World Library  
Flag as Inappropriate
Email this Article

Neumann boundary condition

Article Id: WHEBN0000735443
Reproduction Date:

Title: Neumann boundary condition  
Author: World Heritage Encyclopedia
Language: English
Subject: Mixed boundary condition, Robin boundary condition, Green's function, List of partial differential equation topics, Boundary conditions
Collection: Boundary Conditions
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Neumann boundary condition

In mathematics, the Neumann (or second-type) boundary condition is a type of boundary condition, named after Carl Neumann.[1] When imposed on an ordinary or a partial differential equation, it specifies the values that the derivative of a solution is to take on the boundary of the domain. In engineering applications, the following would be considered Neumann boundary conditions:

  • In thermodynamics, where a surface has a prescribed heat flux, such as a perfect insulator (where flux is zero) or an electrical component dissipating a known power.

For an ordinary differential equation, for instance:

y'' + y = 0~

the Neumann boundary conditions on the interval [a, \, b] take the form:

y'(a)= \alpha \ \text{and} \ y'(b) = \beta

where \alpha and \beta are given numbers.

  • For a partial differential equation, for instance:
\nabla^2 y + y = 0

where \nabla^2 denotes the Laplacian, the Neumann boundary conditions on a domain \Omega \subset \mathbb{R}^n take the form:

\frac{\partial y}{\partial \mathbf{n}}(\mathbf{x}) = f(\mathbf{x}) \quad \forall \mathbf{x} \in \partial \Omega.

where \mathbf{n} denotes the (typically exterior) normal to the boundary \partial \Omega and f is a given scalar function.

The normal derivative which shows up on the left-hand side is defined as:

\frac{\partial y}{\partial \mathbf{n}}(\mathbf{x})=\nabla y(\mathbf{x})\cdot \mathbf{n}(\mathbf{x})

where \nabla is the gradient (vector) and the dot is the inner product.

It becomes clear that the boundary must be sufficiently smooth such that the normal derivative can exist, since for example at corner points of the boundary the normal vector is not well defined.

Many other boundary conditions are possible. For example, there is the Cauchy boundary condition, or the mixed boundary condition which is a combination of the Neumann and Dirichlet conditions.

See also

References

  1. ^ Cheng, A. H. -D.; Cheng, D. T. (2005). "Heritage and early history of the boundary element method". Engineering Analysis with Boundary Elements 29 (3): 268.  
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.