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Neumann Problem

   Also found in: Wikipedia 0.01 sec.
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.



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He covers first-order equations, linear second-order equations, elements of Fourier analysis, the wave equation, the heat equation, Dirichlet and Neumann problems, existence theorems, and a selection of the aforesaid advanced topics.
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.
Fourier series, integrals, and transforms are followed by their rigorous application to wave and diffusion equations as well as to Dirichlet and Neumann problems.
 
 
 
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