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A method for approximately solving non-linear operator equations. It consists of generalizing the equation to be solved, <img align="absmidd If the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l

...nary differential equations with deviated arguments, ordinary differential equations with deviating arguments'' Ordinary differential equations connecting the argument, the unknown function and its derivatives generally taken for different values of the argument. Examp

...athematical physics where the boundaries of the working regions are moving and undergo considerable deformations. A substantial part of an algorithm for h ...the cells (the enumeration is analogous to that of matrix elements by rows and columns).

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...a12]]], [[#References|[a13]]], [[#References|[a14]]], [[#References|[a2]]] and the numerous references therein).

...0107360 Boundary value problem, numerical methods for partial differential equations Please remove this comment and the {{TEX|auto}} line below,

Please remove this comment and the {{TEX|auto}} line below, ...transport equation (cf. [[Transport equations, numerical methods|Transport equations, numerical methods]])

Please remove this comment and the {{TEX|auto}} line below, from above and from below. An estimate from above is an inequality of the form $ a \leq

...rd "system" in their names, such as systems analysis, system-technology, and general systems theory, among others. ...s to the description of complex systems are united by a common methodology and a common object of study.

Please remove this comment and the {{TEX|auto}} line below, in the problem of an extremum for the functional

$#C+1 = 201 : ~/encyclopedia/old_files/data/D031/D.0301730 Difference scheme, variational Please remove this comment and the {{TEX|auto}} line below,

Please remove this comment and the {{TEX|auto}} line below, ...ving problems of optimal control with constraints on the phase coordinates and control functions, based on variation in the state space (see [[#References

$#C+1 = 153 : ~/encyclopedia/old_files/data/F042/F.0402060 Functional equation Please remove this comment and the {{TEX|auto}} line below,

$#C+1 = 201 : ~/encyclopedia/old_files/data/F042/F.0402070 Functional equation, methods of solution of a Please remove this comment and the {{TEX|auto}} line below,

Please remove this comment and the {{TEX|auto}} line below, and range of values in a Banach space $ Y $.

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...e to a solution of the given equation. The stability problem of functional equations originates from such a fundamental question.

Please remove this comment and the {{TEX|auto}} line below, A method for solving functional equations of the type:

Please remove this comment and the {{TEX|auto}} line below, ...e, two problems are usually studied: the [[Cauchy problem|Cauchy problem]] and the [[Mixed problem|mixed problem]].

Please remove this comment and the {{TEX|auto}} line below, ...ense, this name, adopted also by A.L. Cauchy, was used in the 19th century and in the 20th century (cf. [[#References|[a2]]], [[#References|[a3]]], [[#Ref

...mula]] [[#References|[a1]]], [[Spline interpolation|spline interpolation]] and Fourier transformation (cf. [[Fourier transform|Fourier transform]]) [[#Ref ...fferential equations. The complete error analysis of the finite difference and finite volume schemes in [[#References|[a9]]] is based on the Bramble–Hil

...m of analytical expressions are only known for a few types of differential equations. ...f the oldest methods for the approximate solution of ordinary differential equations is their expansion into a Taylor series. In this method the solution of an

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...the study of many classes of partial and functional differential equations and has solved problems that had been left open by the previous classical metho

Please remove this comment and the {{TEX|auto}} line below, ...s it is typical to assume that the solution of the original problem exists and has sufficiently many derivatives. Because of the absence of other assumpti

Please remove this comment and the {{TEX|auto}} line below, ...f the original problem, but such a discretization is nevertheless implicit and is realized relative to a known model.

...ue of its derivative, at a given point of the extremal. In the case of the functional and let $ X ^{*} $

...old_files/data/B017/B.0107380 Boundary value problem, partial differential equations Please remove this comment and the {{TEX|auto}} line below,

Please remove this comment and the {{TEX|auto}} line below, ...ly related to those of a [[Generalized derivative|generalized derivative]] and a [[Generalized function|generalized function]]. The concept of a generaliz

Please remove this comment and the {{TEX|auto}} line below, ...terpolation|interpolation]] of functions. The interpolation of functionals and operators is also widely used in constructing numerical methods.

