pde

PDE stands for partial differential equation. It is an equation that involves an unknown function of several independent variables and its partial derivatives. PDEs describe how physical, geometric, or probabilistic quantities change with respect to multiple factors. The order of a PDE is the highest order of any partial derivative appearing in the equation. Common examples include the heat equation, the wave equation, and Laplace’s equation.

PDEs are often categorized as linear or nonlinear, and by the type determined by the sign structure of the principal part of the equation: elliptic, parabolic, or hyperbolic. Elliptic equations, such as Laplace’s equation, model equilibrium states. Parabolic equations, such as the heat equation, describe diffusion-like processes. Hyperbolic equations, such as the wave equation, model wave propagation. Many problems are posed with boundary conditions (Dirichlet, Neumann, Robin) and, in time-dependent problems, initial conditions.

Solution methods include analytical techniques such as separation of variables, characteristics for first-order PDEs, Fourier or Laplace transforms, and Green’s functions. Numerical methods, such as finite difference, finite element, and finite volume schemes, are widely used for complex geometries and nonlinear problems. Existence and uniqueness theory, regularity, and stability are central concerns in the mathematical study of PDEs.

PDEs arise in physics (electrostatics, fluid dynamics, quantum mechanics), engineering (heat conduction, elasticity, acoustics), biology (pattern formation), and finance (pricing of options). They are used to model and predict phenomena across many scales and disciplines.

History includes early work by Euler and Lagrange that led to foundational equations. In the 19th and 20th centuries, PDE theory matured with boundary-value problems and methods developed by Poisson, Laplace, Courant, Hilbert, and others. Modern computational PDEs underpin many simulations in science and engineering.