Lecture Notes. - Department of Mathematics, Leipzig University, 2012. - 205 pages.Introduction
Examples
Equations from variational problems
Exercises
Equations of first order
Linear equations
Quasilinear equations
Nonlinear equations in two variables
Nonlinear equations in Rn
Hamilton-Jacobi theory
Exercises
Classification
Linear equations of second order
Quasilinear equations of second order
Systems of first order
Systems of second order
Theorem of Cauchy-Kovalevskaya
Exercises
Hyperbolic equations
One-dimensional wave equation
Higher dimensions
Inhomogeneous equation
A method of Riemann
Initial-boundary value problems
Exercises
Fourier transform
Definition, properties
Exercises
Parabolic equations
Poisson’s formula
Inhomogeneous heat equation
Maximum principle
Initial-boundary value problem
Black-Scholes equation
Exercises
Elliptic equations of second order
Fundamental solution
Representation formula
Boundary value problems
Green’s function
Inhomogeneous equation
Exercises
