Ph.D. Comprehensive Examination Topics
Applied Mathematics
2009
FINITE-DIMENSIONAL LINEAR OPERATORS
- Spectral theory
- Fredholm alternative
- Generalized inversion
- Singular value decomposition
- Minimax principle
INTEGRAL AND DIFFERENTIAL OPERATORS
- Contraction Mapping Theorem
- Compact operators
- Hilbert-Schmidt theory
- Spectral theory
- Distributions
- Green's functions
- Resolvent operators
- Method of images
- Discrete and integral transforms
CALCULUS OF VARIATIONS
- Euler-Lagrange equations
- Hamilton's principle
- Approximation techniques
PARTIAL DIFFERENTIAL EQUATIONS
- Poisson equation
- Wave Equation
- Heat Equation
- Transform Methods for PDE's
PERTURBATION METHODS
- Regular Perturbations
- Perturbation of eigenvalues
- Multiple Scales
- Averaging
- Homogenization Theory
- Singular Perturbation Theory
- Boundary Layers
- Poincare-Lindstedt
Text
- Principles of Applied Mathematics: Transformation and
Approximation, by J. P. Keener, Addison-Wesley (2000) [ISBN 0-7382-0129-4]
Other Texts
- Applied Mathematics: A contemporary Approach, by D. Logan, Wiley
- Linear Algebra and its Applicaitons, by G. Strang, Harcourt, Brace, Jovanovich (1988) [ISBN 0-15-551005-3]
- Boundary Value Problems of Mathematical Physics, Vol. 1, by I. Stakgold, Macmillan (1967-1968)
- Methods of Mathematical Physics, by R. Courant and D. Hilbert, Interscience Publishers (1953-1962)
- Applied Mathematics Lecture Notes, J. Olver www.math.umn.edu/~olver/appl.html
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