Boundary conditions (sometimes also referred to as general conditions ) are generally circumstances that can only be influenced with great effort or not at all or that result from the problem and must therefore be considered as given quantities ( data parameters ), for example in scientific experiments or in mathematical experiments Calculations.
In many cases, the term boundary condition is also used as a synonym for “ secondary condition ”.
Boundary conditions and differential equations
In the area of differential equations , boundary conditions are specific information for calculating the solution function on a domain . For this purpose, the values of the function on the edge (in the topological sense ) of are given.
In the simplest case is an interval and the boundary conditions are given function values .
If instead of two values only at one edge point of the interval - mostly - values for and additionally for derivatives of are given, one speaks of an initial value problem and the given values are called its initial conditions .
Partial differential equations are mostly considered on Sobolew spaces . In these spaces, functions that match up to zero sets are considered to be equal. Since the boundary of an area is usually a null set, the concept of boundary condition is problematic. Solutions to this problem are Sobolew's embedding theorems or - more generally - trace operators .
Boundary value problems do not always have a solution (see example below); if they exist, the solution is not always unique. The calculation of an approximate solution for a boundary value problem with the means of numerical mathematics is often time-consuming and usually results in the solution of very large systems of equations .
Let be the given differential equation . The solution set to this equation is .
- Find the solution with and the solution is .
- Periodic boundary condition: Find the solution with and There are infinitely many solutions of the form with any .
- The solution is sought with and there is no solution.
Types of boundary conditions
- Prescribe values of the solution; in the case of an ordinary differential equation defined on the interval one prescribes and and then speaks of Dirichlet boundary conditions .
- Applying conditions to the derivatives , i.e. and prescribing, then one speaks of Neumann boundary conditions (for ordinary differential equations, as explained above, of initial conditions).
- A special case are periodic boundary conditions , here (in the example of an ordinary differential equation considered on the interval ): or .
Artificial boundary conditions
In the case of unrestricted areas, the numerical solution usually requires a restriction of the area. Here, boundary conditions are to be specified that do not exist in the actual problem, i.e. are artificial.
In Business Administration and Economics , the boundary conditions correspond to short or not by the makers influenced data parameters such as the environmental conditions of the weather or the laws .