boundary condition

Boundary conditions are the circumstances that can be influenced only with great difficulty or not at all, and therefore must be used in calculations as given magnitudes in general.

Boundary conditions and differential equations

In the field of differential equations, boundary conditions are specific information for calculating the solution function on a set of definitions D. For this purpose, the values ​​of the function on the boundary ( in the topological sense) will be given the desired domain of definition D. In the simplest case, an interval, and the boundary conditions are predetermined function values ​​. Are here instead of two values ​​only at a boundary point of the interval - usually a - set values ​​for u and also for derivatives of u, it is called an initial value problem and calls the preset values ​​its initial conditions.

For partial differential equations we consider the differential equation mostly on Sobolev spaces. In these rooms are functions that correspond to zero quantities considered equal. Since the boundary of a face is usually a null set, the concept of boundary condition is problematic. Solutions to this problem are sobolewsche embedding theorems or - more generally - track operators.

Boundary value problems do not always have a solution ( see example), in the case of its existence, the solution is not always unique. The calculation of an approximate solution of a boundary value problem with means of numerical mathematics is often complex and usually runs out on to the solution of very large systems of equations.

Example

Let the given differential equation. The solution set of this equation.

  • Wanted is the solution and the solution is.
  • Periodic boundary condition: Find with the solution and there are infinitely many solutions of the form with any.
  • Wanted is the solution and it is no solution.

Types of boundary conditions

There are different ways to impose on the edge of the considered values ​​. One possibility is to prescribe values ​​of the solution in the case of a defined on the interval ordinary differential equation and therefore, we speak of Dirichlet boundary conditions. On the other hand, one can impose conditions on the derivatives, so and pretend this is called Neumann boundary conditions ( for ordinary differential equations, as stated above, from initial conditions ). A special case are periodic boundary conditions must be here (in the example a considered on the interval ordinary differential equation ) or apply.

Artificial boundary conditions

In unbounded domains the numerical solution usually requires a limitation of the area. Here boundary conditions are then pretend the words are not present in the true problem, artificially.

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