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

→ Main article : Dirichlet boundary condition

Let and be real numbers. Boundary data or boundary conditions of a function of the form ${\displaystyle \alpha }$ ${\displaystyle \beta }$ ${\displaystyle u\colon [a,b]\to \mathbb {R} }$

${\displaystyle u(a)=\alpha \quad {\text{und}}\quad u(b)=\beta }$

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. ${\displaystyle \alpha =\beta =0}$

So we are looking for a function that is a solution to the following problem: ${\displaystyle u}$

${\displaystyle (N){\begin{cases}f(x,u(x),u'(x),u''(x))=0,\quad x\in (a,b)&\\u(a)=\alpha ,~u(b)=\beta .&\end{cases}}}$

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 . ${\displaystyle f}$ ${\displaystyle \alpha ,\beta }$ ${\displaystyle (N)}$

Storm Liouville RWP

→ Main article : Sturm-Liouville problem

Let be a self-adjoint linear differential operator of the 2nd order boundary operators with let ${\displaystyle r,p,q\in {\mathcal {C}}([a,b],\mathbb {R} )}$
${\displaystyle Lu:=(pu')'+qu}$
${\displaystyle {\alpha _{0}}^{2}+{\alpha _{1}}^{2}>0,~{\beta _{0}}^{2}+{\beta _{1}}^{2}>0}$
${\displaystyle R_{a}u:=\alpha _{0}u(a)+\alpha _{1}p(a)u'(a)}$
${\displaystyle R_{b}u:=\beta _{0}u(b)+\beta _{1}p(b)u'(b)}$

${\displaystyle (*){\begin{cases}(Lu)(x)=r(x)&\\R_{u}(a)=\eta _{a},~R_{u}(b)=\eta _{b}&\end{cases}}}$

is called Sturm-Liouville-RWP.

Storm Liouville EWP

See also : Eigenvalue Problem

${\displaystyle (P_{\lambda }){\begin{cases}(Lu)(x)=\lambda u(x)&\\R_{u}(a)=R_{u}(b)=0&\end{cases}}}$

Those for which it is not clearly solvable are called eigenvalues . The corresponding solutions are called eigenfunctions. ${\displaystyle \lambda \in \mathbb {R} }$ ${\displaystyle (P_{\lambda })}$

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 . ${\displaystyle \Omega \subset \mathbb {R} ^{d}}$ ${\displaystyle f}$ ${\displaystyle \Omega }$ ${\displaystyle g}$ ${\displaystyle u\colon \Omega \rightarrow \mathbb {R} ^{n}}$ ${\displaystyle L\colon u\mapsto L(u)}$

Dirichlet problem

With the Dirichlet problem , function values ​​are specified on the edge.

${\displaystyle L(u)(x)=f(x)}$ For ${\displaystyle x\in \Omega ,}$

${\displaystyle u(x)=g(x)}$ For ${\displaystyle x\in \partial \Omega .}$

Neumann problem

Instead of functional values, derivative values ​​are prescribed for the Neumann problem .

${\displaystyle L(u)(x)=f(x)}$ For ${\displaystyle x\in \Omega ,}$

${\displaystyle {\frac {\partial u}{\partial n}}(x)=g(x)}$ For ${\displaystyle x\in \partial \Omega .}$

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.

${\displaystyle L(u)(x)=f(x)}$ For ${\displaystyle x\in \Omega ,}$

${\displaystyle u(x)={\frac {\partial u}{\partial n}}(x)}$ For ${\displaystyle x\in \partial \Omega .}$

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 .