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Institutional Subscription. Free Shipping Free global shipping No minimum order. Powered by. You are connected as. Connect with:. As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations linearization.

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This works well up to some accuracy and some range for the input values, but some interesting phenomena such as solitons , chaos , [10] and singularities are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic.

Although such chaotic behavior may resemble random behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology. Some authors use the term nonlinear science for the study of nonlinear systems. This is disputed by others:. Using a term like nonlinear science is like referring to the bulk of zoology as the study of non -elephant animals.

For example, an antilinear map is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle.

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Nonlinear algebraic equations , which are also called polynomial equations , are defined by equating polynomials of degree greater than one to zero. For example,. For a single polynomial equation, root-finding algorithms can be used to find solutions to the equation i. However, systems of algebraic equations are more complicated; their study is one motivation for the field of algebraic geometry , a difficult branch of modern mathematics.

It is even difficult to decide whether a given algebraic system has complex solutions see Hilbert's Nullstellensatz. Nevertheless, in the case of the systems with a finite number of complex solutions, these systems of polynomial equations are now well understood and efficient methods exist for solving them. A nonlinear recurrence relation defines successive terms of a sequence as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the logistic map and the relations that define the various Hofstadter sequences.

Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX Nonlinear Autoregressive Moving Average with eXogenous inputs model and the related nonlinear system identification and analysis procedures.

## Nonlinear Ordinary Differential Equations in Transport Processes, Volume 42 - 1st Edition

A system of differential equations is said to be nonlinear if it is not a linear system. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the Navier—Stokes equations in fluid dynamics and the Lotka—Volterra equations in biology. One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions.

In linear problems, for example, a family of linearly independent solutions can be used to construct general solutions through the superposition principle. A good example of this is one-dimensional heat transport with Dirichlet boundary conditions , the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.

First order ordinary differential equations are often exactly solvable by separation of variables , especially for autonomous equations. For example, the nonlinear equation. The equation is nonlinear because it may be written as. Note that if the u 2 term were replaced with u , the problem would be linear the exponential decay problem. Second and higher order ordinary differential equations more generally, systems of nonlinear equations rarely yield closed-form solutions, though implicit solutions and solutions involving nonelementary integrals are encountered.

## Free Nonlinear Ordinary Differential Equations In Transport Processes 1968

Common methods for the qualitative analysis of nonlinear ordinary differential equations include:. The most common basic approach to studying nonlinear partial differential equations is to change the variables or otherwise transform the problem so that the resulting problem is simpler possibly even linear. Sometimes, the equation may be transformed into one or more ordinary differential equations , as seen in separation of variables , which is always useful whether or not the resulting ordinary differential equation s is solvable.

Another common though less mathematic tactic, often seen in fluid and heat mechanics, is to use scale analysis to simplify a general, natural equation in a certain specific boundary value problem.

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For example, the very nonlinear Navier-Stokes equations can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation. Other methods include examining the characteristics and using the methods outlined above for ordinary differential equations. A classic, extensively studied nonlinear problem is the dynamics of a pendulum under the influence of gravity. Using Lagrangian mechanics , it may be shown [14] that the motion of a pendulum can be described by the dimensionless nonlinear equation.

Another way to approach the problem is to linearize any nonlinearities the sine function term in this case at the various points of interest through Taylor expansions. This is a simple harmonic oscillator corresponding to oscillations of the pendulum near the bottom of its path. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state. This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find exact phase portraits and approximate periods.

From Wikipedia, the free encyclopedia. For the journal, see Nonlinear Dynamics journal. This article is about "nonlinearity" in mathematics, physics and other sciences. For video and film editing, see Non-linear editing system. For other uses, see Nonlinearity disambiguation. This article includes a list of references , but its sources remain unclear because it has insufficient inline citations.