DirichletCondition

DirichletCondition[beqn,pred]
represents a Dirichlet boundary condition given by equation beqn, satisfied on the part of the boundary of the region given to NDSolve where pred is True.

DetailsDetails

  • DirichletCondition is used together with differential equations to describe boundary conditions in functions such as NDSolve.
  • In NDSolve[eqns,{u1,u2,},{x1,x2,}Ω], are the independent variables, are the dependent variables, and Ω is the region with boundary Ω.
  • Locations where Dirichlet conditions might be specified are shown in blue. They appear (in light blue) on the boundary Ω of the region Ω and also possibly (in dark blue) on interior boundaries of Ω, and they specify that solution values at those points satisfy the condition beqn.
  • DirichletCondition expressions should be included with the equations eqns.
  • Any logical combination of equalities and inequalities in the independent variables , may be used for the predicate pred.
  • DirichletCondition[u1r,pred] is used to prescribe that values of on the boundary Ω should be r. In general, the boundary equation beqn needs to be affine linear in the dependent variables where and r can depend on any of the independent variables .
  • For time-dependent equations, both beqn and pred may depend on time.
  • Typically at least one Dirichlet-type boundary condition needs to be specified to make the differential equation uniquely solvable. Dirichlet conditions are also called essential boundary conditions.
  • Dirichlet conditions are enforced at each point in the discretization of Ω where pred is True.
  • DirichletCondition[{eqn1,eqn2,},pred] is equivalent to .
  • DirichletCondition[eqn,{pred1,pred2,}] is equivalent to .

ExamplesExamplesopen allclose all

Basic Examples  (1)Basic Examples  (1)

Solve on the unit disk with Dirichlet boundary condition :

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Specify multiple Dirichlet conditions for and for :

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Introduced in 2014
(10.0)