An equation containing the partial derivative (or partial differential) of an unknown function.The highest order of the partial derivative of the unknown function in the equation is called the order of the equation.The second order partial differential equations are the most widely used in mathematics, physics and engineering technology. These equations are customarily called mathematical physical equations.[1]
The physical quantity of the objective world generally changes with time and space position, so it can be expressed as time coordinate t and space coordinate
Function of
The change law of this physical quantity is often shown in its order of time and space coordinatesRate of changeThe relationship between, that is, the function u with respect to t and
The equality between the partial derivatives of order.
For example, in a uniform heat transfer object, if the generating term of internal heat source is ignored, the temperature u will meet the following equation:
。(1)
Such a class includes unknown functions andpartial derivativeThe equation of is called partial differential equation.Generally speaking, if
Is the independent variable, and the general form of the partial differential equation with u as the unknown function is:
。(2)
Here F is the function of its argument,
The highest order of the included partial derivative is called the order of the partial differential equation.
The system of equations composed of several partial differential equations is calledPartial differential equations, 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 underdetermined.
If a partial differential equation (system) is related to all unknown functions andderivativesAre linear, then calledLinear partial differential equation(group).Otherwise, it is calledNonlinear partial differential equation(group).In nonlinear partial differential equations (systems), if the highest derivative of an unknown function is linear, it is called quasilinear partial differential equations (systems).
Let Ω be a region in the space of independent variables R, and u be a function defined on this region with continuous derivatives of order | α |.If it can make equation (2) identical on Ω, then u is said to be a classical solution of the equation in Ω, which is called the classical solution for short.Without misunderstanding, it is called solution.
The theory of partial differential equations studies whether an equation (system) has a solution that satisfies some supplementary conditions（Existence of solutions）, how many solutions (uniqueness or degree of freedom of the solution), various properties of the solution, solution methods, etc., and also try to use partial differential equations to explain and predict natural phenomena and apply it to various sciences and engineering technologies.The formation and development of the theory of partial differential equations are closely related to the development of physics and other natural sciences, and promote and promote each other.The development of other mathematical branches, such as analysis, geometry, algebra, topology and other theories, also has a profound impact on partial differential equations.[2]
In the process of rapid development of science and technology, it is not enough to describe many problems that people study with a function of independent variable. Many problems have multiple variable functions to describe.
For example, from the physical point of view, physical quantities have different properties. Temperature, density, etc. are described numerically as pure quantities;Velocity, the gravitational force of electric field, etc., not only have different values, but also have directions. These quantities are called vectors;The quantity described by the tension state of an object at a point is calledtensor, etc.These quantities are not only related to time, but also related to space coordinates, which requires a function of multiple variables.
It should be pointed out that all possible physical phenomena can only be idealized by some multi variable functions, such as the density of the medium. In fact, the density "at one point" does not exist.And we regard the density at a point as the ratio of the mass and volume of the material when the volume is infinitely reducedlimitThis is idealized.The temperature of the medium is the same.In this way, the ideal multi variableFunctional equationThis equation is called partial differential equation.
origin
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CalculusThe discipline of equations came into being in the 18th century,EulerIn his work, he first proposed the second order equation of string vibration, and soon afterwards, French mathematiciansDarumbelHe also put forward special partial differential equations in his book On Dynamics.However, these works did not attract much attention at that time.
In 1746, in his paper Research on Curves Formed when Tensioned Strings Vibrate, d'Alembert proposed to prove that an infinite number of curves different from sinusoidal curves are vibration modes.Thus, the study of string vibration created the discipline of partial differential equations.
Daniel Bernoulli, a Swiss mathematician at Euler's time, also studiedmathematical physics The general method of solving the vibration problem of elastic system is proposed, which has a great influence on the development of partial differential equations.LagrangeAlso discussedFirst order partial differential equation, enriching the content of this discipline.
The rapid development of partial differential equations was in the 19th century, when the research of mathematical physics problems flourished, and many mathematicians made contributions to the solution of mathematical physics problems.
French mathematicians should be mentioned hereFourier He was an outstanding mathematical scholar when he was young.In the research of heat flow《Analytic theory of heat》, in which he proposedthree-dimensional spaceThe heat equation of is a kind of partial differential equation.His research has a great influence on the development of partial differential equations.
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Second order linear and nonlinear partial differential equations are always important research objects.
These equations are usually divided into three types: elliptic, hyperbolic and parabolic. The basic problems established and discussed around these three types of equations are the existence, uniqueness, stability and asymptotic properties of solutions to various boundary value problems, initial value problems and mixed problems, as well as the solution methods.
The development of modern physics, mechanics and engineering technology has produced many new nonlinear problems, which often lead to other related problems besides the above equations, such as mixed equation, degenerate equation and high-order partial differential equation. These problems are usually very complex and difficult. So far, they have been important research topics.
