Search results
Results from the WOW.Com Content Network
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.
In general solutions of equations involving the p-Laplacian do not have second order derivatives in classical sense, thus solutions to these equations have to be understood as weak solutions. For example, we say that a function u belonging to the Sobolev space W 1 , p ( Ω ) {\displaystyle W^{1,p}(\Omega )} is a weak solution of
The Laplace operator is a second-order differential operator in the n-dimensional Euclidean space, defined as the divergence of the gradient (). Thus if f {\displaystyle f} is a twice-differentiable real-valued function , then the Laplacian of f {\displaystyle f} is the real-valued function defined by:
An illustration of the five-point stencil in one and two dimensions (top, and bottom, respectively). In numerical analysis, given a square grid in one or two dimensions, the five-point stencil of a point in the grid is a stencil made up of the point itself together with its four "neighbors".
The Hodge Laplacian, also known as the Laplace–de Rham operator, is a differential operator acting on differential forms. (Abstractly, it is a second order operator on each exterior power of the cotangent bundle.) This operator is defined on any manifold equipped with a Riemannian- or pseudo-Riemannian metric.
Verbally, the second version is the second derivative in the direction of the gradient. In the case of the infinity Laplace equation Δ ∞ u = 0 {\displaystyle \Delta _{\infty }u=0} , the two definitions are equivalent.
In the discrete case, the standard central difference approximation of the second derivative is used on a uniform grid. These formulas are used to derive the expressions for eigenfunctions of Laplacian in case of separation of variables , as well as to find eigenvalues and eigenvectors of multidimensional discrete Laplacian on a regular grid ...
The continuous -Laplace operator is a second-order differential operator that can be well-translated to finite weighted graphs. It allows the translation of various partial differential equations, e.g., the heat equation, to the graph setting.