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  2. Laplace's equation - Wikipedia

    en.wikipedia.org/wiki/Laplace's_equation

    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.

  3. Green's function for the three-variable Laplace equation

    en.wikipedia.org/wiki/Green's_function_for_the...

    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 ...

  4. Bateman transform - Wikipedia

    en.wikipedia.org/wiki/Bateman_transform

    In the mathematical study of partial differential equations, the Bateman transform is a method for solving the Laplace equation in four dimensions and wave equation in three by using a line integral of a holomorphic function in three complex variables. It is named after the mathematician Harry Bateman, who first published the result in (Bateman ...

  5. Relaxation (iterative method) - Wikipedia

    en.wikipedia.org/wiki/Relaxation_(iterative_method)

    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 ...

  6. Hankel transform - Wikipedia

    en.wikipedia.org/wiki/Hankel_transform

    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

  7. Walk-on-spheres method - Wikipedia

    en.wikipedia.org/wiki/Walk-on-spheres_method

    [1] [2] The WoS method was first introduced by Mervin E. Muller in 1956 to solve Laplace's equation, [1] and was since then generalized to other problems. It relies on probabilistic interpretations of PDEs, and simulates paths of Brownian motion (or for some more general variants, diffusion processes ), by sampling only the exit-points out of ...

  8. Laplace transform applied to differential equations - Wikipedia

    en.wikipedia.org/wiki/Laplace_transform_applied...

    In mathematics, the Laplace transform is a powerful integral transform used to switch a function from the time domain to the s-domain. The Laplace transform can be used in some cases to solve linear differential equations with given initial conditions. First consider the following property of the Laplace transform:

  9. Separation of variables - Wikipedia

    en.wikipedia.org/wiki/Separation_of_variables

    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.

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