Diffusions and Parabolic Evolution Equations
We begin with a study of the classic 1-D diffusion equation (also called heat equation) and its self-similar solutions. This is the simplest example of a linear parabolic partial differential equations. Well-posedness of an initial value problem with periodic data is then discussed. Subsequently, the exposition switches to the complex domain, and we introduce a simple version of the general Schrödinger equation. This makes it possible to study the diffraction problem and the so-called Fresnel zones. Multidimensional parabolic equation follow, and the general reflection method is explained. The chapter concludes with a study of the moving boundary problem and the standard physical problem of particle motion in a potential well.