Linear equations of second order
Classification [ edit ] Notation [ edit ] When writing PDEs, it is common to denote partial derivatives using subscripts. For example: � � = ∂ � ∂ � , � � � = ∂ 2 � ∂ � 2 , � � � = ∂ 2 � ∂ � ∂ � = ∂ ∂ � ( ∂ � ∂ � ) . In the general situation that u is a function of n variables, then u i denotes the first partial derivative relative to the i -th input, u ij denotes the second partial derivative relative to the i -th and j -th inputs, and so on. The Greek letter Δ denotes the Laplace operator ; if u is a function of n variables, then Δ � = � 11 + � 22 + ⋯ + � � � . In the physics literature, the Laplace operator is often denoted by ∇ 2 ; in the mathematics literature, ∇ 2 u may also denote the Hessian matrix of u . Equations of first order [ edit ] Main article: First-order partial differential equation Linear and nonlinear equations ...