## What are Boundary Conditions?

Differential equations have many solutions and it’s usually impossible to find them all. To narrow down the set of answers from a family of functions to a particular solution, conditions are set. These conditions can be initial conditions (which define a starting point at the extreme of an interval) or **boundary conditions **(which define bounds that constrain the whole solution). Different types of boundary conditions can be imposed on the boundary of a domain.

One way to think of the difference between the two is that initial conditions deal with time, while boundary conditions deal with space. Boundaries can describe all manner of shapes: e.g. triangles, circles, polygons.

## Types of Boundary Conditions

The **five types of boundary conditions** are:

- Dirichlet (also called Type I),
- Neumann (also called Type II, Flux, or Natural),
- Robin (also called Type III),
- Mixed,
- Cauchy.

Dirichlet and Neumann are the most common.

**Dirichlet**: Specifies the function’s value on the boundary. For example, you could specify Dirichlet boundary conditions for the interval domain [a, b], giving the unknown at the endpoints a and b. For two dimensions, the boundary conditions stretch along an entire curve; for three dimensions, they must cover a surface. This type of problem is called a*Dirichlet Boundary Value Problem.*.**Neumann**: Similar to the Dirichlet, except the boundary condition specifies the derivative of the unknown function. For example, we could specify u′(a) = α which imposes a Neumann boundary condition at the right endpoint of the interval domain [a, b].**Robin**: A weighted combination of the function’s value and its derivative. For example, for unknown u(x) on the interval domain [a, b] we could specify the Robin condition u(a) −2u′(a) = 0.**Mixed**: Similar to the Robin, except that parts of the boundary are specified by different conditions. For example, on the interval [a, b], the unknown u′(x) at x = a could be specified by a Neumann condition and the unkownn u(x) at x = b could be specified by a Dirichlet condition. [1]**Cauchy**: Similar to the Robin, except that while the Robin condition implies only one constraint, the Cauchy condition implies two.

**Homogeneous boundary conditions** are set to zero; Otherwise they are called *inhomogeneous*.

## Examples

When solving a differential equation, the values for the conditions depend on the problem you’re trying to solve.

As an example, let’s say you wanted to find the equation for a straight line on a curve-length function between two points (a, A) and (b, B). The function could be set up as with the points as boundary conditions [2]:

Where y′ = a constant.

Watch the following video from MIT courseware that overviews boundary conditions and shows two more examples:

## References

[1] Introduction to Boundary Value Problems. Retrieved March 20, 2021 from: https://people.sc.fsu.edu/~jpeterson/bvp_notes.pdf

[2] Open University. Introduction to the calculus of variations.