Boundary value problem
Boundary value problems (short: RWP ) also boundary value problem (short: RWA ) or English boundary value problem (short: BVP ) is an important class of problems in mathematics in which solutions to a given differential equation (DGL) are sought that are based on the The edge of the definition range should assume predetermined function values ( boundary conditions ). The counterpart to this is the initial value problem , in which the solution is given for any point in the domain.
Ordinary differential equation
Dirichlet problem
Let and be real numbers. Boundary data or boundary conditions of a function of the form
are called boundary conditions of the first kind or Dirichlet boundary conditions. If so we speak of homogeneous Dirichlet boundary conditions. Otherwise we speak of inhomogeneous boundary conditions.
So we are looking for a function that is a solution to the following problem:
Here is a prescribed function and are the prescribed boundary conditions. Sufficient conditions for the existence (and uniqueness) of solutions of can be found in the article Dirichlet problem .
Storm Liouville RWP
Let be a self-adjoint linear differential operator of the 2nd order
boundary operators with let
is called Sturm-Liouville-RWP.
Storm Liouville EWP
Those for which it is not clearly solvable are called eigenvalues . The corresponding solutions are called eigenfunctions.
Partial differential equations
Be open and restricted, be a Lebesgue-measurable function, describe the boundary specifications. Solutions are sought in each case . 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 to the Poisson equation .
Dirichlet problem
With the Dirichlet problem , function values are specified on the edge.
- For
- For
Neumann problem
Instead of functional values, derivative values are prescribed for the Neumann problem .
- For
- For
Skewed boundary condition
The skewed boundary condition represents a combination of the two previous problems. Here, the function sought on the boundary should be equal to its normal derivative on the boundary.
- For
- For
Tools
Green's functions are an important theoretical aid for the investigation of boundary value problems .
In numerics , as a method for approximate solution z. B. the FDM ( finite difference method ), the FEM ( finite element method ), the shooting method and the multi-target method are 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 that is firmly clamped at both ends (= edge)
- vibrating membrane (the edge is a circular ring here)
- Equations of motion of satellites in Kepler orbits , see also orbit determination
- the chain line of a chain hanging between two points or the seabed and the ship
- the shape of the radii of the 3 lamellas that form when 2 independent soap bubbles first pair
- the deformation of a trampoline surface when it bounces on.
Conversely, experiments with material models - made of a spring network, rubber blanket, soap bubble - can be used to solve mathematically formulated tasks or to illustrate them:
- Gravitational potential represented by the central indentation of a rubber blanket clamped horizontally at the edge, (elliptical) circling movement by a small rolling ball
- Tension optics
literature
- Martin Hermann : Numerics of ordinary differential equations. Initial and boundary value problems. Oldenbourg, Munich et al. 2004, ISBN 3-486-27606-9 .