Neumann boundary condition

A Neumann boundary condition ( after Carl Gottfried Neumann ) referred to in the context of differential equations (more precisely, boundary value problems ) values ​​that are specified on the boundary of the domain for the normal derivative of the solution. Other boundary conditions are, for example, Dirichlet boundary conditions or oblique boundary conditions.

• 2.1 The Neumann problem
• 2.2 Identification of necessary conditions
• 2.3 Example of a partial differential equation

Ordinary differential equation

The Neumann problem

In the case of an ordinary differential equation of the domain of the function is a closed interval. Consequently, there is the edge of the definition range only from the right and the left end of the interval. Due to the freedom in ordinary differential equations make Neumann boundary conditions for equations of second or higher order sense. In this case, sees a Neumann problem, that is, a differential equation with Neumann boundary condition as follows:

Here, the right-hand side of the differential equation is a prescribed function, and are prescribed real numbers for the values ​​of the first derivative of a solution to the interval ends. Finally, a solution of the given Regularitätsklasse is searched.

Example of an ordinary differential equation

We choose as our interval and consider the following problem:

With the theory of linear ordinary differential equations with constant coefficients, we obtain first as a general solution of the differential equation:

The derivative

And two arbitrary real constants and. We use the boundary conditions to fix these constants. We obtain a linear system of equations in the unknowns and:

Remarkably, this system is not uniquely solvable, but there is a solution given for arbitrary real by

Partial Differential Equations

The Neumann problem

For a partial differential equation, the sole indication of Neumann boundary conditions is only useful for elliptic equations on a bounded domain, since the other types also require specifications of the initial values. Here, Neumann boundary conditions are prescribed on the boundary of the domain. The derivative of the solution in the direction of the outward normal so it is prescribed. Thus, the derivative in the direction of the outward normal is sensible to the area, must be necessarily presupposed here is that there is a boundary.

Here, we define the Neumann problem for a quasilinear partial differential equation:

Here, the function of the prescribed derivative in the direction of the outward normal to dar. of our solution alone, the question of the solvability of such a problem is very challenging and is the focus of current research. It is also very difficult to give a general solution method.

Determination of necessary conditions

It should be noted however, that only represents the validity of the Gaussian integral theorem another (necessary) condition on the data and solutions of our Neumann problem. We have purposely only apply the Gaussian integral theorem to the vector field.

If we consider, for example, a solution of a simple linear Neumann problem with the Laplace operator:

We obtain using the Gaussian integral theorem the condition on the data and:

Therefore the validity of the equation

Necessary for the solvability of this Neumann problem. For other problem is to consider, where appropriate, helpful other suitable vector fields.

Example of a partial differential equation

We consider in this example in the field with the regular edge

The following boundary value problem:

Herein, the Laplace operator. First, we note that a solution of the problem is. To find other solutions, we can purely formal follow the example to Dirichlet boundary conditions for partial differential equations and obtain a product approach:

But we must note that we can not actually require the zeros of freedom here, since the cosine function is known to have a zero at. This means that we do not know whether our formal solution is also really solve our Neumann problem. If we use this, however, we realize that we are lucky and our actually is solution of our problem.

Generalization for partial differential equations

It is often advisable to general boundary value problems such as

Look at. In this case, a directional derivative into an outer direction. That is, it applies to everyone. However, we note that the direction vector is a date of the problem.