Neumann boundary condition

A Neumann boundary condition (according to Carl Gottfried Neumann ) describes values in connection with differential equations (more precisely: boundary value problems ) that are given on the boundary of the domain for the normal derivative of the solution. In the case of Neumann boundary value problems, derivative values ​​are not given, but rather function values. Further boundary conditions are, for example, Dirichlet boundary conditions (in which the function values ​​are specified on the boundary) or skewed boundary conditions .

Ordinary differential equation

The Neumann problem

In the case of an ordinary differential equation , the domain of the function is a closed interval. Consequently, the edge of the definition area consists only of the right and left end of the interval. Due to the freedom in ordinary differential equations, Neumann boundary conditions are only useful for equations of the second or higher order. In this case, a Neumann problem, i.e. H. a differential equation with Neumann boundary condition as follows:

${\displaystyle y\in C^{2}(a,b)\cap C^{1}[a,b]}$

${\displaystyle y^{\prime \prime }=f(x,y,y')}$

${\displaystyle y^{\prime }(a)=\alpha ,\quad y'(b)=\beta }$

Here, the right side of the differential equation is a prescribed function, and prescribed real numbers are the values ​​of the first derivative of a solution at the interval ends. Finally, a solution is sought from the specified regularity class . ${\displaystyle f}$ ${\displaystyle \alpha }$ ${\displaystyle \beta }$ ${\displaystyle y}$

Example of an ordinary differential equation

We choose as our interval and consider the following problem: ${\displaystyle [0,\pi ]}$

${\displaystyle y\in C^{2}(0,\pi )\cap C^{1}[0,\pi ]}$

${\displaystyle y''=-y}$

${\displaystyle y'(0)=0,\quad y'(\pi )=0}$

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

${\displaystyle y(x)=C\cos x+D\sin x}$

with the derivative

${\displaystyle y'(x)=-C\sin x+D\cos x}$

and two freely selectable real constants and . We use the boundary conditions to fix these constants. We get a linear system of equations in the unknowns and : ${\displaystyle C}$ ${\displaystyle D}$ ${\displaystyle C}$ ${\displaystyle D}$

${\displaystyle D=0,}$

${\displaystyle -D=0.}$

Remarkably, this system is not uniquely solvable, but for any real thing a solution is given by ${\displaystyle C}$

${\displaystyle y(x)=C\cos x.}$

Partial differential equations

The Neumann problem

In the case of a partial differential equation , the sole specification of Neumann boundary conditions only makes sense for elliptical equations in a restricted area , since the other types also require specifications for the initial values. Neumann boundary conditions are prescribed on the edge of the area . So the derivation of the solution in the direction of the external normal is prescribed. In order for the derivation in the direction of the outer normal to the area to be meaningful, it must be assumed that it is a -rand. ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ ${\displaystyle \partial \Omega }$ ${\displaystyle C^{1}}$

Here we define the Neumann problem for a quasi-linear partial differential equation:

${\displaystyle u\in C^{2}(\Omega )\cap C^{1}({\overline {\Omega }})}$

${\displaystyle \sum _{i,j=1}^{n}a_{ij}(x,u,\nabla u){\frac {\partial ^{2}}{\partial x_{i}\partial x_{j}}}u=f(x,u,\nabla u)}$

${\displaystyle {\frac {\partial u(x)}{\partial \mathbf {n} }}=g(x),\qquad x\in \partial \Omega }$

Here, the function of the required derivative in the direction of the outward normal to our solution. But the question of the solubility of such a problem is very demanding and is the focus of current research. It is also very difficult to give a general solution method. ${\displaystyle g\colon \partial \Omega \rightarrow \mathbb {R} }$ ${\displaystyle \mathbf {n} (x)}$ ${\displaystyle \partial \Omega }$

Determination of necessary conditions

It should be noted, however, that the validity of the Gaussian integral theorem alone represents a further (necessary) condition for the data and for the solutions to our Neumann problem. We only have to apply the Gaussian integral theorem to the vector field . ${\displaystyle f(x)=\nabla u(x)}$