Please remove this comment and the {{TEX|auto}} line below, ...tions. The best known of these are the [[Löwner equation|Löwner equation]] and the Löwner–Kufarev equation. In the discrete case — for lattice domain

...rder in function spaces makes it possible to study within the framework of functional analysis problems that are essentially connected with inequalities between ...opediaofmath.org/legacyimages/s/s084/s084260/s08426022.png" /> is a wedge, and every wedge with vertex at the origin generates a quasi-order in <img align

Please remove this comment and the {{TEX|auto}} line below, a minimum of a functional, eigen values and eigen vectors of an equation $ A u = \lambda u $,

$#C+1 = 202 : ~/encyclopedia/old_files/data/T093/T.0903900 Transport equations, numerical methods Please remove this comment and the {{TEX|auto}} line below,

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...mation is functionally parametrized by a pair of solutions of the equation and the transform vanishes if the solutions coincide.

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...ad influences on many parts of mathematics and different areas of sciences and technology ([[#References|[a4]]], [[#References|[a7]]], [[#References|[a8]]

Please remove this comment and the {{TEX|auto}} line below, Iteration methods for the solution of non-linear equations.

Please remove this comment and the {{TEX|auto}} line below, ...ight makes it possible to enlarge or restrict ordinary non-weighted normed and semi-normed function spaces consisting of functions with infinite ordinary

Please remove this comment and the {{TEX|auto}} line below, ...tic aspects), depending on input functions or parameters, called controls, and usually subject to constraints. The sought controls, as well as the realiza

Please remove this comment and the {{TEX|auto}} line below, A system of ordinary differential equations in the numerical solution of which by explicit methods of Runge–Kutta or

Please remove this comment and the {{TEX|auto}} line below, ...statistical models for vertical-heterogeneous random fields of turbulence and [[Internal waves|internal waves]] (cf. also [[Turbulence, mathematical prob

Please remove this comment and the {{TEX|auto}} line below, ...respective boundary value problem. In direct methods, the extremum of the functional is found directly. The optimization methods thus employed are a development

Please remove this comment and the {{TEX|auto}} line below, defined and acting on a closed subset $ G $

Please remove this comment and the {{TEX|auto}} line below, ...ing them, to an examination of simpler algebraic problems. The development and systematic use of operational calculus began with the work of O. Heaviside

Please remove this comment and the {{TEX|auto}} line below, is considered to be fixed for the given quadrature formula and is called the weight function; the function $ f ( x) $

Please remove this comment and the {{TEX|auto}} line below, ...atics that are applied to find the extrema (maxima or minima) of functions and functionals.

...everal classes of functions that arise in the solution of both theoretical and applied problems in various branches of mathematics. ...ons of mathematical physics, which arise when solving partial differential equations by the

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...ymplectic manifolds, Hamiltonian vector fields, symplectic diffeomorphisms and Lagrangean submanifolds.

Please remove this comment and the {{TEX|auto}} line below, A branch of modern computational mathematics, devoted to the design and investigation of methods which allow one to find an approximate solution, w

Please remove this comment and the {{TEX|auto}} line below, and of the strain tensor $ \epsilon _ {ij} ( x, t) $,

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...nti-holomorphic involution, cf. also [[Riemann surface|Riemann surface]]), and with the theory of multi-dimensional systems. The theory of operator vessel

Please remove this comment and the {{TEX|auto}} line below, and the law of motion of the dividing boundary $ \xi = \xi ( t) $(

Please remove this comment and the {{TEX|auto}} line below, and $ A _ {1} ( t) $,

Please remove this comment and the {{TEX|auto}} line below, ...he relation between algebraic-topological properties of topological spaces and extremal properties of functions (functionals) on them. Morse theory is a b

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...o that differing ages imply different reproduction and survival capacities and, also, different behaviours.

