微分方程

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此词条暂由Yuling翻译,未经人工整理和审校,带来阅读不便,请见谅。 由CecileLi初步审校

模板:Distinguish

文件:Elmer-pump-heatequation.png
Visualization of heat transfer in a pump casing, created by solving the heat equation. Heat is being generated internally in the casing and being cooled at the boundary, providing a steady state temperature distribution.

Visualization of heat transfer in a pump casing, created by solving the heat equation. Heat is being generated internally in the casing and being cooled at the boundary, providing a steady state temperature distribution.

通过求解热力学方程,我们建立了泵壳内传热的可视化模型。热量在内部产生并在边界冷却,从而为整体提供稳定的温度分布。

In mathematics, a differential equation is an equation that relates one or more functions and their derivatives.[1] In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common, therefore differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common, therefore differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

在数学上,微分方程 Differential Equation是可以将一个或多个函数及其导数相互关联的方程。在实际应用中,函数通常代表物理量,导数代表其变化率,而微分方程则定义了两者之间的关系。由于这种关系十分普遍,因此微分方程在包括工程学、物理学、经济学和生物学在内的许多学科中得到了广泛的应用。


Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly.

Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly.

微分方程的研究主要包括对微分方程解(满足每个方程的函数集)及其解的性质的研究。只有最简单的微分方程才能直接用公式求解;然而,有时无需精确计算便可以确定给定微分方程的解的许多性质。


Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.

Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.

一般地,当闭式解不存在时,可以用计算机求方程的近似解。动力系统理论着重于对由微分方程描述的系统进行定性分析。同时,现在已经得出了许多数值方法来计算给定精度下微分方程的解。


History

历史


Differential equations first came into existence with the invention of calculus by Newton and Leibniz. In Chapter 2 of his 1671 work Methodus fluxionum et Serierum Infinitarum,[2] Isaac Newton listed three kinds of differential equations:

Differential equations first came into existence with the invention of calculus by Newton and Leibniz. In Chapter 2 of his 1671 work Methodus fluxionum et Serierum Infinitarum, Isaac Newton listed three kinds of differential equations:

微分方程是在牛顿和莱布尼茨发明微积分后才出现的。艾萨克 · 牛顿在他1671年的著作《无限的循环与系列》的第二章中列出了三种微分方程:


[math]\displaystyle{ \begin{align} & \frac {dy}{dx} = f(x) \\[5pt] & \frac {dy}{dx} = f(x,y) \\[5pt] & x_1 \frac {\partial y}{\partial x_1} + x_2 \frac {\partial y}{\partial x_2} = y \end{align} }[/math]

In all these cases, y is an unknown function of x (or of [math]\displaystyle{ x_1 }[/math] and [math]\displaystyle{ x_2 }[/math]), and f is a given function.

In all these cases, y y is an unknown function of (or of [math]\displaystyle{ x_1 }[/math] and [math]\displaystyle{ x_2 }[/math]), and is a given function.

在这些例子中,y是自变量 x(或者是[math]\displaystyle{ x_1 }[/math][math]\displaystyle{ x_2 }[/math])的未知函数,并且 f 是一个给定的函数。


He solves these examples and others using infinite series and discusses the non-uniqueness of solutions.

He solves these examples and others using infinite series and discusses the non-uniqueness of solutions.

他利用无穷级数来求解这些以及其他例子,并讨论了解的非唯一性。


Jacob Bernoulli proposed the Bernoulli differential equation in 1695.[3] This is an ordinary differential equation of the form

Jacob Bernoulli proposed the Bernoulli differential equation in 1695. This is an ordinary differential equation of the form

雅可比·伯努利在1695年提出了伯努利微分方程。这种方程是常微分方程 Ordinary Differential Equation的一种形式,


[math]\displaystyle{ y'+ P(x)y = Q(x)y^n\, }[/math]


for which the following year Leibniz obtained solutions by simplifying it.[4]

for which the following year Leibniz obtained solutions by simplifying it.

莱布尼茨于第二年将方程简化从而得到了方程的解。


Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange.[5][6][7][8] In 1746, d’Alembert discovered the one-dimensional wave equation, and within ten years Euler discovered the three-dimensional wave equation.[9]

Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange. In 1746, d’Alembert discovered the one-dimensional wave equation, and within ten years Euler discovered the three-dimensional wave equation.

