# 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 ${\ displaystyle \ alpha}$${\ displaystyle \ beta}$${\ displaystyle u \ colon [a, b] \ to \ mathbb {R}}$

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

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

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

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

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