Nonlinearity & Functional Analysis: Lectures on Nonlinear Problems in Mathematical Analysis

On discrete analogues of nonlinear implicit differential equations

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Examples of nonlinear recurrence relations are the logistic map and the relations that define the various Hofstadter sequences.

Constructive inverse function theorems

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. 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. 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. 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.

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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. Please help to improve this article by introducing more precise citations. March Learn how and when to remove this template message.

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Main articles: Algebraic equation and System of polynomial equations. See also: List of nonlinear partial differential equations. Main article: Pendulum mathematics.

Nonlinear Functional Analysis, A First Course by S. Kesavan | | Booktopia

MIT News. Retrieved March Annals of Biomedical Engineering. Bibcode : Nonli.. Bibcode : Chaos.. Bibcode : NatSR Berlin: Springer. Retrieved 20 January Bibcode : Natur.

Journal of Symbolic Computation. Diederich Hinrichsen and Anthony J.

Pritchard Let be a Banach space and let be the set of continuous linear functionals on it; is a vector space with respect to the usual operations of adding functions and multiplying them by a number, it becomes a Banach space if one introduces the norm. The space is called the dual of cf. If is finite-dimensional, then every linear functional is of the form. It turns out that the formula also holds when is a Hilbert space Riesz' theorem. Namely, in this case , where is a certain vector in. This formula shows that a Hilbert space essentially coincides with its dual. For a Banach space the situation is far more complicated: One can construct , and these spaces may turn out to be all different.

At the same time, there always exists a canonical imbedding of into , namely, to each one can associate the functional , where ,. The spaces for which are called reflexive. Generally, in the case of a Banach space even the existence of non-trivial that is, non-zero linear functionals is not a simple question.

This question is easily solved affirmatively with the help of the Hahn—Banach theorem. The dual space is, in a certain sense, "better" than the original space. For example, along with the norm one can introduce another weak topology in which, in terms of convergence, is such that if for all. In this topology the unit ball in is compact which is never the case for infinite-dimensional spaces in the topology generated by a norm. This makes it possible to study in more detail a number of geometric questions about sets in the dual space for example, establishing the structure of convex sets, etc.

For a number of specific spaces the dual space can be found explicitly. However, for the majority of Banach spaces, and especially for topological vector spaces, the functionals are elements of a new kind which cannot be expressed simply in terms of classical analysis. The elements of the dual space are called generalized functions.

For many questions in functional analysis and its applications an essential role is played by a triple of spaces , where is the original Hilbert space, is a topological vector space in particular, a Hilbert space with a different inner product and is its dual space, the elements of which can be taken as generalized functions. The space itself is then called a rigged Hilbert space.

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The study of linear functionals on in many respects promotes a deeper understanding of the nature of the original space. On the other hand, in many questions it is necessary to study general functions , that is, non-linear functionals in the case of an infinite-dimensional cf. Non-linear functional. Since the unit ball in such a space is non-compact, its study often encounters essential difficulties, although, for example, such concepts as the differentiability of , its analyticity, etc. One can consider a set of functions having definite properties as a new topological vector space of functions of "an infinite number of variables".

Such functions also appear in constructing infinite tensor products of spaces of functions of one variable. The study of such spaces, of the operators on them, etc. The main objects of study in functional analysis are operators , where and are topological vector for the most part, normed or Hilbert spaces and, above all, linear operators cf. Linear operator. When and are finite-dimensional, the linearity of an operator implies that it is of the form. Thus, in the finite-dimensional case to each linear operator corresponds, in terms of fixed bases in and , a matrix.

The study of linear operators in this case is a topic of linear algebra. The situation becomes much more complicated when and become infinite-dimensional even Hilbert spaces. First of all, two classes of operators arise here: continuous operators, for which the function is continuous they are also called bounded, since the continuity of an operator between Banach spaces is equivalent to its boundedness , and unbounded operators, where there is no such continuity. The operators of the first type are simpler, but those of the second type are met more often, e.

The important especially for quantum mechanics class of self-adjoint operators on a Hilbert space has been studied most of all cf. Self-adjoint operator.

Other classes of operators on , closely connected with the self-adjoint operators the so-called unitary and normal operators, cf. Unitary operator ; Normal operator , have also been well studied. Among the general facts about bounded operators acting in a Banach space , one can select the construction of a functional calculus of analytic functions. Namely, the operator is called the resolvent of the operator , where is the identity operator and.

The points for which the inverse operator exists are called the regular points of , the complement of the set of regular points is called the spectrum of. The spectrum is never empty and lies in the disc ; the eigen values of , of course, belong to , but the spectrum, generally speaking, does not entirely consist of them. If is an analytic function defined in a neighbourhood of , and if is some closed contour enclosing and lying in the domain of analyticity of , then one puts.

If is a polynomial, then is obtained by simply replacing in this polynomial by. The correspondence has the important homomorphism properties:. Thus, under definite conditions on one can define, for example, , , , etc. Among the special classes of operators acting on a Banach space the most important role is played by the so-called completely-continuous or compact operators cf. Completely-continuous operator ; Compact operator. If is compact, then the equation is a given vector and is the desired vector has been well studied. The analogues of all the facts which hold for linear equations in finite-dimensional spaces are also valid for this equation the so-called Fredholm theory.

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