历史上,让·勒朗·达朗贝尔,欧拉,丹尼尔·伯努利和约瑟夫·路易斯·拉格朗日等都研究过弦(比如乐器的弦)振动问题。1746年,达朗贝尔发现了一维波动方程,十年之内,欧拉又发现了三维波动方程。


The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point. Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed Lagrange's method and applied it to mechanics, which led to the formulation of Lagrangian mechanics.

The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point. Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed Lagrange's method and applied it to mechanics, which led to the formulation of Lagrangian mechanics.

欧拉-拉格朗日方程式是欧拉和拉格朗日在18世纪50年代结合他们对等时降线问题的研究而发明的。这是一个不考虑起始点的曲线求解问题,其中一个加权的粒子将在一个给定的时间内下降到一个固定的点。拉格朗日在1755年解决了这个问题,并将其寄给欧拉。二人都进一步发展了拉格朗日的方法并将其应用于力学,从而促使了拉格朗日力学的形成。 ==Yuling讨论) independent of the starting point 这里翻译不太好。


In 1822, Fourier published his work on heat flow in Théorie analytique de la chaleur (The Analytic Theory of Heat),[10] in which he based his reasoning on Newton's law of cooling, namely, that the flow of heat between two adjacent molecules is proportional to the extremely small difference of their temperatures. Contained in this book was Fourier's proposal of his heat equation for conductive diffusion of heat. This partial differential equation is now taught to every student of mathematical physics.

In 1822, Fourier published his work on heat flow in Théorie analytique de la chaleur (The Analytic Theory of Heat), in which he based his reasoning on Newton's law of cooling, namely, that the flow of heat between two adjacent molecules is proportional to the extremely small difference of their temperatures. Contained in this book was Fourier's proposal of his heat equation for conductive diffusion of heat. This partial differential equation is now taught to every student of mathematical physics.

1822年,傅立叶在《热的分析理论》中发表了他关于热流的研究成果,其中他以牛顿的冷却定律为基础进行推导,即两个相邻分子之间的热流与它们之间微小的温差成正比。这本书中包含了傅立叶关于热传导扩散的热方程式的建议。现在,每一个学习数学物理的学生都需要学习这类偏微分方程。

Example

示例


In classical mechanics, the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow these variables to be expressed dynamically (given the position, velocity, acceleration and various forces acting on the body) as a differential equation for the unknown position of the body as a function of time.

In classical mechanics, the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow these variables to be expressed dynamically (given the position, velocity, acceleration and various forces acting on the body) as a differential equation for the unknown position of the body as a function of time.

在经典力学中,物体运动是由其不断随时间变化的位置和速度来描述的。这些变量的表达在牛顿定律中是动态的(给定位置、速度、加速度和作用在物体上的各种力) ,并以时间函数的形式给出了未知物体位置的微分方程。


In some cases, this differential equation (called an equation of motion) may be solved explicitly.

In some cases, this differential equation (called an equation of motion) may be solved explicitly.

在某些情况下,这种微分方程(称为运动方程)可以精确地求解。


An example of modeling a real-world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity and air resistance. The ball's acceleration towards the ground is the acceleration due to gravity minus the deceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the ball's velocity. This means that the ball's acceleration, which is a derivative of its velocity, depends on the velocity (and the velocity depends on time). Finding the velocity as a function of time involves solving a differential equation and verifying its validity.

An example of modeling a real-world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity and air resistance. The ball's acceleration towards the ground is the acceleration due to gravity minus the deceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the ball's velocity. This means that the ball's acceleration, which is a derivative of its velocity, depends on the velocity (and the velocity depends on time). Finding the velocity as a function of time involves solving a differential equation and verifying its validity.

使用微分方程来模拟现实世界问题的一个例子是仅考虑重力和空气阻力来确定球在空中落下的速度。球对地面的加速度是重力加速度减去由于空气阻力提供的加速度。重力被认为是常数,空气阻力可以被模拟为与球的速度成正比。这意味着球的加速度,也就是其速度的导数,取决于速度(而速度取决于时间)。找到时间的函数--速度--需要解决一个微分方程问题并验证其正确性。

Types

微分方程的类型


Differential equations can be divided into several types. Apart from describing the properties of the equation itself, these classes of differential equations can help inform the choice of approach to a solution. Commonly used distinctions include whether the equation is: Ordinary/Partial, Linear/Non-linear, and Homogeneous/heterogeneous. This list is far from exhaustive; there are many other properties and subclasses of differential equations which can be very useful in specific contexts.

