The highest order of the partial derivative of an unknown function in an equation is called the order of the equation.

Second-order partial differential equations are widely used in mathematics, physics and engineering technology, and are traditionally called mathematical physical equations.

The physical quantity of the objective world generally changes with time and space position, so it can be expressed by time coordinate T and space coordinate.

Function of

The change law of this physical quantity is often expressed by time and space coordinates, that is, the function u is about t and.

The equation between the partial derivatives of.

For example, in a uniform heat transfer object, the temperature u satisfies the following equation:

( 1)

Such an equation containing an unknown function and its partial derivative is called a partial differential equation. Generally speaking, if

Is an independent variable, and the general form of a partial differential equation with u as an unknown function is

(2)

Where f is a function of its independent variable,

The highest order number of partial derivatives is called the order of partial differential equations.

An equation group composed of several partial differential equations is called a partial differential equation group, and its unknown functions can also be several. When the number of equations exceeds the number of unknown functions, the partial differential equations are said to be overdetermined; When the number of equations is less than the number of unknown functions, it is called underdetermination.

If a partial differential equation (group) is linear with respect to all unknown functions and their derivatives, it is called a linear partial differential equation (group). Otherwise it is called nonlinear partial differential equation (group). In a nonlinear partial differential equation (group), if the highest derivative of an unknown function is linear, it is called a quasi-linear partial differential equation (group).

Let Ω be a region in the independent variable space r, and u be a function whose derivative of order | α| defined in this region is continuous. If equation (2) can be made equal on ω, then it is said that U is the classical solution of the equation on ω, which is called classical solution for short. If there is no misunderstanding, it is called solution.

The theory of partial differential equations studies whether an equation (group) has a solution (existence of solution), how many solutions (uniqueness or degree of freedom of solution), and various properties of solution and so on. And try to explain and predict natural phenomena with partial differential equations as much as possible, and apply them to various sciences and engineering technologies. The formation and development of partial differential equation theory are closely related to the development of physics and other natural sciences, and promote each other. The development of other branches of mathematics, such as analysis, geometry, algebra and topology, has also had a far-reaching impact on partial differential equations.

Today, with the rapid development of science and technology, it is not enough to describe many problems studied by people with a function of independent variables, and many problems are described with multiple functions.

For example, from a physical point of view, physical quantities have different properties, such as temperature and density. Described by a numerical value called a scalar; Speed, electric field attraction, etc. , not only has different values, but also has a direction. These quantities are called vectors. The quantity described by the tension state of an object at a point is called tensor, and so on. These quantities are not only related to time, but also to spatial coordinates, and should be expressed by functions of multivariate variables.

It should be pointed out that all possible physical phenomena can only be expressed by functions of some multivariate variables, such as the density of media, but in fact there is no density at all. And we regard the density at a point as the limit of the mass-volume ratio of matter when the volume is infinitely reduced, which is idealized. So is the temperature of the medium. In this way, an ideal multivariate function equation for studying some physical phenomena is produced, which is a partial differential equation.