### 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.