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In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties.This is often written as = or =, where = = is the Laplace operator, [note 1] is the divergence operator (also symbolized "div"), is the gradient operator (also symbolized "grad"), and (,,) is a twice-differentiable real-valued function.
Laplace obtained these equations by simplifying the fluid dynamics equations, but they can also be derived from energy integrals via Lagrange's equation. For a fluid sheet of average thickness D , the vertical tidal elevation ζ , as well as the horizontal velocity components u and v (in the latitude φ and longitude λ directions, respectively ...
The Laplace pressure is determined from the Young–Laplace equation given as [2] = (+), where and are the principal radii of curvature and (also denoted as ) is the surface tension. Although signs for these values vary, sign convention usually dictates positive curvature when convex and negative when concave.
The Laplace–Beltrami operator also can be generalized to an operator (also called the Laplace–Beltrami operator) which operates on tensor fields, by a similar formula. Another generalization of the Laplace operator that is available on pseudo-Riemannian manifolds uses the exterior derivative , in terms of which the "geometer's Laplacian" is ...
Relaxation methods are used to solve the linear equations resulting from a discretization of the differential equation, for example by finite differences. [ 2 ] [ 3 ] [ 4 ] Iterative relaxation of solutions is commonly dubbed smoothing because with certain equations, such as Laplace's equation , it resembles repeated application of a local ...
Using the Green's function for the three-variable Laplace operator, one can integrate the Poisson equation in order to determine the potential function. Green's functions can be expanded in terms of the basis elements (harmonic functions) which are determined using the separable coordinate systems for the linear partial differential equation ...
Bessel functions describe the radial part of vibrations of a circular membrane.. Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, [1] are canonical solutions y(x) of Bessel's differential equation + + = for an arbitrary complex number, which represents the order of the Bessel function.
The cylindrical harmonics for (k,n) are now the product of these solutions and the general solution to Laplace's equation is given by a linear combination of these solutions: (,,) = | | (,) (,) where the () are constants with respect to the cylindrical coordinates and the limits of the summation and integration are determined by the boundary ...
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