or 4 and the variables denote coordinates in space dimensions, whereas represents the time variable. In this case, we usually replace by the most common notation: . Further, we shall use the subscript notation for the partial derivatives, viz.
A more general notation for partial derivatives of a sufficiently smooth function (see Definition 1.1 below) is given by
where , denotes the partial derivative of order with respect to the variable , and is a multi‐index of integers with .
Definition 1.1
A function of one real variable is said to be of class on an open interval if its derivatives exist and are continuous on . A function of real variables is said to be of class on a set if all of its partial derivatives of order , i.e. with the multi‐index and , exist and are continuous on .
As we mentioned, a key defining property of a PDE is that there are derivatives with respect to more than one independent variable and a PDE is a relation between an unknown function and its partial derivatives:
(1.3.1)
Example 1.10
The one‐space dimensional, homogeneous, heat, and wave equations (here ) are among the simplest PDEs:
Other examples are
The most general PDE of first order in two independent variables, and is of the form
(1.3.2)
Likewise, the most general PDE of second order in two independent variables can be written as
(1.3.3)
As stated in Remark 1.2, when Eqs. (1.3.1)–(1.3.3) are considered in bounded domains , in order to obtain a unique solution one should supply boundary conditions: conditions at the boundary of the domain , denoted, e.g. by or (as well as conditions for , initial conditions; denoted, e.g. by or ; in the time‐dependent cases), as in the theory of ODEs. and are expressions of and its partial derivatives, stated on the whole or a part of the boundary of (or, in case of , for ), and are associated with the underlying PDE. Below we shall discuss the choice of relevant initial and boundary conditions for a PDE.
A solution of a PDE of type (1.3.1)–(1.3.3) is a function that identically satisfies the corresponding PDE, and the associated initial and boundary conditions, in some region of the variables , or (and ). Note that a solution of an equation of order has to be
