Download or read book Development of New Methods for the Solution of Nonlinear Differential Equations by the Method of Lie Series and Extension to New Fields written by W. Groebner and published by . This book was released on 1970 with total page 120 pages. Available in PDF, EPUB and Kindle. Book excerpt: Chapter 1 of the report gives an application of the well experienced method of Lie series to the theory of Lie groups. First, find a representation of these functions by Lie series. The operators are commutative and contain a matrix of functions w sub ik(x;y) the infinitesimal transformations and the connected Lie algebra, or Lie ring more generally, of a given Lie group are derived. Linear infinitesimal operators are developed in detail the construction of the invariants belonging to these groups with the help of Lie series is demonstrated. A new method for finding subgroups is shown. A new derivation of the Campbell-Baker-Hausdorff-Formula and improvement to the Cayley Theorem is given. Chapter 2 clears the connection between the perturbation formulas of Groebner (1960) and Alexseev (1961) for the solution of ordinary differential equations. These formulas are generalized and iteration methods are given, which include the Methods of Picard, Groebner-Knapp, Poincare, Chen, as special cases. Chapter 3 generalizes an iterated integral equation of Chen and indicates an iteration method based on this generalization. A compound form combining the generalization with Groebner's perturbation formula is furnished. (Author).
Download or read book Development of New Methods for the Solution of Nonlinear Differential Equations by the Method of Lie Series and Extension to New Fields written by Wolfgang Gröbner and published by . This book was released on 1970 with total page 116 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Bibliography of Scientific and Industrial Reports written by and published by . This book was released on 1970 with total page 922 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book U S Government Research Development Reports written by and published by . This book was released on 1971 with total page 1368 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Development of New Methods for the Solution of Differential Equations by the Method of Lie Series written by W. Groebner and published by . This book was released on 1969 with total page 164 pages. Available in PDF, EPUB and Kindle. Book excerpt: The report summarizes the recent work in the application of the LIE-series method to the solution of ordinary and partial differential equations. The power series method which is a special case of the Lie series method of chapter III is described in chapter II. Chapter III deals with the numerical evaluation of the Lie series perturbation formula. In chapter IV we prove Grobner's integral equation which leads to short proofs of the formulas of chapter III and to various generalizations of the method. A survey of these is presented at the end of this summary. Chapter V generalizes the concept of Runge-Kutta to methods with multiple nodes, which is possible with the use of the Lie differential operator D. Chapter VI deals with the step-size control and chapter VII shows the application of generalized Lie series to the calculation of switch-on transients occurring in the telegraphic equation.
Download or read book Recent Developments in the Solution of Nonlinear Differential Equations written by Bruno Carpentieri and published by BoD – Books on Demand. This book was released on 2021-09-08 with total page 374 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nonlinear differential equations are ubiquitous in computational science and engineering modeling, fluid dynamics, finance, and quantum mechanics, among other areas. Nowadays, solving challenging problems in an industrial setting requires a continuous interplay between the theory of such systems and the development and use of sophisticated computational methods that can guide and support the theoretical findings via practical computer simulations. Owing to the impressive development in computer technology and the introduction of fast numerical methods with reduced algorithmic and memory complexity, rigorous solutions in many applications have become possible. This book collects research papers from leading world experts in the field, highlighting ongoing trends, progress, and open problems in this critically important area of mathematics.
Download or read book U S Government Research Development Reports written by and published by . This book was released on 1966-11 with total page 1014 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Government Reports Announcements written by and published by . This book was released on 1971-02-25 with total page 238 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics written by Victor A. Galaktionov and published by CRC Press. This book was released on 2006-11-02 with total page 538 pages. Available in PDF, EPUB and Kindle. Book excerpt: Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics is the first book to provide a systematic construction of exact solutions via linear invariant subspaces for nonlinear differential operators. Acting as a guide to nonlinear evolution equations and models from physics and mechanics, the book focuses on the existence of new exact solutions on linear invariant subspaces for nonlinear operators and their crucial new properties. This practical reference deals with various partial differential equations (PDEs) and models that exhibit some common nonlinear invariant features. It begins with classical as well as more recent examples of solutions on invariant subspaces. In the remainder of the book, the authors develop several techniques for constructing exact solutions of various nonlinear PDEs, including reaction-diffusion and gas dynamics models, thin-film and Kuramoto-Sivashinsky equations, nonlinear dispersion (compacton) equations, KdV-type and Harry Dym models, quasilinear magma equations, and Green-Naghdi equations. Using exact solutions, they describe the evolution properties of blow-up or extinction phenomena, finite interface propagation, and the oscillatory, changing sign behavior of weak solutions near interfaces for nonlinear PDEs of various types and orders. The techniques surveyed in Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics serve as a preliminary introduction to the general theory of nonlinear evolution PDEs of different orders and types.
