boundary condition

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. ${\ displaystyle u}$ ${\ displaystyle D}$${\ displaystyle D}$

In the simplest case is an interval and the boundary conditions are given function values . ${\ displaystyle D = (a, b)}$ ${\ displaystyle u (a) = c_ {1}; \; u (b) = c_ {2}}$

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 . ${\ displaystyle a}$${\ displaystyle u}$${\ displaystyle u}$

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 .

example

Let be the given differential equation . The solution set to this equation is . ${\ displaystyle y '' (x) = - y (x)}$${\ displaystyle a \ sin (x) + b \ cos (x)}$

• Find the solution with and the solution is .${\ displaystyle y (0) = 1}$${\ displaystyle y (\ pi / 2) = 0}$ ${\ displaystyle \ Rightarrow}$${\ displaystyle y = \ cos (x)}$
• Periodic boundary condition: Find the solution with and There are infinitely many solutions of the form with any .${\ displaystyle y (0) = 0}$${\ displaystyle y (\ pi) = 0}$ ${\ displaystyle \ Rightarrow}$${\ displaystyle a \ sin (x)}$${\ displaystyle a}$
• The solution is sought with and there is no solution.${\ displaystyle y (0) = 0}$${\ displaystyle y (2 \ pi) = 1}$ ${\ displaystyle \ Rightarrow}$

Types of boundary conditions

There are different ways of prescribing values on the edge of the area under consideration :

• 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 .${\ displaystyle \ left [a, b \ right]}$${\ displaystyle y (a)}$${\ displaystyle y (b)}$
• 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).${\ displaystyle y ^ {\ prime} (a)}$${\ displaystyle y ^ {\ prime} (b)}$
• A special case are periodic boundary conditions , here (in the example of an ordinary differential equation considered on the interval ): or .${\ displaystyle \ left [a, b \ right]}$${\ displaystyle y (a) = y (b)}$${\ displaystyle y ^ {\ prime} (a) = y ^ {\ prime} (b)}$

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.

Economics

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 .