System of coupled partial differential equation operators op j, Neumann boundary values Γ N j and Dirichlet and periodic boundary condition equations Γ D and Γ P DirichletCondition may also be given in a PDE equation as well. Making a NeumannValue part of a PDE equation solves this problem without ambiguity. It is not possible to derive (unambiguously) from the Neumann value with which PDE equation the value should be associated. One would like to be able to unambiguously specify any given Neumann value to any given single PDE of that system of PDEs. Neumann values are mathematically tied to the PDE.įor practical reasons, in NDSolve and related functions, NeumannValue needs to be given as a part of the equation. Generalized Neumann values, on the other hand, are specified by giving a value, since the equation satisfied is implicit in the value. Dirichlet boundary conditions are specified as equations. They can be specified independently of the equation. In most cases, Dirichlet boundary conditions need not be associated with a particular equation. Other boundary conditions are conceivable, but currently not implemented. Periodic boundary conditions make the dependent variables behave according to a given relation between two distinct parts of the boundary. Instead of making use of integration by parts to obtain equation (11), the divergence theorem and Green's identities can also be used. Specifies the value of the boundary integral integrand (11) in the weak form and thus the name NeumannValue.
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