# Dynamical Systems Method for Solving Operator Equations

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## Dynamical Systems Method for Solving Nonlinear Operator Equations

Author | : Alexander G. Ramm |

Publsiher | : Elsevier |

Total Pages | : 304 |

Release | : 2006-09-25 |

ISBN 10 | : 9780080465562 |

ISBN 13 | : 0080465560 |

Language | : EN, FR, DE, ES & NL |

**Dynamical Systems Method for Solving Nonlinear Operator Equations Book Review:**

Dynamical Systems Method for Solving Nonlinear Operator Equations is of interest to graduate students in functional analysis, numerical analysis, and ill-posed and inverse problems especially. The book presents a general method for solving operator equations, especially nonlinear and ill-posed. It requires a fairly modest background and is essentially self-contained. All the results are proved in the book, and some of the background material is also included. The results presented are mostly obtained by the author. Contains a systematic development of a novel general method, the dynamical systems method, DSM for solving operator equations, especially nonlinear and ill-posed Self-contained, suitable for wide audience Can be used for various courses for graduate students and partly for undergraduates (especially for RUE classes)

## Dynamical Systems Method and Applications

Author | : Alexander G. Ramm,Nguyen S. Hoang |

Publsiher | : John Wiley & Sons |

Total Pages | : 576 |

Release | : 2013-06-07 |

ISBN 10 | : 111819960X |

ISBN 13 | : 9781118199602 |

Language | : EN, FR, DE, ES & NL |

**Dynamical Systems Method and Applications Book Review:**

Demonstrates the application of DSM to solve a broad range of operator equations The dynamical systems method (DSM) is a powerful computational method for solving operator equations. With this book as their guide, readers will master the application of DSM to solve a variety of linear and nonlinear problems as well as ill-posed and well-posed problems. The authors offer a clear, step-by-step, systematic development of DSM that enables readers to grasp the method's underlying logic and its numerous applications. Dynamical Systems Method and Applications begins with a general introduction and then sets forth the scope of DSM in Part One. Part Two introduces the discrepancy principle, and Part Three offers examples of numerical applications of DSM to solve a broad range of problems in science and engineering. Additional featured topics include: General nonlinear operator equations Operators satisfying a spectral assumption Newton-type methods without inversion of the derivative Numerical problems arising in applications Stable numerical differentiation Stable solution to ill-conditioned linear algebraic systems Throughout the chapters, the authors employ the use of figures and tables to help readers grasp and apply new concepts. Numerical examples offer original theoretical results based on the solution of practical problems involving ill-conditioned linear algebraic systems, and stable differentiation of noisy data. Written by internationally recognized authorities on the topic, Dynamical Systems Method and Applications is an excellent book for courses on numerical analysis, dynamical systems, operator theory, and applied mathematics at the graduate level. The book also serves as a valuable resource for professionals in the fields of mathematics, physics, and engineering.

## Dynamical Systems Method and Applications

Author | : Alexander G. Ramm,Nguyen S. Hoang |

Publsiher | : John Wiley & Sons |

Total Pages | : 576 |

Release | : 2011-12-20 |

ISBN 10 | : 1118024281 |

ISBN 13 | : 9781118024287 |

Language | : EN, FR, DE, ES & NL |

**Dynamical Systems Method and Applications Book Review:**

Demonstrates the application of DSM to solve a broad range of operator equations The dynamical systems method (DSM) is a powerful computational method for solving operator equations. With this book as their guide, readers will master the application of DSM to solve a variety of linear and nonlinear problems as well as ill-posed and well-posed problems. The authors offer a clear, step-by-step, systematic development of DSM that enables readers to grasp the method's underlying logic and its numerous applications. Dynamical Systems Method and Applications begins with a general introduction and then sets forth the scope of DSM in Part One. Part Two introduces the discrepancy principle, and Part Three offers examples of numerical applications of DSM to solve a broad range of problems in science and engineering. Additional featured topics include: General nonlinear operator equations Operators satisfying a spectral assumption Newton-type methods without inversion of the derivative Numerical problems arising in applications Stable numerical differentiation Stable solution to ill-conditioned linear algebraic systems Throughout the chapters, the authors employ the use of figures and tables to help readers grasp and apply new concepts. Numerical examples offer original theoretical results based on the solution of practical problems involving ill-conditioned linear algebraic systems, and stable differentiation of noisy data. Written by internationally recognized authorities on the topic, Dynamical Systems Method and Applications is an excellent book for courses on numerical analysis, dynamical systems, operator theory, and applied mathematics at the graduate level. The book also serves as a valuable resource for professionals in the fields of mathematics, physics, and engineering.

