# Wolfram Language & System 10.0 (2014)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

# NeumannValue

NeumannValue[val,pred]
represents a Neumann boundary value, val, specified on the part of the boundary of the region given to NDSolve where pred is True.

## DetailsDetails

• NeumannValue is used within partial differential equations to specify boundary values in functions such as NDSolve.
• In NDSolve[eqns,{u1,u2,},{x1,x2,}Ω], are the independent variables, are the dependent variables, and Ω is the region with boundary Ω.
• Locations where Neumann values might be specified are shown in green. They appear on the boundary Ω of the region Ω and specify a flux across those edges in the direction of the outward normal.
• is used to specify the flux over the part of the boundary Ω where pred is true, such that holds. is the outward-facing unit normal of Ω. The coefficients g and q can depend on any of the independent variables .
• When no boundary condition is specified on a part of the boundary Ω, then the flux term over that part is taken to be , so not specifying a boundary condition at all is equivalent to specifying a Neumann 0 condition.
• Any logical combination of equalities and inequalities in the independent variables may be used for the predicate pred.
• NeumannValue can be used to specify both Neumann and Robin boundary conditions:
•  ·(c ∇u+α u-γ)=0 natural (Neumann 0) no conditions specified or NeumannValue[0, pred] ·(c ∇u+α u-γ)=g Neumann NeumannValue[g,pred] ·(c ∇u+α u-γ)=g-q u Robin (generalized Neumann) NeumannValue[g-q u,pred]
• For systems, corresponds to the condition being satisfied on the parts of the region boundary where pred is True.
• For time dependent equations, both val and pred may depend on time.
• Neumann values are enforced as integrated conditions over each boundary cell in the discretization of Ω where pred is True. Boundary cells are points in 1D, edges in 2D, and faces in 3D.
• Neumann boundary conditions can be used to enforce boundary loads in structural mechanics settings.

## ExamplesExamplesopen allclose all

### Basic Examples  (1)Basic Examples  (1)

Solve a Laplace equation over a disk with Dirichlet and Neumann conditions for and for :

 Out[1]=
 Out[2]=