Differential Equations and Stability Theory: An Introduction
9. F. Verhulst, Nonlinear Differential Equations and Dynamical Systems
10. M. Hirsch and S. Smale, Differential Equations, Dynamical Systems and Linear Algebra
11. L. C. Piccinini, G. Stampacchia, G. Vidossich, Ordinary Differential Equations in R^N
12. C. Corduneanu, Principles of Differential and Integral Equations

Partial Differential Equations

1. F. John, Partial Differentail Equations (4^th Ed.)
2. G. Folland, Introduction to Partial Differential Equations
3. E.C. Zachmanoglou and D. Thoe, Introduction to Partial Differential Equations with Applications
4. Colton, Partial Differential Equations: An Introduction
5. P. Duchateau and D. Zachmann, Partial Differential Equations (Schaum’s Outline Series)
6. R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. II
7. P. Garabedian, Partial Differential Equations
8. L. C. Evans, Partial Differential Equations
   
9. Pavel, Nicolae H., Introduction to Partial Differential Equations-a graduate textbook.  Zip Publishing, 1313 Chespeake Ave, Columbus OH 43212.  2007. vi+220 pp. ISBN: 978-0-9778102-3-9 35-01.
   www.zippublishing.com