Partial Differential Equations - Period 4+5

An overwhelming number of physical phenomena can be described by partial differential equations (PDEs). This course discusses these equations and methods for their solution. For first order equations we discuss the method of characteristics, and the solution of such PDEs by methods from ordinary differential equations. For second order equations, in particular for the heat and wave equation, we discuss the method of separation of variables. This ties in with the remarkable result of Fourier that almost any periodic function can be represented as a sum of sines and cosines, called its Fourier series. An analogous representation for non-periodic functions is provided by the Fourier transform, to be discussed briefly in part 2 of the course, together with some theoretical background for Fourier series. We discuss some of the background for generalised Fourier series: the role of eigenvalue problems and some basic spectral theory. Potential methods and fundamental solutions will be discussed for the standard examples: heat, wave and Poisson equation. Harmonic functions will be discussed in relation to mean value properties.

Contact hours listed under a course description may vary due to the combination of lecture-based and independent work required for each course therefore, CEA's recommended credits are based on the ECTS credits assigned by VU Amsterdam. 1 ECTS equals 28 contact hours assigned by VU Amsterdam.