Differential equations can be divided into several types. Apart from describing the properties of the equation itself, these classes of differential equations can help inform the choice of approach to a solution. Commonly used distinctions include whether the equation is: Ordinary/Partial, Linear/Non-linear, and Homogeneous/heterogeneous. This list is far from exhaustive; there are many other properties and subclasses of differential equations which can be very useful in specific contexts.

微分方程可分为以下几种类型。除了描述方程本身的性质之外,微分方程的多种类型为我们选择何种解决方案提供了多种指导。常见的微分方程有: 常微分/偏微分方程、线性/非线性方程和齐次/非齐次方程。微分方程还有许多类型,以及许多在特定的情况下实用的其它性质和子类。


Ordinary differential equations

常微分方程


An ordinary differential equation (ODE) is an equation containing an unknown function of one real or complex variable x, its derivatives, and some given functions of x. The unknown function is generally represented by a variable (often denoted y), which, therefore, depends on x. Thus x is often called the independent variable of the equation. The term "ordinary" is used in contrast with the term partial differential equation, which may be with respect to more than one independent variable.

An ordinary differential equation (ODE) is an equation containing an unknown function of one real or complex variable , its derivatives, and some given functions of . The unknown function is generally represented by a variable (often denoted y), which, therefore, depends on x. Thus x is often called the independent variable of the equation. The term "ordinary" is used in contrast with the term partial differential equation, which may be with respect to more than one independent variable.

常微分方程是只含有一个实变量或复变量的未知函数,其导数以及此函数的一些方程。未知函数因变量(通常由 y 表示),其常常随 x的变化而变化 。因此 x 通常被称为方程式的自变量。“常微分方程”一词与偏微分方程一词相比,后者涉及一个以上的独立变量。


Linear differential equations are the differential equations that are linear in the unknown function and its derivatives. Their theory is well developed, and in many cases one may express their solutions in terms of integrals.

Linear differential equations are the differential equations that are linear in the unknown function and its derivatives. Their theory is well developed, and in many cases one may express their solutions in terms of integrals.

线性微分方程是指方程中未知函数及其导数都是线性的微分方程。关于这些方程的理论发展得很好,在多数情况下可以用积分来表示它们的解。


Most ODEs that are encountered in physics are linear. Therefore, most special functions may be defined as solutions of linear differential equations (see Holonomic function).

Most ODEs that are encountered in physics are linear. Therefore, most special functions may be defined as solutions of linear differential equations (see Holonomic function).

物理学中遇到的大多数常微分方程都是线性的。因此,大多数特殊函数可以定义为线性微分方程的解(见完整性函数)。


As, in general, the solutions of a differential equation cannot be expressed by a closed-form expression, numerical methods are commonly used for solving differential equations on a computer.

As, in general, the solutions of a differential equation cannot be expressed by a closed-form expression, numerical methods are commonly used for solving differential equations on a computer.

一般地,微分方程的解不能用解析解表示,而会在计算机上利用数值方法求解。

Partial differential equations

偏微分方程


A partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. (This is in contrast to ordinary differential equations, which deal with functions of a single variable and their derivatives.) PDEs are used to formulate problems involving functions of several variables, and are either solved in closed form, or used to create a relevant computer model.

A partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. (This is in contrast to ordinary differential equations, which deal with functions of a single variable and their derivatives.) PDEs are used to formulate problems involving functions of several variables, and are either solved in closed form, or used to create a relevant computer model.

偏微分方程 Partial Differential Equation是一种包含多元函数及其偏导数的微分方程函数(这与处理单变量函数及其导数的常微分方程不同)。偏微分方程可用于描述涉及多元函数的问题求闭式解,或者用于创建相关的计算机模型。


PDEs can be used to describe a wide variety of phenomena in nature such as sound, heat, electrostatics, electrodynamics, fluid flow, elasticity, or quantum mechanics. These seemingly distinct physical phenomena can be formalized similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems. Stochastic partial differential equations generalize partial differential equations for modeling randomness.

PDEs can be used to describe a wide variety of phenomena in nature such as sound, heat, electrostatics, electrodynamics, fluid flow, elasticity, or quantum mechanics. These seemingly distinct physical phenomena can be formalized similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems. Stochastic partial differential equations generalize partial differential equations for modeling randomness.

