boundarycondition

Boundary condition is a constraint required to determine a solution to a differential equation defined on a domain. It prescribes the behavior of the unknown function on the boundary ∂Ω or relates the function to its derivatives there. Together with the differential equation, boundary conditions specify a boundary value problem or an initial value problem for ordinary or partial differential equations.

The most common types are Dirichlet, Neumann, and Robin conditions. Dirichlet conditions specify the value of the unknown function on the boundary, u|∂Ω = g. Neumann conditions specify the value of the normal derivative on the boundary, ∂u/∂n|∂Ω = h, representing a prescribed flux. Robin conditions combine the function and its normal derivative in a linear relation, a u + b ∂u/∂n = r on ∂Ω. Periodic boundary conditions require the function and its derivatives to match at corresponding points on opposite parts of the boundary.

In ordinary differential equations, boundary conditions may fix the function value or its derivatives at specific points, turning an equation into a boundary value problem. For partial differential equations, boundary conditions are essential for determining a unique solution and for ensuring stability in numerical methods. The feasibility and properties of solutions often depend on the type of boundary condition and its compatibility with the equation.

Applied contexts include physics, engineering, and applied mathematics. Dirichlet conditions model fixed states such as specified temperatures, Neumann conditions model insulated or flux-controlled boundaries, and Robin conditions model convective or impedance-like interactions. In numerical simulations, boundary conditions must be carefully discretized to preserve accuracy and stability.