Neumann Problem

Neumann problem

[′nȯi‚män ‚präb·ləm]
(mathematics)
The determination of a harmonic function within a finite region of three-dimensional space enclosed by a closed surface when the normal derivatives of the function on the surface are specified.

Neumann Problem

 

(also the second boundary value problem of potential theory), a boundary value problem posed for second-order partial differential equations. In the simplest cases, particularly for the Laplace equation, the Neumann problem consists in finding in some region a solution of the equation having a given normal derivative on the boundary of the region. The problem was first systematically studied in 1877 by C. Neumann.

References in periodicals archive ?
For example, the three classical boundary value problems: the Dirichlet problem, the Neumann problem, and the mixed Dirichlet-Neumann problem can be reduced to (1.
They cover the Cauchy problem, the Dirichlet problem, the Neumann problem, the Neumann problem for a nonlocal nonlinear diffusion equation, nonlocal p-Laplacian evolution problems, the nonlocal total variation flow, and nonlocal models for sandpiles.
The Neumann problem is uniquely solvable in the weak sense if and only if for any z [not member of] [[bar.
The principal focus will be on what happens to the eigenstructure of the Neumann problem ([sigma] = 0) as [sigma] proceeds along rays emanating from the origin toward the point at infinity in the complex plane.
If u is the (unique) harmonic solution of the Dirichlet- Neumann problem with boundary values of u equal to 0 on [[gamma].
In this paper we give necessary and sufficient conditions on the existence of global weak solutions to the Neumann problem (1.
The case of three dimensions will be however treated separately, because the 3D Neumann problem is significantly more complex, especially when it comes to devising efficient numerical methods.
The approximation subspace for the Neumann problem is
k] can be reversed and used to approximate the solution to the Neumann problem
GAMMA]] can be evaluated by solving a Neumann problem on the subdomain [[omega].
Fourier series, integrals, and transforms are followed by their rigorous application to wave and diffusion equations as well as to Dirichlet and Neumann problems.