The direct current conduction problems, such as electrolysis and computation of resistances of grounding plates, involve a steady current passing through a conductive medium. Current density J is related to the electric field E as J = σE, where σ is the conductivity of the medium. Combining the continuity equation ∇ · J = Q, where Q is the current source, with the definition of the electric potential V yields the elliptic Poisson's equation:
–∇ · (σ∇V) = Q.
Use the Dirichlet boundary condition to assign values of the electric potential V to the boundaries. Typically, the boundaries are metallic conductors. Use the Neumann boundary conditions if the value of the normal component of the current density (n · (σ∇V)) is known. You also can specify a generalized Neumann condition defined by n · (σ∇V) + qV = g, where q is interpreted as film conductance for thin plates. For details, see Conductive Media DC.
|PDE||Solve partial differential equations in 2-D regions|
Solve the Laplace equation for a geometry consisting of two circular metallic conductors placed on a plane.
Solve a simple elliptic PDE in the form of Poisson's equation on a unit disk.
Solve a nonlinear problem on a unit disk using the PDE app and the command line.