## Methods of Hilbert Spaces in the Theory of Nonlinear Dynamical Systems

Author | : K Kowalski |

Publsiher | : World Scientific |

Total Pages | : 140 |

Release | : 1994-07-26 |

ISBN 10 | : 9814502057 |

ISBN 13 | : 9789814502054 |

Language | : EN, FR, DE, ES & NL |

**Methods of Hilbert Spaces in the Theory of Nonlinear Dynamical Systems Book Review:**

This book is the first monograph on a new powerful method discovered by the author for the study of nonlinear dynamical systems relying on reduction of nonlinear differential equations to the linear abstract Schrödinger-like equation in Hilbert space. Besides the possibility of unification of many apparently completely different techniques, the “quantal” Hilbert space formalism introduced enables new original methods to be discovered for solving nonlinear problems arising in investigation of ordinary and partial differential equations as well as difference equations. Applications covered in the book include symmetries and first integrals, linearization transformations, Bäcklund transformations, stroboscopic maps, functional equations involving the case of Feigenbaum-Cvitanovic renormalization equations and chaos. Contents:IntroductionOrdinary Differential Equations:Evolution Equation in Hilbert SpaceOperator Evolution EquationsSymmetries and First IntegralsAlternative Linearization ApproachesPartial Differential Equations:Evolution Equation in Hilbert SpaceOperator Evolution EquationsSymmetries and First IntegralsDifference Equations:Evolution Equation in Hilbert SpaceOperator Evolution EquationsFunctional EquationsApplications:First IntegralsLinearization TransformationsBäcklund TransformationsFeigenbaum-Cvitanovic Renormalization EquationsChaosAppendices:Hilbert SpacesQuantum MechanicsBose Operators and Coherent StatesPosition and Momentum OperatorsFunctional DerivativeBibliographySymbol IndexSubject Index Readership: Researchers in the field of nonlinear dynamical systems and advanced graduate students. keywords:Nonlinear Dynamical Systems;Classical Mechanics;Carleman Linearization;Koopman Approach;Hilbert Space “… a systematic and detailed presentation of the Hilbert space approach to the theory of nonlinear dynamical systems, a far-reaching generalization of the Carleman embedding.” Mathematical Reviews

## Inverse Problems

Author | : Alexander G. Ramm |

Publsiher | : Springer Science & Business Media |

Total Pages | : 442 |

Release | : 2006-01-20 |

ISBN 10 | : 0387232184 |

ISBN 13 | : 9780387232188 |

Language | : EN, FR, DE, ES & NL |

**Inverse Problems Book Review:**

Inverse Problems is a monograph which contains a self-contained presentation of the theory of several major inverse problems and the closely related results from the theory of ill-posed problems. The book is aimed at a large audience which include graduate students and researchers in mathematical, physical, and engineering sciences and in the area of numerical analysis.

## Inverse Problems

Author | : Alexander G. Ramm |

Publsiher | : Springer |

Total Pages | : 442 |

Release | : 2006-01-20 |

ISBN 10 | : 9780387232188 |

ISBN 13 | : 0387232184 |

Language | : EN, FR, DE, ES & NL |

**Inverse Problems Book Review:**

Inverse Problems is a monograph which contains a self-contained presentation of the theory of several major inverse problems and the closely related results from the theory of ill-posed problems. The book is aimed at a large audience which include graduate students and researchers in mathematical, physical, and engineering sciences and in the area of numerical analysis.

