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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.
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 ...
Diagram illustrating the image method for Laplace's equation for a sphere of radius R. The green point is a charge q lying inside the sphere at a distance p from the origin, the red point is the image of that point, having charge −qR/p, lying outside the sphere at a distance of R 2 /p from the origin. The potential produced by the two charges ...
In mathematics, a Dirichlet problem asks for a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region. [1] The Dirichlet problem can be solved for many PDEs, although originally it was posed for Laplace's equation. In that case the ...
Spherical harmonics originate from solving Laplace's equation in the spherical domains. Functions that are solutions to Laplace's equation are called harmonics . Despite their name, spherical harmonics take their simplest form in Cartesian coordinates , where they can be defined as homogeneous polynomials of degree ℓ {\displaystyle \ell } in ...
For example, a time-independent solution of the heat equation solves Laplace's equation. That is, if parabolic and hyperbolic PDEs are associated with modeling dynamical systems then the solutions of elliptic PDEs are associated with steady states. Informally, this is reflective of the above regularity theorem, as steady states are generally ...
They have solved numerous problems which exhibit circular cylindrical symmetry employing the toroidal functions. The above expressions for the Green's function for the three-variable Laplace operator are examples of single summation expressions for this Green's function. There are also single-integral expressions for this Green's function.
Examples of archetypal well-posed problems include the Dirichlet problem for Laplace's equation, and the heat equation with specified initial conditions. These might be regarded as 'natural' problems in that there are physical processes modelled by these problems.