偏微分方程可以用来描述自然界中各种各样的现象,如声音、热量、静电、电动力学、流体流动、弹性和量子力学等。这些看起来截然不同的物理现象其实都可以用相似的偏微分方程表达。正如常微分方程常被用于对一维动力系统进行建模一样,偏微分方程常被用于对多维系统进行建模。随机偏微分方程延伸了偏微分方程在模拟随机性上的应用。

Non-linear differential equations

非线性微分方程


A non-linear differential equation is a differential equation that is not a linear equation in the unknown function and its derivatives (the linearity or non-linearity in the arguments of the function are not considered here). There are very few methods of solving nonlinear differential equations exactly; those that are known typically depend on the equation having particular symmetries. Nonlinear differential equations can exhibit very complicated behavior over extended time intervals, characteristic of chaos. Even the fundamental questions of existence, uniqueness, and extendability of solutions for nonlinear differential equations, and well-posedness of initial and boundary value problems for nonlinear PDEs are hard problems and their resolution in special cases is considered to be a significant advance in the mathematical theory (cf. Navier–Stokes existence and smoothness). However, if the differential equation is a correctly formulated representation of a meaningful physical process, then one expects it to have a solution.

A non-linear differential equation is a differential equation that is not a linear equation in the unknown function and its derivatives (the linearity or non-linearity in the arguments of the function are not considered here). There are very few methods of solving nonlinear differential equations exactly; those that are known typically depend on the equation having particular symmetries. Nonlinear differential equations can exhibit very complicated behavior over extended time intervals, characteristic of chaos. Even the fundamental questions of existence, uniqueness, and extendability of solutions for nonlinear differential equations, and well-posedness of initial and boundary value problems for nonlinear PDEs are hard problems and their resolution in special cases is considered to be a significant advance in the mathematical theory (cf. Navier–Stokes existence and smoothness). However, if the differential equation is a correctly formulated representation of a meaningful physical process, then one expects it to have a solution.

非线性微分方程是微分方程的一种,但它不是关于未知函数及其导数的线性方程(这里不考虑函数本身的线性或非线性)。能够精确求解非线性微分方程的方法很少; 那些已有的方法通常依赖于方程具有某种特定的对称性。非线性微分方程在更长的时间段内表现出非常复杂的行为,具有混沌特性。即使非线性微分方程也有解的存在性、唯一性和可扩展性等基本问题以及初边值问题的适定性问题,但对其研究也是一个难题(可参考纳维-斯托克斯方程的存在性和光滑性)。然而,如果微分方程是一个有意义物理过程的正确表述,那么人们期望它有一个解析解。

Linear differential equations frequently appear as approximations to nonlinear equations. These approximations are only valid under restricted conditions. For example, the harmonic oscillator equation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations (see below).

Linear differential equations frequently appear as approximations to nonlinear equations. These approximations are only valid under restricted conditions. For example, the harmonic oscillator equation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations (see below).

线性微分方程经常作为非线性方程的近似形式出现。这些近似只有某些限制条件下才有效。例如,谐振子方程是非线性摆方程的近似这一情况只有对于小幅度振荡是有效的(见下文)。

模板:Anchor Equation order

方程的阶

Differential equations are described by their order, determined by the term with the highest derivatives. An equation containing only first derivatives is a first-order differential equation, an equation containing the second derivative is a second-order differential equation, and so on.[11][12] Differential equations that describe natural phenomena almost always have only first and second order derivatives in them, but there are some exceptions, such as the thin film equation, which is a fourth order partial differential equation.

Differential equations are described by their order, determined by the term with the highest derivatives. An equation containing only first derivatives is a first-order differential equation, an equation containing the second derivative is a second-order differential equation, and so on. Differential equations that describe natural phenomena almost always have only first and second order derivatives in them, but there are some exceptions, such as the thin film equation, which is a fourth order partial differential equation.

微分方程的阶数是由它们的导数的最高阶决定的。只含有一阶导数的方程是一阶微分方程,含有二阶导数的方程是二阶微分方程,等等。描述自然现象的微分方程几乎总是只有一阶和二阶导数,但也有一些例外,例如薄膜方程,它是一个四阶偏微分方程。

Examples

示例


In the first group of examples u is an unknown function of x, and c and ω are constants that are supposed to be known. Two broad classifications of both ordinary and partial differential equations consist of distinguishing between linear and nonlinear differential equations, and between homogeneous differential equations and heterogeneous ones.