For the discussion and solution of partial differential equations, it is often necessary to apply the theories and methods of functional analysis, algebra and topology, differential geometry and other mathematical branches.On the other hand, due to the rapid development of electronic computers, all kinds of equations can be solved numerically, and many important facts have been revealed. Therefore, the research on numerical solutions will develop faster on the basis of many important achievements.[3]
Physical string vibration
String vibration is a mechanical movement. Of course, the basic law of mechanical movement is F=ma of particle mechanics, but string is not a particle, so the law of particle mechanics is not applicable to the study of string vibration.However, if we divide the string into several minimax segments, and each segment is abstractly regarded as a particle, then we can apply the basic laws of particle mechanics.
Strings refer to thin and long elastic materials. For example, strings used in string instruments are thin, soft and elastic.When playing, the string is always tight with a tension that is tens of thousands of times greater than the weight of the string.When the player pokes with a thin piece or pulls on the string with a bow, although only a section of string that he touches vibrates, it spreads to make the whole string vibrate due to the effect of tension.
usedifferentialIt can be obtained that the displacement of a point on the chord is a partial differential equation with the position and time of this point as independent variables, which belongs to the mathematical physics equationwave equation , that isHyperbolic partial differential equation。
There are generally infinite solutions to partial differential equations, but when solving specific physical problems, you must select the required solutions from them, so you must also know the additional conditions.Because partial differential equations are expressions of the common law of the same kind of phenomena, it is not enough to grasp and understand the particularity of specific problems only to know this common law, so in terms of physical phenomena, the particularity of each specific problem lies in the specific conditions of the research object, namely, the initial conditions and boundary conditions.
For a stringed instrument with the same string, if one plucks the string with a slice and the other pulls on the string with a bow, the sound they make is different.The reason is that the vibration at the "initial" moment of "pulling" or "pulling" is different, so the subsequent vibration is also different.
Celestial prediction
In astronomy, there are similar situations of string vibration. If you want to predict the movement of celestial bodies through calculation, you must know the mass of these celestial bodiesNewton's lawIn addition to the general formula of, we must also know the initial state of the celestial system we are studying, that is, at a certain starting time, the distribution of these celestial bodies and their velocities.When solving any mathematical physics equation, there will always be similar additional conditions.
Of course, there are still "problems without initial conditions" in the objective reality, such as fixed field problems (electrostatic field, stable concentration distribution, stable temperature distribution, etc.), and "problems without boundary conditions".
Mathematical Application
Mathematically, initial conditions and boundary conditions are called definite solution conditions.
The partial differential equation itself expresses the commonness of the same kind of physical phenomenon, and is the basis for solving problems;The definite solution condition reflects the individuality of the specific problem, and it puts forward the specific situation of the problem.The equation and the definite solution condition are integrated into one, which is calledDefinite solution problem。
To find the definite solution of partial differential equation, we can first find itsgeneral solutionAnd then determine the function with the definite solution condition.But generally speaking, it is not easy to find the general solution in practice, and it is more difficult to determine the function with the definite solution conditions.
The solution of partial differential equation can also be usedSeparation coefficient method, also calledFourier series；You can also use the method of separating variables, also calledFourier transformation Or Fourier integral.The separation coefficient method can solve the definite solution problem in bounded space, and the separation variable method can solve the definite solution problem in unbounded space;It can also be usedLaplace transform Method to solve the definite solution of mathematical and physical equations in one-dimensional space.Applying Laplace transform to the equation can be converted intoordinary differential equation In addition, the initial conditions are also taken into account. After solving the ordinary differential equation, inversion can be carried out.
It should be pointed out that although there are all kinds of solutions above for the definite solution of partial differential equation, we can not ignore that many problems of definite solution can not be solved strictly due to some reasons, and we can only use the approximate method to find the approximate solution that meets the actual needs.
The variational method is to transform the definite solution problem into a variational problem, and then to find the approximate solution of the variational problem;
The finite difference method is to transform the problem of definite solution into algebraic equation, and then calculate it with computer.
There is also a more meaningful simulation method, which uses another physical problem experimental research to replace the definite solution of a physical problem under study.Although physical phenomena are different in nature, they are abstractly expressed as the same definite solution problem in mathematics. For example, the problem of stable temperature distribution in an irregular shape object is mathematicallyLaplace equationBecause it is difficult to solve the boundary value problem of, we can do the corresponding electrostatic field or steady current field experimental research, measure the potential everywhere in the field, and thus solve the problem of temperature distribution in the studied steady temperature field.
along withPhysical ScienceWith the expansion of the phenomenon studied in both breadth and depth, the application of partial differential equations is more extensive.From the perspective of mathematics itself, the solution of partial differential equations promotes the development of mathematics in function theory, variational method, series expansion, ordinary differential equations, algebra, differential geometry and other aspects.From this point of view, partial differential equations become the center of mathematics.
