Wolfram Language & System 11.0 (2016)|Legacy Documentation

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PeriodicBoundaryCondition

PeriodicBoundaryCondition[u[x1,],pred,f]
represents a periodic boundary condition for all xtarget on the boundary of the region given to NDSolve where pred is True.

PeriodicBoundaryCondition[a+b u[x1,],pred,f]
represents a generalized periodic boundary condition .

DetailsDetails

  • PeriodicBoundaryCondition is used together with differential equations to describe boundary conditions in functions such as NDSolve.
  • In NDSolve[eqns,{u1,u2,},{x1,x2,}Ω], xi are the independent variables, uj are the dependent variables, and Ω is the region with boundary Ω.
  • Locations where periodic boundary conditions might be specified are shown in blue. They appear on the boundary Ω of the region Ω and specify the relation of the solution at those locations to the locations shown in green. The function f maps from the blue locations to the green locations.
  • In the special case of rectangular region Ω, a boundary equation u[,xi,min,]u[,xi,max,] is taken to be equivalent to PeriodicBoundaryCondition[u[,xi,],xixi,max,f] with f=TranslationTransform[{,0,xi,min-xi,max,0,}].  »
  • Any logical combination of equalities and inequalities in the independent variables x1, may be used for the predicate pred.
  • In PeriodicBoundaryCondition[a+b u[x1,],pred,f], for any point xtarget in the part of Ω where pred is True, then xsource=f[xtarget] should be a point in Ω where pred is not True.
  • With PeriodicBoundaryCondition[a+b u[x1,],pred,f], in NDSolve the system matrices are modified so that the solution values u[xtarget] approximately satisfy u[xtarget]==a+b u[xsource] for all xtarget on the boundary of Ω where pred is True.
  • In PeriodicBoundaryCondition[a+b u[x1,],pred,f], both a and b are scalar values that may depend on any of the independent variables xi, including time.
  • Antiperiodic boundaries may be specified using PeriodicBoundaryCondition[-u[x1,],pred,f].

ExamplesExamplesopen allclose all

Basic Examples  (2)Basic Examples  (2)

Find 5 eigenvalues and eigenvectors of a Laplacian with periodic boundary conditions:

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Compare the eigenvalues with the expected analytical eigenvalues:

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Visualize the eigenfunctions:

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Inspect to see that the bounds are periodic:

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Solve a Poisson equation with periodic boundary conditions on curved boundaries:

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Visualize the solution:

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Introduced in 2016
(11.0)