In the first group of examples u is an unknown function of x, and c and ω are constants that are supposed to be known. Two broad classifications of both ordinary and partial differential equations consist of distinguishing between linear and nonlinear differential equations, and between homogeneous differential equations and heterogeneous ones.

在第一组示例中,待求解的ux的函数,cω是应该已知的常数。常微分方程和偏微分方程这两种广义分类下还要区分微分方程的线性和非线性,以及区分微分方程的齐次和非齐次。


  • Heterogeneous first-order linear constant coefficient ordinary differential equation:

非齐次一阶常系数常微分方程


[math]\displaystyle{ \frac{du}{dx} = cu+x^2. }[/math]



  • Homogeneous second-order linear ordinary differential equation:

齐次二阶线性常微分方程

[math]\displaystyle{ \frac{d^2u}{dx^2} - x\frac{du}{dx} + u = 0. }[/math]


  • Homogeneous second-order linear constant coefficient ordinary differential equation describing the harmonic oscillator:

用于描述简谐振动的齐次二阶常系数常系数微分方程

[math]\displaystyle{ \frac{d^2u}{dx^2} + \omega^2u = 0. }[/math]


  • Heterogeneous first-order nonlinear ordinary differential equation:

非齐次一阶非线性常微分方程

[math]\displaystyle{ \frac{du}{dx} = u^2 + 4. }[/math]


  • Second-order nonlinear (due to sine function) ordinary differential equation describing the motion of a pendulum of length L:

用于描述摆长为L的钟摆运动的二阶非线性(因正弦函数产生)常微分方程


[math]\displaystyle{ L\frac{d^2u}{dx^2} + g\sin u = 0. }[/math]


In the next group of examples, the unknown function u depends on two variables x and t or x and y.

In the next group of examples, the unknown function u depends on two variables x and t or x and y.

在下一组例子中,未知函数u依赖于两个变量xt或者xy


  • Homogeneous first-order linear partial differential equation:

齐次一阶线性偏微分方程


[math]\displaystyle{ \frac{\partial u}{\partial t} + t\frac{\partial u}{\partial x} = 0. }[/math]



  • Homogeneous second-order linear constant coefficient partial differential equation of elliptic type, the Laplace equation:

齐次二阶线性常系数椭圆形偏微分方程,也称为拉普拉斯方程


[math]\displaystyle{ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0. }[/math]



  • Homogeneous third-order non-linear partial differential equation :

齐次三阶非线性偏微分方程


[math]\displaystyle{ \frac{\partial u}{\partial t} = 6u\frac{\partial u}{\partial x} - \frac{\partial^3 u}{\partial x^3}. }[/math]

Existence of solutions

解的存在性


Solving differential equations is not like solving algebraic equations. Not only are their solutions often unclear, but whether solutions are unique or exist at all are also notable subjects of interest.

Solving differential equations is not like solving algebraic equations. Not only are their solutions often unclear, but whether solutions are unique or exist at all are also notable subjects of interest.

解微分方程不同于解代数方程。方程解的情况往往是不确定的,而且解是否唯一或是否存在也是值得关注的问题。


For first order initial value problems, the Peano existence theorem gives one set of circumstances in which a solution exists. Given any point [math]\displaystyle{ (a,b) }[/math] in the xy-plane, define some rectangular region [math]\displaystyle{ Z }[/math], such that [math]\displaystyle{ Z = [l,m]\times[n,p] }[/math] and [math]\displaystyle{ (a,b) }[/math] is in the interior of [math]\displaystyle{ Z }[/math]. If we are given a differential equation [math]\displaystyle{ \frac{dy}{dx} = g(x,y) }[/math] and the condition that [math]\displaystyle{ y=b }[/math] when [math]\displaystyle{ x=a }[/math], then there is locally a solution to this problem if [math]\displaystyle{ g(x,y) }[/math] and [math]\displaystyle{ \frac{\partial g}{\partial x} }[/math] are both continuous on [math]\displaystyle{ Z }[/math]. This solution exists on some interval with its center at [math]\displaystyle{ a }[/math]. The solution may not be unique. (See Ordinary differential equation for other results.)

