Applications of PDEs in many variables

Question

One reason that solving systems of partial differential equations is so important is the many applications of PDEs in science and engineering (eg. the heat equation, the wave equation, etc.). Often these use numerical methods. However, most of these applications that I am aware of involve systems of PDEs in only a few variables, often corresponding to physical dimensions and/or time.

What are some important applications of systems of PDEs in many variables, or even when the number of variables is growing as a parameter of the problem?

Answer 1

Take a look at Ian Mitchell's A Toolbox of Level Set Methods:

... a software package for solving time-dependent Hamilton-Jacobi partial differential equations (PDEs) in the MATLAB programming environment. Level set methods are often used for simulation of dynamic implicit surfaces in graphics, fluid and combustion simulation, image processing, and computer vision. Hamilton-Jacobi and related PDEs arise in fields such as control, robotics, differential games, dynamic programming, mesh generation, stochastic differential equations, financial mathematics, and verification. The algorithms in the toolbox can be used in any number of dimensions, although computational cost and visualization difficulty make dimensions four and higher a challenge.