## Mathematical Reviews

Author | : Anonim |

Publsiher | : Unknown |

Total Pages | : 329 |

Release | : 2007 |

ISBN 10 | : |

ISBN 13 | : UOM:39015078588616 |

Language | : EN, FR, DE, ES & NL |

**Mathematical Reviews Book Review:**

## Seventh Conference on Probability and Statistics in Atmospheric Sciences of the American Meteorological Society November 2 6 1981 Monterey California

Author | : Anonim |

Publsiher | : Unknown |

Total Pages | : 235 |

Release | : 1981 |

ISBN 10 | : |

ISBN 13 | : UCSD:31822028410587 |

Language | : EN, FR, DE, ES & NL |

**Seventh Conference on Probability and Statistics in Atmospheric Sciences of the American Meteorological Society November 2 6 1981 Monterey California Book Review:**

## Linear Operator Equations

Author | : Anonim |

Publsiher | : Unknown |

Total Pages | : 329 |

Release | : 2021 |

ISBN 10 | : 981446967X |

ISBN 13 | : 9789814469678 |

Language | : EN, FR, DE, ES & NL |

**Linear Operator Equations Book Review:**

## Ordinary Differential Equations and Dynamical Systems

Author | : Gerald Teschl |

Publsiher | : American Mathematical Soc. |

Total Pages | : 356 |

Release | : 2012-08-30 |

ISBN 10 | : 0821883283 |

ISBN 13 | : 9780821883280 |

Language | : EN, FR, DE, ES & NL |

**Ordinary Differential Equations and Dynamical Systems Book Review:**

This book provides a self-contained introduction to ordinary differential equations and dynamical systems suitable for beginning graduate students. The first part begins with some simple examples of explicitly solvable equations and a first glance at qualitative methods. Then the fundamental results concerning the initial value problem are proved: existence, uniqueness, extensibility, dependence on initial conditions. Furthermore, linear equations are considered, including the Floquet theorem, and some perturbation results. As somewhat independent topics, the Frobenius method for linear equations in the complex domain is established and Sturm-Liouville boundary value problems, including oscillation theory, are investigated. The second part introduces the concept of a dynamical system. The Poincare-Bendixson theorem is proved, and several examples of planar systems from classical mechanics, ecology, and electrical engineering are investigated. Moreover, attractors, Hamiltonian systems, the KAM theorem, and periodic solutions are discussed. Finally, stability is studied, including the stable manifold and the Hartman-Grobman theorem for both continuous and discrete systems. The third part introduces chaos, beginning with the basics for iterated interval maps and ending with the Smale-Birkhoff theorem and the Melnikov method for homoclinic orbits. The text contains almost three hundred exercises. Additionally, the use of mathematical software systems is incorporated throughout, showing how they can help in the study of differential equations.

## Numerical Mathematics

Author | : Anonim |

Publsiher | : Unknown |

Total Pages | : 329 |

Release | : 2007 |

ISBN 10 | : |

ISBN 13 | : UOM:39015072605630 |

Language | : EN, FR, DE, ES & NL |

**Numerical Mathematics Book Review:**

## Nonlinear Dynamical Systems in Engineering

Author | : Vasile Marinca,Nicolae Herisanu |

Publsiher | : Springer Science & Business Media |

Total Pages | : 396 |

Release | : 2012-01-05 |

ISBN 10 | : 364222735X |

ISBN 13 | : 9783642227356 |

Language | : EN, FR, DE, ES & NL |

**Nonlinear Dynamical Systems in Engineering Book Review:**

This book presents and extend different known methods to solve different types of strong nonlinearities encountered by engineering systems. A better knowledge of the classical methods presented in the first part lead to a better choice of the so-called “base functions”. These are absolutely necessary to obtain the auxiliary functions involved in the optimal approaches which are presented in the second part. Every chapter introduces a distinct approximate method applicable to nonlinear dynamical systems. Each approximate analytical approach is accompanied by representative examples related to nonlinear dynamical systems from to various fields of engineering.