For first order initial value problems, the Peano existence theorem gives one set of circumstances in which a solution exists. Given any point [math]\displaystyle{ (a,b) }[/math] in the xy-plane, define some rectangular region [math]\displaystyle{ Z }[/math], such that [math]\displaystyle{ Z = [l,m]\times[n,p] }[/math] and [math]\displaystyle{ (a,b) }[/math] is in the interior of [math]\displaystyle{ Z }[/math]. If we are given a differential equation [math]\displaystyle{ \frac{dy}{dx} = g(x,y) }[/math] and the condition that [math]\displaystyle{ y=b }[/math] when [math]\displaystyle{ x=a }[/math], then there is locally a solution to this problem if [math]\displaystyle{ g(x,y) }[/math] and [math]\displaystyle{ \frac{\partial g}{\partial x} }[/math] are both continuous on [math]\displaystyle{ Z }[/math]. This solution exists on some interval with its center at [math]\displaystyle{ a }[/math]. The solution may not be unique. (See Ordinary differential equation for other results.)

对于一阶初值问题,皮亚诺存在性定理给出了一组解存在的情况。给定的x-y平面上的任意点 [math]\displaystyle{ (a,b) }[/math] ,定义矩形区域 [math]\displaystyle{ Z }[/math] ,如,[math]\displaystyle{ Z = [l,m]\times[n,p] }[/math] 而且 [math]\displaystyle{ (a,b) }[/math][math]\displaystyle{ Z }[/math] 内部一点。如果我们给出一个微分方程 [math]\displaystyle{ \frac{dy}{dx} = g(x,y) }[/math] 和当[math]\displaystyle{ x=a }[/math][math]\displaystyle{ y=b }[/math],如果[math]\displaystyle{ g(x,y) }[/math][math]\displaystyle{ \frac{\partial g}{\partial x} }[/math][math]\displaystyle{ Z }[/math]上是连续的,那么这个问题就有一个局部解。这个解在以 [math]\displaystyle{ a }[/math] 为中心的某些区间上存在,其可能不是唯一的。(其他结果请参见常微分方程。)


However, this only helps us with first order initial value problems. Suppose we had a linear initial value problem of the nth order:

However, this only helps us with first order initial value problems. Suppose we had a linear initial value problem of the nth order:

然而,这只能帮助我们解决一阶初始值问题。假设我们有一个n阶线性初始值问题:


[math]\displaystyle{ f_{n}(x)\frac{d^n y}{dx^n} + \cdots + f_{1}(x)\frac{d y}{dx} + f_{0}(x)y = g(x) }[/math]

such that

such that

其中有

[math]\displaystyle{ y(x_{0})=y_{0}, y'(x_{0}) = y'_{0}, y''(x_{0}) = y''_{0}, \cdots }[/math]


For any nonzero [math]\displaystyle{ f_{n}(x) }[/math], if [math]\displaystyle{ \{f_{0},f_{1},\cdots\} }[/math] and [math]\displaystyle{ g }[/math] are continuous on some interval containing [math]\displaystyle{ x_{0} }[/math], [math]\displaystyle{ y }[/math] is unique and exists.[13]

For any nonzero [math]\displaystyle{ f_{n}(x) }[/math], if [math]\displaystyle{ \{f_{0},f_{1},\cdots\} }[/math] and [math]\displaystyle{ g }[/math] are continuous on some interval containing [math]\displaystyle{ x_{0} }[/math], [math]\displaystyle{ y }[/math] is unique and exists.

对于任意非零 [math]\displaystyle{ f_{n}(x) }[/math] ,如果[math]\displaystyle{ \{f_{0},f_{1},\cdots\} }[/math][math]\displaystyle{ g }[/math]在某个包含[math]\displaystyle{ x_{0} }[/math]的区间上连续,则[math]\displaystyle{ y }[/math]是存在且唯一的。

Related concepts

相关概念


  • A delay differential equation (DDE) is an equation for a function of a single variable, usually called time, in which the derivative of the function at a certain time is given in terms of the values of the function at earlier times.

延迟微分方程(DDE)是一元函数的方程,变量通常为时间,其中函数在一定时间点的微分会被较早时间点的函数值表达。

积分微分方程(IDE)结合了微分方程和积分方程。

随机微分方程(SDE)中的未知量处于随机过程,并且涉及一些已知的随机过程,例如,扩散方程中的维纳过程。

随机偏微分方程(SPDE)是一种含空间和时间噪声过程的广义随机微分方程,它通常应用于量子场论以及统计力学中。

微分代数方程(DAE)是一种含微分和代数项的微分方程,通常以隐式形式给出。

Connection to difference equations

与差分方程之间的联系


The theory of differential equations is closely related to the theory of difference equations, in which the coordinates assume only discrete values, and the relationship involves values of the unknown function or functions and values at nearby coordinates. Many methods to compute numerical solutions of differential equations or study the properties of differential equations involve the approximation of the solution of a differential equation by the solution of a corresponding difference equation.

