Main navigation
- Programs
- Subjects
- Universities
- Destinations
- Advice
Our comprehension of nature's fundamental mechanisms relies heavily on partial differential equations (PDEs). For instance, Einstein's equations define space-time geometry and its interaction with matter. The motion of fluids and elastic materials follows PDEs originating from Euler and Cauchy's work. Light propagation and electromagnetic waves across different mediums are represented by Maxwell's equations. Increasingly, PDEs play vital roles in social and life sciences - the Black-Scholes equation forms the foundation of option pricing in finance, while reaction-diffusion systems model neural activity and population changes.
PDE research has spawned novel mathematical concepts. The examination of heat diffusion equations, for example, yielded Fourier series and eventually the entire discipline of Fourier analysis. Hilbert's exploration of eigenvalue problems from vibration and quantum mechanics PDEs developed into contemporary functional analysis and operator theory.