MATH 132B: Partial Differential Equations and Integral Equations
Spring quarter, 2005
Instructor: Bo Li
Topics
(An * denotes an optional topic.)
- Examples and classification of integral equations
- Integral equations of the second kind
- Fredholm integral equations of the second kind: iteration
- Volterra integral equations of the second kind: iteration.
Existence and uniqueness of solutions
- Fredholm integral equations of the second kind: degenerate kernels
- Elements of Hilbert Space
- Inner-product, the Cauchy-Schwarz inequality, examples
- Banach space, convergence, Neumann series
- Hilbert space, orthognality, Parseval identity, Bessel inequality,
complete orthonormal basis
- Elements of Operator Theory
- Bounded linear operators on Banach spaces
- Self-adjoint operators, spectral theory
- compact linear operators, spectral theory, Fredholm Alternatives
- Revisit integral equations of the second kind
- Application of the Fredholm Alternatives, Fredholm solution kernels
- Symmetric kernels: eigenvalues and eigenfunctions, Hilbert-Schmidt theorem
- Methods of finding eigenvalues and eigenfunctions
- Integral equations of first kind
- Eigenvalues and eigenfunctions for Fredholm integral equations of the
first kind
- Expansion and Schimidt-Picard theory of Fredholm integral equations of the first kind
- An iterative method for Fredholm integral equations of the first kind
- Examples of Volterra integral equations of the first kind that can be
transformed to that of the second kind
- Singular integral equations
- Weakly singular kernels: Iteration. Abel's integral
- Cauchy singular kernels and Hilbert transforms
- Fourier transform methods
- *Linear systems of integral equations
- *Nonlinear integral equations
- Nonlinear integral equations of the second kind
- A parameter-embedding method
- Nonlinear systems of integral equations of the second kind
- Hammmerstein type nonlinear integral equations
- *Appliecations
- The boundary-value problem of Poisson equation
- Linear elasticity
- Potential flows
- Electrostatics
Last updated by Bo Li on May 2, 2005.