The theory of differential equations is closely related to the theory of difference equations, in which the coordinates assume only discrete values, and the relationship involves values of the unknown function or functions and values at nearby coordinates. Many methods to compute numerical solutions of differential equations or study the properties of differential equations involve the approximation of the solution of a differential equation by the solution of a corresponding difference equation.

微分方程理论与差分方程理论密切相关。在差分方程理论中,坐标系中只假定存在离散值,计算中会涉及到未知函数或已知函数的值以及坐标附近的值。许多求微分方程数值解或研究微分方程性质的方法,都会涉及通过相应差分方程的解来逼近微分方程的解。

Applications 应用

The study of differential equations is a wide field in pure and applied mathematics, physics, and engineering. All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the rigorous justification of the methods for approximating solutions. Differential equations play an important role in modeling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i.e. do not have closed form solutions. Instead, solutions can be approximated using numerical methods.

The study of differential equations is a wide field in pure and applied mathematics, physics, and engineering. All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the rigorous justification of the methods for approximating solutions. Differential equations play an important role in modeling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i.e. do not have closed form solutions. Instead, solutions can be approximated using numerical methods.

微分方程的研究可以应用于许多领域,如理论数学、应用数学、物理学和工程学,它们都与各种类型的微分方程的性质有关。理论数学关注解的存在性和唯一性,而应用数学则强调求解方法的严格准确性。从天体运动到桥梁设计,再到神经元之间的相互作用,微分方程在几乎所有物理、技术或生物过程的建模中都扮演着重要的角色。用于解决实际问题的微分方程,不一定是直接可解的,如可能不存在闭式解。但我们可以用数值方法来近似得到方程的解。


Many fundamental laws of physics and chemistry can be formulated as differential equations. In biology and economics, differential equations are used to model the behavior of complex systems. The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application. However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations. Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena. As an example, consider the propagation of light and sound in the atmosphere, and of waves on the surface of a pond. All of them may be described by the same second-order partial differential equation, the wave equation, which allows us to think of light and sound as forms of waves, much like familiar waves in the water. Conduction of heat, the theory of which was developed by Joseph Fourier, is governed by another second-order partial differential equation, the heat equation. It turns out that many diffusion processes, while seemingly different, are described by the same equation; the Black–Scholes equation in finance is, for instance, related to the heat equation.

Many fundamental laws of physics and chemistry can be formulated as differential equations. In biology and economics, differential equations are used to model the behavior of complex systems. The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application. However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations. Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena. As an example, consider the propagation of light and sound in the atmosphere, and of waves on the surface of a pond. All of them may be described by the same second-order partial differential equation, the wave equation, which allows us to think of light and sound as forms of waves, much like familiar waves in the water. Conduction of heat, the theory of which was developed by Joseph Fourier, is governed by another second-order partial differential equation, the heat equation. It turns out that many diffusion processes, while seemingly different, are described by the same equation; the Black–Scholes equation in finance is, for instance, related to the heat equation.

许多物理和化学的基本定律都可以用微分方程来表示。在生物学和经济学中,微分方程被用来模拟复杂系统的行为。微分方程理论最初是与其起源并得到应用的科学一起发展起来的89。然而,有时完全不同的科学领域,却可能产生相同的微分方程。当这种情况发生时,方程后面的数学理论可以被看作是不同现象背后的统一原则。例如,光和声在大气中的传播,或是池塘表面的水波的传播。所有这些过程都可以用相同的二阶偏微分方程来描述,即波动方程。我们把光和声音想象成与水波相似的形式。由约瑟夫·傅里叶提出的热传导的理论由另一个二阶偏微分方程——热方程所支配。事实证明,许多扩散过程,虽然看上去形式不同,但都可以同一个方程来描述;。例如,金融学中的布莱克-斯科尔斯方程就与热方程有关。


==Yuling讨论) "results found application" 翻译为“方程解的搜索”,可能不太准确


The number of differential equations that have received a name, in various scientific areas is a witness of the importance of the topic. See List of named differential equations.

