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Encyclopedia > 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. In the problems most frequently considered, only one of the infinitely many solutions of the differential equation satisfies the boundary conditions.

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.

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.

In partial differential equations boundary conditions are usually defined on a continuous perimeter or surface, rather than at discrete points. Results from FactBites:

 Boundary Condition File (971 words) Defaults are applied based on algorithms that detect the topological proximity of a wall to the irregular flow walls (levees, domain boundaries, etc.) The defaults provide for regular free-flow through the interior of the domain, direct flow near discontinuities, and no-flow on the domain boundaries. A Specified head boundary for an element is a user supplied input that the model is constrained to force the element head to match. Because of the method of matrix manipulation used to implement this boundary condition, it is not assured to conserve mass.
 Hydrus-2D Tutorials (763 words) Infiltration is initiated with a constant head boundary condition and is maintained for 25 days, with the duration of the solute pulse being 10 days. The drain is represented by a circle to which a seepage face boundary condition is applied. Initial and boundary conditions are specified, and graphical displays of the results using contour and spectrum maps, including animation, are provided, for a more complex transport domain than in the previous example.
More results at FactBites »

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