Course description of AMS 212A

AMS 212A focuses on analytical methods for partial differential equations (PDEs). The topics include

• Review of chain rule, types of ODEs that can be solved analytically.
• General introduction to PDEs and their physical background.
• Method of separation of variables. Fourier expansion. Applications to heat equation, wave equation and Laplace equation.
• Scaling and non-dimensionalization.
• Sturm-Liouville problem. Theory of Sturm-Liouville problem.
• Eigenfunction expansion. Bessel functions. Oscillation of a hanging chain.
• Non-homogeneous PDEs and forced systems.
• Green's functions for ODEs. Green's functions for PDEs. Applications to heat equation, wave equation and Laplace equation.
• First-order PDEs, introduction to the method of characteristics.
• Method of Characteristics for semilinear and quasilinear equations.
• Conservation laws. Weak solutions and shock waves. Compression shocks, expansion shocks and entropy condition.
• Traffic flow problem. Propagation of traffic jam vs trajectory of vehicles.
• Classification and canonical forms of second order PDEs. Method for transforming a PDE to its canonical form.