Topics
This list of topics is approximate and is subject to revision during the term.
- What is a partial differential equation (PDE)? What is a solution to a PDE?
- How to solve a "PDE that is almost an ODE".
- Classification by order. Classification by linearity.
- Some nice properties of linear equations.
- Initial and boundary conditions.
- What is a conservation law? What is a constitutive relation?
- Derivation of the advection equation.
- Derivation of the diffusion equation.
- Derivation of the heat equation.
- Derivation of the wave equation for a vibrating string.
- What are the principal goals in the study of PDEs? What do we mean by a well-posed problem?
- First order linear PDEs in two variables.
- Classification of second order linear PDEs in two variables.
- Homogeneous wave equation on the whole line.
- Operator factorization approach.
- d'Alembert's formula.
- Domain of dependence. Domain of influence.
- Energy methods.
- Energy for the wave equation.
- Energy for the heat equation.
- Maximum principle for the heat equation.
- Homogeneous heat equation on the whole line. Integral transforms. How to solve PDEs by the method of Laplace transform.
- How do we proceed if the PDE contains a source term? Duhamel's principle.
- A maximum principle for the Laplace equation.
- Series solutions of ODEs.
- How to construct an orthogonal basis for our function spaces. Sturm-Liouville problems.
- Fourier sine and cosine series. Full Fourier series.
- Pointwise convergence of Fourier series.
- The method of separation of variables.
- The method of eigenfunction expansion.