The number of differential equations that have received a name, in various scientific areas is a witness of the importance of the topic. See List of named differential equations.

事实上,同一类型的微分方程可以应用于不同领域这样的现象屡见不鲜,这足以证明微分方程这一课题的重要性。参见已命名的微分方程列表。

See also 另请参见

复微分方程

精确微分方程

泛函微分方程

初始条件

积分方程

求解常微分方程的数值方法

求解偏微分方程的数值方法

关于解的存在性和唯一性的皮卡德–林德洛夫定理

递推关系,也称为差分方程

抽象微分方程

微分方程组

References 参考文献

  1. Dennis G. Zill (15 March 2012). A First Course in Differential Equations with Modeling Applications. Cengage Learning. ISBN 1-285-40110-7. https://books.google.com/books?id=pasKAAAAQBAJ&printsec=frontcover#v=snippet&q=%22ordinary%20differential%22&f=false. 
  2. Newton, Isaac. (c.1671). Methodus Fluxionum et Serierum Infinitarum (The Method of Fluxions and Infinite Series), published in 1736 [Opuscula, 1744, Vol. I. p. 66].
  3. Bernoulli, Jacob (1695), "Explicationes, Annotationes & Additiones ad ea, quae in Actis sup. de Curva Elastica, Isochrona Paracentrica, & Velaria, hinc inde memorata, & paratim controversa legundur; ubi de Linea mediarum directionum, alliisque novis", Acta Eruditorum
  4. Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems, Berlin, New York: Springer-Verlag, ISBN 978-3-540-56670-0
  5. Frasier, Craig (July 1983). "Review of The evolution of dynamics, vibration theory from 1687 to 1742, by John T. Cannon and Sigalia Dostrovsky" (PDF). Bulletin (New Series) of the American Mathematical Society. 9 (1).
  6. Wheeler, Gerard F.; Crummett, William P. (1987). "The Vibrating String Controversy". Am. J. Phys. 55 (1): 33–37. Bibcode:1987AmJPh..55...33W. doi:10.1119/1.15311.
  7. For a special collection of the 9 groundbreaking papers by the three authors, see First Appearance of the wave equation: D'Alembert, Leonhard Euler, Daniel Bernoulli. - the controversy about vibrating strings (retrieved 13 Nov 2012). Herman HJ Lynge and Son.
  8. For de Lagrange's contributions to the acoustic wave equation, can consult Acoustics: An Introduction to Its Physical Principles and Applications Allan D. Pierce, Acoustical Soc of America, 1989; page 18.(retrieved 9 Dec 2012)
  9. Speiser, David. Discovering the Principles of Mechanics 1600-1800, p. 191 (Basel: Birkhäuser, 2008).
  10. Fourier, Joseph (1822) (in French). Théorie analytique de la chaleur. Paris: Firmin Didot Père et Fils. OCLC 2688081. https://archive.org/details/bub_gb_TDQJAAAAIAAJ. 
  11. Weisstein, Eric W. "Ordinary Differential Equation Order." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/OrdinaryDifferentialEquationOrder.html
  12. Order and degree of a differential equation -{zh-cn:互联网档案馆; zh-tw:網際網路檔案館; zh-hk:互聯網檔案館;}-存檔,存档日期2016-04-01., accessed Dec 2015.
  13. Zill, Dennis G. (2001). A First Course in Differential Equations (5th ed.). Brooks/Cole. ISBN 0-534-37388-7. 

Further reading 延伸阅读

  • Abbott, P.; Neill, H. (2003). Teach Yourself Calculus. pp. 266–277. 
  • Blanchard, P.; Devaney, R. L.; Hall, G. R. (2006). Differential Equations. Thompson. 
  • Boyce, W.; DiPrima, R.; Meade, D. (2017). Elementary Differential Equations and Boundary Value Problems. Wiley. 
  • Ince, E. L. (1956). Ordinary Differential Equations. Dover. 
  • Polyanin, A. D.; Zaitsev, V. F. (2003). Handbook of Exact Solutions for Ordinary Differential Equations (2nd ed.). Boca Raton: Chapman & Hall/CRC Press. ISBN 1-58488-297-2. 
  • Porter, R. I. (1978). "XIX Differential Equations". Further Elementary Analysis. 

External links 外部链接

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