differential equations. Laplace's equation is also a special case of the Helmholtz equation. The general theory of solutions to Laplace's equation is known...

For fluids in general, the speed of sound c is given by the Newton–Laplace equation: c = K s ρ , {\displaystyle c={\sqrt {\frac {K_{s}}{\rho }}},} where...

In physics, the Young–Laplace equation (/ləˈplɑːs/) is an equation that describes the capillary pressure difference sustained across the interface between...

of probability was developed mainly by Laplace. Laplace formulated Laplace's equation, and pioneered the Laplace transform which appears in many branches...

side of this equation is the Laplace operator, and the entire equation Δu = 0 is known as Laplace's equation. Solutions of the Laplace equation, i.e. functions...

Lorenz equation Laplace's equation Maxwell's equations Navier-Stokes equation Poisson's equation Reaction–diffusion system Schrödinger equation Wave equation...

Grattan-Guinness, I (1997), "Laplace's integral solutions to partial differential equations", in Gillispie, C. C. (ed.), Pierre Simon Laplace 1749–1827: A Life in...

field v satisfies Laplace's equation. However, the converse is not true; not every vector field that satisfies Laplace's equation is a Laplacian vector...

method of separation of variables for partial differential equations for Laplace's equation in polar form. This means that you write u ( r , θ ) = R (...

states that every weak solution of Laplace's equation is a smooth solution. This contrasts with the wave equation, for example, which has weak solutions...

the Helmholtz equation is the eigenvalue problem for the Laplace operator. It corresponds to the elliptic partial differential equation: ∇ 2 f = − k 2...

in time. The canonical examples of elliptic PDEs are Laplace's Equation and Poisson's Equation. Elliptic PDEs are also important in pure mathematics...

(force) field. It is a generalization of Laplace's equation, which is also frequently seen in physics. The equation is named after French mathematician and...

harmonics originate from solving Laplace's equation in the spherical domains. Functions that are solutions to Laplace's equation are called harmonics. Despite...

function. One of the most well-known of these, the Laplace expansion for the three-variable Laplace equation, is given in terms of the generating function...

mathematician, unified the work of these two scientists to derive the Young–Laplace equation, the formula that describes the capillary pressure difference sustained...

above properties mean, more precisely, that Laplace's equation for u {\displaystyle u} and Laplace's equation for v {\displaystyle v} are the integrability...

tides is described by the horizontal structure equation which is also called Laplace's tidal equation: L Θ n + ε n Θ n = 0 {\displaystyle {L}{\Theta }_{n}+\varepsilon...

to the differential equation This equation is found when the technique of separation of variables is used on Laplace's equation when expressed in parabolic...

differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important...

L2(0, ∞). The Hankel transform can be used to transform and solve Laplace's equation expressed in cylindrical coordinates. Under the Hankel transform,...

subset of ⁠ R n , {\displaystyle \mathbb {R} ^{n},} ⁠ that satisfies Laplace's equation, that is, ∂ 2 f ∂ x 1 2 + ∂ 2 f ∂ x 2 2 + ⋯ + ∂ 2 f ∂ x n 2 = 0 {\displaystyle...

differential equation is frequently encountered in physics and other technical fields. In particular, it occurs when solving Laplace's equation (and related...

Helmholtz equation in spherical coordinates. Bessel's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical...

Springer-Verlag, ISBN 3-540-42627-2. Solomentsev, E.D.; Shikin, E.V. (2001) [1994], "Laplace–Beltrami equation", Encyclopedia of Mathematics, EMS Press...

Dirichlet problem, of finding the harmonic functions (solutions to Laplace's equation); the solution was given by the Dirichlet's principle. Boundary value...

to solve linear differential equations with given initial conditions. First consider the following property of the Laplace transform: L { f ′ } = s L {...

relationship is a form of Poisson's equation. In the absence of unpaired electric charge, the equation becomes Laplace's equation: ∇ 2 ϕ = 0 , {\displaystyle...

Its second-order version can emerge from Laplace's equation in polar coordinates. Euler–Bernoulli beam equation, a fourth-order ODE concerning the elasticity...

gas, or between two immiscible liquids. The Laplace pressure is determined from the Young–Laplace equation given as Δ P ≡ P inside − P outside = γ ( 1...