Boundary value problem

Boundary value problems (short: RWP ) and the boundary value problem (in short: RWA) or English boundary value problem- (short: BVP ) is called in mathematics an important class of problems, in particular where the solutions to a given differential equation ( ODE ) sought which on the edge of the domain of given function values ​​( boundary condition ) should accept. The counterpart to this is the initial value problem, in which only values ​​at an initial time can be specified.

  • 2.1 Dirichlet problem
  • 2.2 Neumann problem
  • 2.3 skewness boundary condition

Ordinary differential equation

Dirichlet problem

Let and be real numbers. Boundary data or boundary conditions of a function of the form

Hot boundary conditions of the first kind or Dirichlet boundary conditions. If we speak of homogeneous Dirichlet boundary conditions. Otherwise we speak of inhomogeneous boundary conditions.

So looking for is a function which is solution of the following problem:

This is a prescribed function and are the prescribed boundary conditions. Sufficient conditions for the existence ( and uniqueness) of solutions can be found in the article Dirichlet problem.

Sturm- Liouville RWP

Be   is a self-adjoint linear differential operator of 2nd order Boundary operators with were

Called Sturm-Liouville - RWP.

Sturm- Liouville EWP

Those for whom is not uniquely solvable, hot eigenvalues. The corresponding solutions are called eigenfunctions.

Partial Differential Equations

Be open and bounded, is a Lebesgue measurable function on, describing the boundary specifications. We are looking for solutions, respectively. The partial differential equation is given by the differential operator. In particular, elliptic differential operators always lead to boundary value problems, such as the Laplace operator on the Poisson equation.

Dirichlet problem

When Dirichlet problem function values ​​are prescribed on the boundary.

Neumann problem

Rather than function values ​​derivative values ​​are required when Neumann problem.

Leaning boundary condition

The oblique boundary condition is a combination of the two foregoing problems dar. Here, the unknown function on the boundary equal to its normal derivative on the boundary should be.

Aid

An important theoretical tool for the study of boundary value problems are the Green's functions.

In the numerical analysis, as a method for the approximate solution, eg FDM (finite difference method), the FEM (finite element method ), the shooting method and the multi-objective method used.

Scientific Application

The modeling of many processes in nature and technology is based on differential equations. Typical simple examples of RWP are

  • Vibrating string, at its two ends ( = edge ) is firmly clamped
  • Vibrating membrane ( the edge here is a circular ring )
  • Equations of motion of satellites in Keplerian orbits, see also orbit determination.
Linear map Differentialoperator Elliptic operator Laplace-Operator Neumann boundary condition Acoustic membrane Kepler orbit
672144
de