Boundary condition

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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. In the problems most frequently considered, only one of the infinitely many solutions of the differential equation satisfies the boundary conditions. In a physical model simulation, the boundary conditions describe the behavior of the simulation at the edges of the simulation region.

There are many kinds of possible conditions, depending on the formulation of the problem, number of variables involved, and (crucially) the mathematical nature of the equation. Conditions imposed at a time t = 0 are called initial conditions. One may also impose limiting conditions, for example under the limit of time t → +∞. Conditions in problems with a physical science origin usually match what is expected to determine a unique, well-defined physical situation.

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. Another example from this kind is a whip—a string held fixed at one end, with the other end free to vibrate. Physical analysis of the loose end implies that the appropriate boundary condition is that the solution's derivative at this end is equal to 0 all time.

The general picture is of a boundary (in one or several parts) where solutions are specified. For partial differential equations boundary conditions are usually defined on a continuous perimeter or surface, rather than at discrete points.

Famous in potential theory (typical of elliptic PDEs) 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.

See also

Bibliography

  • A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, Chapman & Hall/CRC Press, Boca Raton, 2003 (2nd edition). ISBN 1-58488-297-2
  • A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002. ISBN 1-58488-299-9ja:境界条件

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