# 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 .

# Details  • 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].

# Examples

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## Basic Examples(2)

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

 In:= Compare the eigenvalues with the expected analytical eigenvalues:

 In:= Out= Visualize the eigenfunctions:

 In:= Out= Inspect to see that the bounds are periodic:

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

 In:= In:= Visualize the solution:

 In:= Out= Visualize the periodic solution:

 In:= Out= ## Neat Examples(1)

Introduced in 2016
(11.0)