Boundary condition

In mathematics, boundary conditions are imposed on the solutions of ordinary differential equations and partial differential equations, to fit the solutions to the actual problem. Usually the appropriate solution is unique, while the set of all solutions is infinite. There are many kinds of possible condition, depending on the formulation of the problem, number of variables involved, and (crucially) the mathematical nature of the equation.

For example when a vibrating string is modelled, we assume that the two ends are held fixed: this accords with physical intuition. With the function to be found representing the displacement as function of position on the string, this implies the solution should take the value 0 at two points through all time.

The general picture is of a boundary (in one or several parts) where solutions are specified.

Famous in potential theory (an elliptic PDE) are the Dirichlet and Neumann boundary conditions, on a boundary enclosing a compact region. For a wave (hyperbolic) PDE one assumes waves propagate from an initial disturbance along some surface.

There are very many types of possible conditions.

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