# 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 and related functions where pred is True.

# Details  • DirichletCondition is used together with differential equations to describe boundary conditions in functions such as DSolve, NDSolve, DEigensystem, NDEigensystem, and GreenFunction.
• 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 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 x1, may be used for the predicate pred.
• DirichletCondition[u1r,pred] is used to prescribe that values of ui on the boundary Ω should be r. In general, the boundary equation beqn needs to be affine linear in the dependent variables, i.e. h1 u1+r, where hi and r can depend on any of the independent variables {x1,x2,}.
• 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[eqn1,pred],DirichletCondition[eqn2,pred],}.
• DirichletCondition[eqn,{pred1,pred2,}] is equivalent to {DirichletCondition[eqn,pred1],DirichletCondition[eqn,pred2],}.

# Examples

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## Basic Examples(2)

Solve on the unit disk with Dirichlet boundary condition :

 In:= Out= In:= Out= Specify multiple Dirichlet conditions for and for :

 In:= Out= In:= Out= Solve symbolically on the unit disk with Dirichlet boundary condition :

 In:= Out= In:= Out= ## Possible Issues(3)

Introduced in 2014
(10.0)
|
Updated in 2016
(11.0)