stylistic:style_transfer
Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form where A, B, C, D, E, F, and G are functions of x and y and where u x = ∂ u ∂ x , u x y = ∂ 2 u ∂ x ∂ y and similarly for u x x , u y , u y y . A PDE written in this form is elliptic if with this naming convention inspired by the equation for a planar ellipse. Equations with B 2 − A C = 0 are termed parabolic while those with B 2 − A C > 0 are hyperbolic. The simplest examples of elliptic PDEs are the Laplace equation, Δ u = u x x + u y y = 0 , and the Poisson equation, Δ u = u x x + u y y = f . In a sense, any other elliptic PDE in two variables can be considered to be a generalization of one of these equations, as it can always be put into the canonical form through a change of variables.