PDEs

Partial differential equations (PDEs) are equations that involve unknown functions of several variables and their partial derivatives. They are used to formulate models where the rate of change in one direction depends on changes in other directions, such as time and space, capturing diffusion, wave propagation, and other spatial-temporal processes.

PDEs are classified by the order of the highest derivative, by linearity, and by the type of the principal part of the equation. The majority of common PDEs are linear or nonlinear, and many linear PDEs have constant coefficients. The principal part determines the equation type: elliptic, parabolic, or hyperbolic. Elliptic equations, such as Laplace's equation ∇^2u=0, describe steady-state spatial processes. Parabolic equations, like the heat equation ∂u/∂t = κ∇^2u, model diffusion with time evolution. Hyperbolic equations, such as the classical wave equation ∂^2u/∂t^2 = c^2∇^2u, describe wave propagation with finite speed.

Most PDE problems require initial conditions (specifying the state at an initial time) and boundary conditions (specifying the state on the spatial boundary). A well-posed PDE problem has a solution, the solution is unique, and its behavior changes continuously with the data.

Solutions are sought by analytical methods in simple geometries, using separation of variables, Fourier or Laplace transforms, and Green’s functions. In general, numerical methods—finite difference, finite element, finite volume, and spectral methods—are used for complex domains and nonlinear or variable-coefficient PDEs.

Applications span physics, engineering, chemistry, and beyond: electrostatics, heat conduction, fluid and solid mechanics, quantum mechanics, financial mathematics (e.g., Black-Scholes PDE), biology (reaction-diffusion systems), and environmental modeling.

Historical development spans the 18th to 20th centuries with contributions from d’Alembert, Euler, Lagrange, Laplace, Fourier, and others.