Differential equations §
Differential equations relate a function to its rates of change, and they are the standard language for dynamics: motion, growth, diffusion, oscillation, waves, and fields.
Brief history §
- 17th–18th century: Newton solves the two-body problem with an ODE; Euler and Lagrange develop the theory of equations arising from mechanics and variational principles.
- 19th century: Cauchy and Lipschitz prove existence and uniqueness theorems; Poincaré founds the qualitative theory of dynamical systems.
- 20th century: Sobolev, Lions, and others build the weak-solution and functional-analytic machinery that makes nonlinear PDEs tractable.
Main branches §
- Ordinary differential equations (ODEs): one independent variable; existence and uniqueness (Picard–Lindelöf), linear systems, stability, and dynamical systems.
- Partial differential equations (PDEs): several independent variables; the classical families (elliptic, parabolic, hyperbolic) such as Laplace, heat, and wave equations.
- Qualitative theory: phase portraits, stability of equilibria, bifurcations, and chaos, where explicit formulas are impossible but long-term behavior is still decidable.
- Numerical methods: discretization schemes (finite differences, finite elements, Runge–Kutta) that compute approximate solutions with controlled error.
Where to go next §
- The parent class, with its sibling sub-areas, is Analysis.
- Existence proofs for weak solutions lean on the functional analysis of Functional analysis and, for stochastic equations, on Measure theory.