...ce of the operations, a restricted number of digit places to form a number and a restricted storage capacity. ...e sequence of operations of the computer is independent of the input data, and is called non-linear otherwise.

Please remove this comment and the {{TEX|auto}} line below, ...for solving a problem, the researcher concentrates on certain properties, and his choice of the algorithm depends on these, while the algorithm will also

...the base of the cone and the part of the conical surface between the base and the vertex is called the lateral surface of the cone. ...rac{1}{3}\pi (R^2 + r^2 + Rr) h$, where $R$, $r$ are the radii of the base and $h$ is the height (the distance between the base); the area of the lateral

...primary quantities and the corresponding experimental units of measurement and, associated with the choice of the primary units of measurement, the format ...he mass unit $M$; on the basis of the definition of force in terms of mass and acceleration, the dimension formula for the force $K$ has the form

Please remove this comment and the {{TEX|auto}} line below, ...alls takes place in several channels simultaneously. The basic definitions and notations are as in [[Queue|Queue]].

...ometry in the large]], and find wide and profound application in mechanics and physics [[#References|[4]]].

Please remove this comment and the {{TEX|auto}} line below, A branch of mathematics studying the theory and the methods of solution of multi-step problems of optimal control.

Please remove this comment and the {{TEX|auto}} line below, and a given "terminal" criterion

Please remove this comment and the {{TEX|auto}} line below, ...onnection with work on the construction of atomic reactors, J. von Neumann and S. Ulam suggested using the apparatus of probability theory in the computer

Please remove this comment and the {{TEX|auto}} line below, and $ \mathfrak M _ {1} $,

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...e manufacturing work-cell and expressing the relationship between discrete and continuous events by means of a unified auxiliary variable, as proposed in

...t of convergence arises, for example, in the study of mathematical objects and their approximation by simpler objects. Thus, in order to calculate the are ...r ordinary and partial differential equations there are various convergent difference methods for their numerical solution, which are suitable for use in modern

Please remove this comment and the {{TEX|auto}} line below, ...in applications of probability theory in various areas of natural science and technology, since these processes accurately describe many real phenomena a

Please remove this comment and the {{TEX|auto}} line below, ...the influence of the ideas of Hilbert). One of the most generally followed and definitive approaches to the development of constructive mathematics on thi

...timate the unknown values of these parameters from results of observations and to test hypotheses concerning their values. ...://www.encyclopediaofmath.org/legacyimages/n/n067/n067230/n0672306.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l

...dy of certain algebraic objects, as is done in analytic geometry (with the difference that the transition from geometry to algebra in homology theory is irrevers ...e algebras, the theory of finite-dimensional algebras, the theory of rings and the theory of quadratic forms.

Please remove this comment and the {{TEX|auto}} line below, and let $ C _ {0} ^ \infty ( \Omega ) $

Please remove this comment and the {{TEX|auto}} line below, if and only if $ x = 0 $;

Please remove this comment and the {{TEX|auto}} line below, ...as the geometry of the spectrum and its main parts, spectral multiplicity and the asymptotics of eigenvalues.

...degrees of freedom. Quantum field theory unifies the description of fields and particles, which, in classical physics, appear as two distinct entities. ...d, as this is the only field having a clear content, both in the classical and the quantum case.

Please remove this comment and the {{TEX|auto}} line below, ...ces|[2]]]; engineers had discovered the idea of a feedback law (see below) and found means for its realization.

...diameter is greater than three and ten seventy-firsts and less than three and one seventh" is written as ...as intimately bound up with the general evolution of mathematical concepts and methods.

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...ial differential equations, the calculus of variations, harmonic analysis, and fractals.

Please remove this comment and the {{TEX|auto}} line below, ...pplications. Of particular significance is the connection with the science and practice of economics.

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...' responses to the organization and its offerings by providing information and by supplying reasons for preferring a particular organization's offer.

''a history of the mathematics preceding and relevant to Hilbert’s statement of his 2nd problem, initiating his progra ...had developed deductively a large part of analysis using the real numbers and their properties as a starting point.