Partial differential equations

The diffusion equation is a partial differential equation.

In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928.

The Einstein field equations (EFE; also known as "Einstein's equations") are the set of 10 equations in Albert Einstein's general theory of relativity that describes the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy.

Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits.

In physics, the Navier–Stokes equations /nævˈjeɪ stoʊks/, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous fluid substances.

In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives.

The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in physics—such as sound waves, light waves and water waves.

In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions.

In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differentiability criteria, form a necessary and sufficient condition for a complex function to be complex differentiable, that is, holomorphic.

A continuity equation in physics is an equation that describes the transport of some quantity.

The method of image charges (also known as the method of images and method of mirror charges) is a basic problem-solving tool in electrostatics.

In differential geometry, the Ricci flow (/ˈriːtʃi/) is an intrinsic geometric flow.

The Cauchy momentum equation is a vector partial differential equation put forth by Cauchy that describes the non-relativistic momentum transport in any continuum.

In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties.

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space.

Noether's (first) theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law.

In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself and its derivatives up to a given order.

The mathematical term well-posed problem stems from a definition given by Jacques Hadamard.

In mathematics, and particularly in potential theory, Dirichlet's principle is the assumption that the minimizer of a certain energy functional is a solution to Poisson's equation.

In fluid dynamics, the Euler equations are a set of quasilinear hyperbolic equations governing adiabatic and inviscid flow.

Burgers' equation is a fundamental partial differential equation occurring in various areas of applied mathematics, such as fluid mechanics, nonlinear acoustics, gas dynamics, traffic flow.

Reaction–diffusion systems are mathematical models which correspond to several physical phenomena: the most common is the change in space and time of the concentration of one or more chemical substances: local chemical reactions in which the substances are transformed into each other, and diffusion which causes the substances to spread out over a surface in space.

In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equation for w a complex distribution of the complex variable z in some open set U, with derivatives that are locally L2, and where μ is a given complex function in L∞(U) of norm less than 1, called the Beltrami coefficient.

The Kundu equation is a general form of integrable system that is gauge-equivalent to the mixed nonlinear Schrödinger equation.

In electromagnetism and applications, an inhomogeneous electromagnetic wave equation, or nonhomogeneous electromagnetic wave equation, is one of a set of wave equations describing the propagation of electromagnetic waves generated by nonzero source charges and currents.

The intent of this article is to highlight the important points of the derivation of the Navier–Stokes equations as well as the application and formulation for different families of fluids.

In physics, the Young–Laplace equation (/ˈjʌŋ ləˈplɑːs/) is a nonlinear partial differential equation that describes the capillary pressure difference sustained across the interface between two static fluids, such as water and air, due to the phenomenon of surface tension or wall tension, although usage on the latter is only applicable if assuming that the wall is very thin.

The Benjamin–Bona–Mahony equation (or BBM equation) – also known as the regularized long-wave equation (RLWE) – is the partial differential equation This equation was studied in Benjamin, Bona, and Mahony () as an improvement of the Korteweg–de Vries equation (KdV equation) for modeling long surface gravity waves of small amplitude – propagating uni-directionally in 1+1 dimensions.

In mathematics, Fisher's equation (named after statistician and biologist Ronald Fisher, also known as Kolmogorov–Petrovsky–Piscounov equation, KPP equation or Fisher–KPP equation) is the partial differential equation: It belongs to the class of reaction-diffusion equation: in fact it is one of the simplest semilinear r.

In mathematics and physics, homogenization is a method of studying partial differential equations with rapidly oscillating coefficients, such as where is a very small parameter and is a 1-periodic coefficient: , .

The obstacle problem is a classic motivating example in the mathematical study of variational inequalities and free boundary problems.

In mathematics, the Neumann–Poincaré operator or Poincaré–Neumann operator, named after Carl Neumann and Henri Poincaré, is a non-self-adjoint compact operator introduced by Poincaré to solve boundary value problems for the Laplacian on bounded domains in Euclidean space.

In mathematics, Sobolev spaces for planar domains are one of the principal techniques used in the theory of partial differential equations for solving the Dirichlet and Neumann boundary value problems for the Laplacian in a bounded domain in the plane with smooth boundary.