Partial Differential Equations

Aim

• That the student will achieve in-depth knowledge and skills in the subject area of partial differential equations (PDEs).
• After completing the course, the student should be well acquainted with fundamental theory and solution methods for first-order semilinear PDEs and second-order linear PDEs (in particular Laplace's equation, the heat equation and the wave equation).
• The student should also possess some knowledge about generalized solutions, solitons, and numerical methods for PDEs.

Content

• First-order semilinear PDEs. Characteristic curves.
• Physical background: Conservation laws, the wave equation.
• Laplace's equation: Maximum principles and uniqueness, Poisson's formula, separation of variables, the mean value property, Dirichlet's principle, fundamental solution, Green functions, harmonic polynomials, spherical harmonics.
• The wave equation: Solution on the real line (d'Alembert's formula), solution on the half-line via reflection, solution on a compact interval using separation of variables and Fourier series, wave propagation in higher dimensions.
• The heat equation: Maximum principles and uniqueness, solution on a compact interval using separation of variables and Fourier series, solution in the whole space using Fourier transform, fundamental solution.
• Characteristic curves and classification of second-order linear PDEs.
• Separation of variables in higher dimensions. Eigenvalue problems for the Laplacian.
• Generalized solutions. Weak solutions of conservation laws.
• Introduction to numerical methods for PDEs (finite differences, finite elements).
• Introduction to dispersive PDEs and solitons.
• Introduction to the Schrödinger equation.