Introduction to Partial Differential Equations

• Model simple problems using partial differential equations
• Solve initial/boundary value problems for the diffusion equation in simple geometries
• Solve initial/boundary value problems for the wave equation in simple geometries
• Solve boundary value problems for Laplace’s equation in simple geometries
• Solve first order scalar semilinear and quasilinear equations using the method of characteristics
• Use separation of variables to transform a partial differential equation problem to a set of ordinary differential equation problems
• Use operator factoring to solve initial value problems for second-order linear hyperbolic differential equations in dimension two
• Apply Fourier series to solve problems in partial differential equations
• Analyze PDE problems and retrieve qualitative information about solutions
• Explain basic notions and facts from the theory of distributions
• Set up and solve weak formulations of selected PDE problems

Linear Partial Differential Equations (PDE). Semilinear and quasilinear equations. The wave equation. Diffusion and heat equations. Laplace’s equation and Poisson’s equation. Inhomogeneities/sources. Initial and boundary value problems. Expansion of solutions in orthonormal series. Operator factoring. Method of characteristics. The Fourier transform and relevant function spaces. Basic distribution theory. Propagation of singularities in solutions. Energy conservation and maximum principles. Weak formulation and weak solution of differential equation problems. The Poisson kernel and the heat kernel.

01025/01034/01035/01037, Linear Ordinary Differential Equations (ODE); convergence of infinite series; Fourier series

Lectures and exercises