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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.
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 ...
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.
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 ...
The method of separation of variables is also used to solve a wide range of linear partial differential equations with boundary and initial conditions, such as the heat equation, wave equation, Laplace equation, Helmholtz equation and biharmonic equation.
For example, if = and time is the ... it is possible to solve Laplace's equation ... The two (dis)continuity equations can be solved for and to obtain = ; ...
For example, hitting times of Bessel processes can be computed via an algorithm called "Walk on moving spheres". [12] This problem has applications in mathematical finance. The WoS can be adapted to solve the Poisson and Poisson–Boltzmann equation with flux conditions on the boundary. [13]
The Hankel transform can be used to transform and solve Laplace's equation expressed in cylindrical coordinates. Under the Hankel transform, the Bessel operator becomes a multiplication by . [2] In the axisymmetric case, the partial differential equation is transformed as
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