The definition of a partial differential equation begins with functions of several independent variables. In ordinary differential equations, the unknown function usually depends on one independent variable. In partial differential equations, the unknown function depends on two or more independent variables, and the equation contains one or more partial derivatives of that unknown function. We shall use this page to define the object carefully, introduce PDE notation, and explain what it means for a function to be a solution.

A differential equation that contains, in addition to the dependent variable and the independent variables, one or more partial derivatives of the dependent variable is called a partial differential equation. In general, a partial differential equation may be written in the form $$\begin{equation*} f(x,y,\ldots,u,u_x,u_y,\ldots,u_{xx},u_{xy},\ldots)=0.\quad \text{(1)} \end{equation*}$$ Here $x,y,\ldots$ are independent variables, $u$ is the dependent variable, and $u_x,u_y,\ldots,u_{xx},u_{xy},\ldots$ denote partial derivatives of $u$.