A differential equation involving more than one independent variable and its (resp. their) partial derivatives with respect to those variables is called a partial differential equation (PDE).
Consider a simple PDE of the form:
This equation implies that the function is independent of . Hence the general solution of this equation is , where is an arbitrary function of . The analogous ordinary differential equation is
its general solution is , where is a constant. This example illustrates that general solutions of ODEs involve arbitrary constants, but solutions of PDEs, in contrast, involve arbitrary functions.
In general one can classify PDEs with respect to different criterion, e.g.:
By order of PDE we will understand the order of the highest derivative that occurs. A PDE is said to be linear if it is linear in unknown functions and their derivatives, with coefficients depending on the independent variables. The independent variables typically include one or more space dimensions and sometimes time dimension as well.
Example:
The wave equation
is a one-dimensional, second-order linear PDE. In contrast, the Fisher Equation of the form
where is a two-dimensional, second-order nonlinear PDE.
Gurevich_Svetlana 2008-11-12