Ph.D. Comprehensive Examination Topics
Partial Differential Equations
2004
Topics
- Nonlinear analysis
- Contraction mapping principle
- Implicit function theorem in Banach space
- Fixed point theorems
- Existence-uniqueness theorems for initial value problems for ODE's
- Picard-Lindelof theorem
- Cauchy Peano theorem
- Dependence upon initial conditions and parameters
- Variation of constants formula
- First-Order PDE's
- Cauchy problem
- Method of characteristics
- Weak solutions
- Conservation laws and jump conditions
- General nonlinear equations
- Geometric optics
- The Cauchy Problem
- Caushy-Kovalevski theorem
- Lewy's example
- Linear Equations and Generalized Solutions
- Adjoints
- Weak solutions
- Distributions
- Fundamental solutions and convolutions
- Classical Wave, Laplace and Heat Equations
- Seperation of variables
- Fourier transform
- Green's Functions
- Spherical means
- Energy methods
- Mean values
- Maximum principles
- Eigenfunction expansions
- Perron's method
- Subharmonic functions
- Scale invariance and the similarity method
- Nonlinear Diffusion
- Weak and strong maximum principles
- Comparison principles
- Hilbert Space Theory for Linear PDE
- Sobolev spaces
- Hahn-Banach and Riesz representation theorems
- Lax-Milgram theorem
- Applications of Lax-Milgram to the study of weak solutions of elliptic PDE's
- Regularity theory
- Mollifiers
- Spectral theorem for elliptic operators
References:
Partial Differential Equations methods and applications, Robert McOwen,
Prentice Hall
Shock Waves and Reaction-Diffusion Equations, Smoller, Springer-Verlag
Partial Differential Equations, John, Springer-Verlag
Introduction to Partial Differential Equations, Folland, Princeton University Press
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