An equation containing the partial derivative (or partial differential) of an unknown function. 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 as a function of time coordinate t and space coordinate. The change law of this physical quantity is often manifested in the relationship between its various change rates about time and space coordinates, that is, the function u about t.
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.