For example, if we consider a solution to a simple linear Neumann problem with the Laplace operator : ${\displaystyle \Delta }$

${\displaystyle u\in C^{2}(\Omega )\cap C^{1}({\overline {\Omega }})}$

${\displaystyle \Delta u(x)=f(x),\qquad x\in \Omega }$

${\displaystyle {\frac {\partial u(x)}{\partial \mathbf {n} }}=g(x),\qquad x\in \partial \Omega ,}$

using the Gaussian integral theorem we obtain the condition on the data and : ${\displaystyle f}$ ${\displaystyle g}$

${\displaystyle \int _{\Omega }f(x)\,d^{n}x=\int _{\Omega }\operatorname {div} \nabla u(x)\,d^{n}x=\oint _{\partial \Omega }\nabla u(x)\cdot \mathbf {n} (x)\,d^{n-1}\sigma (x)=\oint _{\partial \Omega }g(x)\,d^{n-1}\sigma (x).}$

Hence the validity of the equation

${\displaystyle \int _{\Omega }f(x)\,d^{n}x=\oint _{\partial \Omega }g(x)\,d^{n-1}\sigma (x)}$

necessary for solving this Neumann problem. For other problems it may be helpful to consider other suitable vector fields.

Example of a partial differential equation

In this example we consider the area with the regular margin ${\displaystyle \Omega =(0,\pi )^{n}=\{x=(x_{1},...,x_{n})\in \mathbb {R} ^{n}\,:\,0<x_{i}<\pi ,\quad i=1,\dots ,n\},}$

${\displaystyle \delta \Omega =\{x=(x_{1},...,x_{n})\in \partial \Omega \,:\,}$for exactly one applies ${\displaystyle i_{0}\in \{1,...,n\}}$ ${\displaystyle x_{i_{0}}\in \{0,\pi \}\}}$

the following boundary value problem:

${\displaystyle u\in C^{2}(\Omega )\cap C^{0}({\overline {\Omega }})}$

${\displaystyle \Delta u(x)=-nu(x),\qquad x\in \Omega }$

${\displaystyle {\frac {\partial u(x)}{\partial \mathbf {n} }}=0,\qquad x\in \delta \Omega .}$

Here denotes the Laplace operator . First, we note that there is a solution to the problem. In order to find further solutions, we can purely formally follow the example of Dirichlet boundary conditions of partial differential equations, and obtain according to a product approach: ${\displaystyle \Delta }$ ${\displaystyle u\equiv 0}$

${\displaystyle u(x)=D\prod _{k=1}^{n}\cos(x_{k}).}$

We have to note, however, that we cannot actually demand zero zero from here , since the cosine function is known to have a zero at . This means that we do not know whether our formal solution is really the solution to our Neumann problem. When we use this, however, we find that we are lucky and ours is indeed the solution to our problem. ${\displaystyle u}$ ${\displaystyle {\tfrac {\pi }{2}}}$ ${\displaystyle u}$

Generalization for partial differential equations

It is often advisable to tackle more general boundary value problems such as

${\displaystyle u\in C^{2}(\Omega )\cap C^{1}({\overline {\Omega }})}$

${\displaystyle \sum _{i,j=1}^{n}a_{ij}(x,u,\nabla u){\frac {\partial ^{2}}{\partial x_{i}\partial x_{j}}}u=f(x,u,\nabla u)}$

${\displaystyle {\frac {\partial u(x)}{\partial \mathbf {\nu } }}=g(x),\qquad x\in \partial \Omega }$

consider. In this case, a directional derivative is in an outward direction. That is, it applies to everyone . We note, however, that the direction vector is a datum of the problem. ${\displaystyle {\frac {\partial }{\partial \mathbf {\nu } }}}$ ${\displaystyle \mathbf {\nu } (x)\cdot \mathbf {n} (x)>0}$ ${\displaystyle x\in \partial \Omega }$ ${\displaystyle \mathbf {\nu } (x)}$

literature

• D. Gilbarg, NS Trudinger: Partial Differential Equations of Second Order , Springer-Verlag, Berlin 1998, ISBN 3-540-41160-7 .