Partial differential equation

A partial differential equation relates an unknown function of multiple variables to its partial derivatives. Such equations describe how quantities vary across space and time.

Partial derivative: A derivative measuring how a multivariable function changes with respect to one variable while others are held fixed. PDEs are built from derivatives taken with respect to selected variables.
Linear partial differential equation: A PDE in which the unknown function and its derivatives appear linearly, with coefficients depending only on independent variables. Linearity enables superposition and supports many exact solution techniques.
Laplace's equation: The elliptic PDE stating that a function's Laplacian is zero. It models steady states without internal sources and exemplifies elliptic behavior.
Computational fluid dynamics: The numerical study of fluid flow using discretized governing equations. Fluid simulations approximate the Navier–Stokes PDEs across complex geometries.
Ordinary differential equation: A differential equation whose unknown function depends on one independent variable. Unlike a PDE, an ODE differentiates with respect to only one independent variable.
Boundary value problem: A differential equation problem whose solution must satisfy specified conditions on the boundary of a domain. Boundary data often select a particular PDE solution from many possibilities.
Nonlinear partial differential equation: A PDE in which the unknown function or its derivatives enter nonlinearly. Nonlinear terms can create steep gradients, interactions, and solutions unavailable to linear theory.
Heat equation: A parabolic PDE describing diffusion of heat or other quantities over time. Its smoothing evolution is the standard example of a parabolic PDE.
Finite element method: A numerical method that approximates solutions by assembling local functions over a mesh. Finite elements handle PDEs on irregular domains and are widely used in engineering.
Elliptic partial differential equation: A PDE class associated with spatial equilibrium and boundary-value behavior. Elliptic equations typically constrain a domain globally rather than propagate signals in time.

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