## Differential Equations and Dynamical Systems

Author | : D. Bahuguna |

Publsiher | : Alpha Science Int'l Ltd. |

Total Pages | : 227 |

Release | : 2005 |

ISBN 10 | : 9788173195884 |

ISBN 13 | : 8173195889 |

Language | : EN, FR, DE, ES & NL |

**Differential Equations and Dynamical Systems Book Review:**

Differential Equations and Dynamical Systems in fifteen chapters from eminent researchers working in the area of differential equations and dynamical systems covers wavelets and their applications, Markovian structural perturbations, conservation laws and their applications, retarded functional differential equations and applications to problems in population dynamics, finite element method and its application to extended Fisher-Kolmogorv equation, generalized K-dV equation, Faedo-Galerkin approximations of solutions to evolution equations, the method of semidiscretization in time and its applications to nonlinear evolution equations, spectral methods, Tikhonov regularization of partial differential equations, mathematical modeling and second order evolution equations.

## The Koopman Operator in Systems and Control

Author | : Alexandre Mauroy,Igor Mezić,Yoshihiko Susuki |

Publsiher | : Springer Nature |

Total Pages | : 556 |

Release | : 2020-02-22 |

ISBN 10 | : 3030357139 |

ISBN 13 | : 9783030357139 |

Language | : EN, FR, DE, ES & NL |

**The Koopman Operator in Systems and Control Book Review:**

This book provides a broad overview of state-of-the-art research at the intersection of the Koopman operator theory and control theory. It also reviews novel theoretical results obtained and efficient numerical methods developed within the framework of Koopman operator theory. The contributions discuss the latest findings and techniques in several areas of control theory, including model predictive control, optimal control, observer design, systems identification and structural analysis of controlled systems, addressing both theoretical and numerical aspects and presenting open research directions, as well as detailed numerical schemes and data-driven methods. Each contribution addresses a specific problem. After a brief introduction of the Koopman operator framework, including basic notions and definitions, the book explores numerical methods, such as the dynamic mode decomposition (DMD) algorithm and Arnoldi-based methods, which are used to represent the operator in a finite-dimensional basis and to compute its spectral properties from data. The main body of the book is divided into three parts: theoretical results and numerical techniques for observer design, synthesis analysis, stability analysis, parameter estimation, and identification; data-driven techniques based on DMD, which extract the spectral properties of the Koopman operator from data for the structural analysis of controlled systems; and Koopman operator techniques with specific applications in systems and control, which range from heat transfer analysis to robot control. A useful reference resource on the Koopman operator theory for control theorists and practitioners, the book is also of interest to graduate students, researchers, and engineers looking for an introduction to a novel and comprehensive approach to systems and control, from pure theory to data-driven methods.

## Dynamic Impulse Systems

Author | : S.T. Zavalishchin,A.N. Sesekin |

Publsiher | : Springer Science & Business Media |

Total Pages | : 260 |

Release | : 2013-03-14 |

ISBN 10 | : 9401588937 |

ISBN 13 | : 9789401588935 |

Language | : EN, FR, DE, ES & NL |

**Dynamic Impulse Systems Book Review:**

A number of optimization problems of the mechanics of space flight and the motion of walking robots and manipulators, and of quantum physics, eco momics and biology, have an irregular structure: classical variational proce dures do not formally make it possible to find optimal controls that, as we explain, have an impulse character. This and other well-known facts lead to the necessity for constructing dynamical models using the concept of a gener alized function (Schwartz distribution). The problem ofthe systematization of such models is very important. In particular, the problem of the construction of the general form of linear and nonlinear operator equations in distributions is timely. Another problem is related to the proper determination of solutions of equations that have nonlinear operations over generalized functions in their description. It is well-known that "the value of a distribution at a point" has no meaning. As a result the problem to construct the concept of stability for generalized processes arises. Finally, optimization problems for dynamic systems in distributions need finding optimality conditions. This book contains results that we have obtained in the above-mentioned directions. The aim of the book is to provide for electrical and mechanical engineers or mathematicians working in applications, a general and systematic treat ment of dynamic systems based on up-to-date mathematical methods and to demonstrate the power of these methods in solving dynamics of systems and applied control problems.

