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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. 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.
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. Two broad classifications of both ordinary and partial differential equations consists of distinguishing between linear and nonlinear differential equations, and between homogeneous differential equations and heterogeneous ones. In the next group of examples, the unknown function u depends on two variables x and t or x and y.
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. For first order initial value problems, the Peano existence theorem gives one set of circumstances in which a solution exists. The solution may not be unique. See Ordinary differential equation for other results.
However, this only helps us with first order initial value problems. Suppose we had a linear initial value problem of the nth order:. 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 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.
- Ordinary Differential Equations with Applications?
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Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i. Instead, solutions can be approximated using numerical methods.
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.
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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.
Mathematical Analysis, Differential Equations and Applications
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.
Nonlinear Differential Equations and Applications | Publons
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. Name of resource. Problem URL. Describe the connection issue. SearchWorks Catalog Stanford Libraries. Delay differential equations : with applications in population dynamics.
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