Boundary value problem

A boundary value problem asks for a function satisfying a differential equation and conditions imposed at the boundary of its domain. Its solutions are constrained by data at multiple boundary points or across a boundary.

Differential equation: An equation relating an unknown function to one or more of its derivatives. The equation supplies the relation that a boundary value problem's unknown function must satisfy.
Green's function: A kernel that represents solutions of a linear differential equation in response to a point source. It constructs solutions from forcing while incorporating the problem's boundary conditions.
Dirichlet problem: A boundary value problem that prescribes a function's values along the boundary. It is the standard formulation when the boundary values themselves are specified.
Electrostatics: The study of electric charges and fields that are stationary or change slowly enough to be treated as static. Electric potentials are often found by solving Poisson's or Laplace's equation with boundary data.
Cauchy problem: A differential equation problem specifying initial data for the unknown function and its derivatives. It emphasizes data on an initial surface, unlike conditions imposed around a domain's boundary.
Boundary condition: A constraint on a solution or its derivatives at the boundary of a domain. Boundary conditions specify the data that distinguish admissible solutions.
Shooting method: A numerical method that converts a boundary value problem into repeated initial value problems. It adjusts unknown initial data until the computed solution meets the far boundary condition.
Neumann problem: A boundary value problem that prescribes the normal derivative of a function along the boundary. It specifies boundary flux or slope rather than the function's value.
Heat conduction: The transfer of thermal energy through matter by microscopic interactions. Fixed temperatures or heat fluxes at surfaces become boundary conditions for temperature equations.
Periodic boundary condition: A condition requiring a function and often its derivatives to match across opposite ends of a domain. It identifies boundary values across a domain rather than prescribing independent endpoint values.

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