## Nonlinear Stochastic Operator Equations

Author | : G. Adomian |

Publsiher | : Unknown |

Total Pages | : 287 |

Release | : 1986 |

ISBN 10 | : |

ISBN 13 | : UOM:39015015714101 |

Language | : EN, FR, DE, ES & NL |

**Nonlinear Stochastic Operator Equations Book Review:**

## Nonlinear Differential Equations and Dynamical Systems

Author | : Ferdinand Verhulst |

Publsiher | : Springer Science & Business Media |

Total Pages | : 277 |

Release | : 2012-12-06 |

ISBN 10 | : 3642971490 |

ISBN 13 | : 9783642971495 |

Language | : EN, FR, DE, ES & NL |

**Nonlinear Differential Equations and Dynamical Systems Book Review:**

Bridging the gap between elementary courses and the research literature in this field, the book covers the basic concepts necessary to study differential equations. Stability theory is developed, starting with linearisation methods going back to Lyapunov and Poincaré, before moving on to the global direct method. The Poincaré-Lindstedt method is introduced to approximate periodic solutions, while at the same time proving existence by the implicit function theorem. The final part covers relaxation oscillations, bifurcation theory, centre manifolds, chaos in mappings and differential equations, and Hamiltonian systems. The subject material is presented from both the qualitative and the quantitative point of view, with many examples to illustrate the theory, enabling the reader to begin research after studying this book.

## The Navier Stokes Problem

Author | : Alexander G. Ramm |

Publsiher | : Morgan & Claypool Publishers |

Total Pages | : 77 |

Release | : 2021-04-06 |

ISBN 10 | : 1636391230 |

ISBN 13 | : 9781636391236 |

Language | : EN, FR, DE, ES & NL |

**The Navier Stokes Problem Book Review:**

The main result of this book is a proof of the contradictory nature of the Navier‒Stokes problem (NSP). It is proved that the NSP is physically wrong, and the solution to the NSP does not exist on ℝ+ (except for the case when the initial velocity and the exterior force are both equal to zero; in this case, the solution 𝑣(𝑥, 𝑡) to the NSP exists for all 𝑡 ≥ 0 and 𝑣(𝑥, 𝑡) = 0). It is shown that if the initial data 𝑣0(𝑥) ≢ 0, 𝑓(𝑥,𝑡) = 0 and the solution to the NSP exists for all 𝑡 ϵ ℝ+, then 𝑣0(𝑥) := 𝑣(𝑥, 0) = 0. This Paradox proves that the NSP is physically incorrect and mathematically unsolvable, in general. Uniqueness of the solution to the NSP in the space 𝑊21(ℝ3) × C(ℝ+) is proved, 𝑊21(ℝ3) is the Sobolev space, ℝ+ = [0, ∞). Theory of integral equations and inequalities with hyper-singular kernels is developed. The NSP is reduced to an integral inequality with a hyper-singular kernel.

## Optimization and Numerical Algebra

Author | : Liqun Qi,Wenyu Sun,Jianzhong Zhang |

Publsiher | : Unknown |

Total Pages | : 388 |

Release | : 2001 |

ISBN 10 | : |

ISBN 13 | : STANFORD:36105111152091 |

Language | : EN, FR, DE, ES & NL |

**Optimization and Numerical Algebra Book Review:**

## International Aerospace Abstracts

Author | : Anonim |

Publsiher | : Unknown |

Total Pages | : 329 |

Release | : 1986 |

ISBN 10 | : |

ISBN 13 | : STANFORD:36105007165058 |

Language | : EN, FR, DE, ES & NL |

**International Aerospace Abstracts Book Review:**