Mathworld-Wiki-bc(s)-in-the-solution-of-O&P-DEs

There are several types of boundary conditions commonly encountered in the solution of partial differential equations (http://mathworld.wolfram.com/DirichletBoundaryConditions.html).

1. Dirichlet boundary conditions specify the value of the function on a surface .
Dirichlet bc(s) is PDE boundary conditions which give the value of the function on a surface, e.g., T=f(r,t).

In mathematics, a Dirichlet boundary condition (often referred to as a first-type boundary condition) imposed on an ODE or a PDE specifies the values a solution is to take on the boundary of the domain. The question of finding solutions to such equations is known as the Dirichlet problem.

In the case of an ODE such as: d2y/dx2 + 3y = 1on the interval [0,1] the Dirichlet boundary conditions take the formy(0) = α1 y(1) = α2 where α1 and α2 are given numbers.

For a PDE on a domain Ω C Rn such as: Δy + y = 0 (Δ denotes the Laplacian), the Dirichlet boundary condition takes the formy(x) = f(x) all x C ∂Ωwhere f is a known function defined on the boundary ∂Ω.
Dirichlet boundary conditions are perhaps the easiest to understand but there are many other conditions possible.

2. Neumann boundary conditions specify the normal derivative of the function on a surface,

Neumann bc(s)s is PDE boundary conditions which give the normal derivative on a surface.

In mathematics, a Neumann boundary condition (named after Carl Neumann) imposed on an ODE or a PDE specifies the values the derivative of a solution is to take on the boundary of the domain.
In the case of an ordinary differential equation, for example such as
d2y/dx2 + 3y = 1
on the interval [0,1], the Neumann boundary condition takes the form
dy(0)/dx=α1

dy(1)/dx=α2

where α1 and α2 are given numbers.
For a partial differential equation on a domain
Ω C Rn
for example : Δ2 y +y =0 (Δ2 denotes the Laplacian), the Neumann boundary condition takes the form: dy(x)/dv = f(x), all x E ∂Ω
Here, 'del' denotes the (typically exterior) normal to the boundary ∂Ω and f is a given scalar function. The normal derivative which shows up on the left-hand side is defined as
dy(x)/dv = ∇ y(x) dot v(x)
where ∇ is the gradient and the dot is the inner product.

3. Cauchy boundary conditions specify a weighted average of first and second kinds.

Bc(s) of a PDE which are a weighted arithmetic mean of Dirichlet boundary conditions (which specify the value of the function on a surface) and Neumann boundary conditions (which specify the normal derivative of the function on a surface).

4. Robin boundary conditions.
Bc(s) of a PDE which, for an elliptic PDE in a region R, Robin bc(s) specify the sum of alpha*u and the normal derivative of u = f at all points of the boundary of R, with alpha and u being prescribed.
Robin boundary condition is another type of hybrid boundary condition; it is a linear combination of Dirichlet and Neumann boundary conditions.

5. Mixed boundary conditions.
In mathematics, a mixed boundary condition for a PDE indicates that different boundary conditions are used on different parts of the boundary of the domain of the equation.
For example, if u is a solution to a PDE on a set Ω with piecewise-smooth boundary and is divided into two parts, Γ1 and Γ2, one can use a Dirichlet boundary condition on Γ1 and a Neumann boundary condition on Γ2,
u for Γ1 = u0
∂u/∂n for Γ2 = g
where u0 and g are given functions defined on those portions of the boundary.