ELLIPTIC EQUATION

any of a class of partial differential equations describing phenomena that do not change from moment to moment, as when a flow of heat or fluid takes place within a medium with no accumulations. The Laplace equation, uxx + uyy = 0, is the sim plest such equation describing this condition in two dimensions. In addition to satisfying a differential equation within the region, the elliptic equation is also determined by its values (boundary values) along the boundary of the region, which represents the effect from outside the region. These conditions can be either those of a fixed temperature distribution at points of the boundary (Dirichlet problem) or those in which heat is being supplied or removed across the boundary in such a way as to maintain a constant temperature distribution throughout (Neumann problem). If the highest-order terms of a second-order partial differential equation with constant coefficients are linear and if the coefficients a, b, c of the uxx, uxy, uyy terms satisfy the inequality b2 - 4ac