Download or read book Selected Works of Ellis Kolchin with Commentary written by Ellis Robert Kolchin and published by American Mathematical Soc.. This book was released on 1999 with total page 660 pages. Available in PDF, EPUB and Kindle. Book excerpt: The work of Joseph Fels Ritt and Ellis Kolchin in differential algebra paved the way for exciting new applications in constructive symbolic computation, differential Galois theory, the model theory of fields, and Diophantine geometry. This volume assembles Kolchin's mathematical papers, contributing solidly to the archive on construction of modern differential algebra. This collection of Kolchin's clear and comprehensive papers--in themselves constituting a history of the subject--is an invaluable aid to the student of differential algebra. In 1910, Ritt created a theory of algebraic differential equations modeled not on the existing transcendental methods of Lie, but rather on the new algebra being developed by E. Noether and B. van der Waerden. Building on Ritt's foundation, and deeply influenced by Weil and Chevalley, Kolchin opened up Ritt theory to modern algebraic geometry. In so doing, he led differential geometry in a new direction. By creating differential algebraic geometry and the theory of differential algebraic groups, Kolchin provided the foundation for a "new geometry" that has led to both a striking and an original approach to arithmetic algebraic geometry. Intriguing possibilities were introduced for a new language for nonlinear differential equations theory. The volume includes commentary by A. Borel, M. Singer, and B. Poizat. Also Buium and Cassidy trace the development of Kolchin's ideas, from his important early work on the differential Galois theory to his later groundbreaking results on the theory of differential algebraic geometry and differential algebraic groups. Commentaries are self-contained with numerous examples of various aspects of differential algebra and its applications. Central topics of Kolchin's work are discussed, presenting the history of differential algebra and exploring how his work grew from and transformed the work of Ritt. New directions of differential algebra are illustrated, outlining important current advances. Prerequisite to understanding the text is a background at the beginning graduate level in algebra, specifically commutative algebra, the theory of field extensions, and Galois theory.
Download or read book A New Lie Series Method for the Numerical Integration of Ordinary Differential Equations with an Application to the Restricted Problem of Three Bodies written by Siegfried Filippi and published by . This book was released on 1967 with total page 42 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Nonlinear Ordinary Differential Equations written by Martin Hermann and published by Springer. This book was released on 2016-05-09 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book discusses the solutions to nonlinear ordinary differential equations (ODEs) using analytical and numerical approximation methods. Recently, analytical approximation methods have been largely used in solving linear and nonlinear lower-order ODEs. It also discusses using these methods to solve some strong nonlinear ODEs. There are two chapters devoted to solving nonlinear ODEs using numerical methods, as in practice high-dimensional systems of nonlinear ODEs that cannot be solved by analytical approximate methods are common. Moreover, it studies analytical and numerical techniques for the treatment of parameter-depending ODEs. The book explains various methods for solving nonlinear-oscillator and structural-system problems, including the energy balance method, harmonic balance method, amplitude frequency formulation, variational iteration method, homotopy perturbation method, iteration perturbation method, homotopy analysis method, simple and multiple shooting method, and the nonlinear stabilized march method. This book comprehensively investigates various new analytical and numerical approximation techniques that are used in solving nonlinear-oscillator and structural-system problems. Students often rely on the finite element method to such an extent that on graduation they have little or no knowledge of alternative methods of solving problems. To rectify this, the book introduces several new approximation techniques.
Download or read book Advance Numerical Techniques to Solve Linear and Nonlinear Differential Equations written by Geeta Arora and published by CRC Press. This book was released on 2024-01-23 with total page 177 pages. Available in PDF, EPUB and Kindle. Book excerpt: Real-world issues can be translated into the language and concepts of mathematics with the use of mathematical models. Models guided by differential equations with intuitive solutions can be used throughout engineering and the sciences. Almost any changing system may be described by a set of differential equations. They may be found just about anywhere you look in fields including physics, engineering, economics, sociology, biology, business, healthcare, etc. The nature of these equations has been investigated by several mathematicians over the course of hundreds of years and, consequently, numerous effective methods for solving them have been created. It is often impractical to find a purely analytical solution to a system described by a differential equation because either the system itself is too complex or the system being described is too vast. Numerical approaches and computer simulations are especially helpful in such systems. The content provided in this book involves real-world examples, explores research challenges in numerical treatment, and demonstrates how to create new numerical methods for resolving problems. Theories and practical applications in the sciences and engineering are also discussed. Students of engineering and applied mathematics, as well as researchers and engineers who use computers to solve problems numerically or oversee those who do, will find this book focusing on advance numerical techniques to solve linear and nonlinear differential equations useful.
Download or read book Nuclear Science Abstracts written by and published by . This book was released on 1975 with total page 1166 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems written by Vasile Marinca and published by Springer Nature. This book was released on 2021-07-14 with total page 479 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents the optimal auxiliary functions method and applies it to various engineering problems and in particular in boundary layer problems. The cornerstone of the presented procedure is the concept of “optimal auxiliary functions” which are needed to obtain accurate results in an efficient way. Unlike other known analytic approaches, this procedure provides us with a simple but rigorous way to control and adjust the convergence of the solutions of nonlinear dynamical systems. The optimal auxiliary functions are depending on some convergence-control parameters whose optimal values are rigorously determined from mathematical point of view. The capital strength of our procedure is its fast convergence, since after only one iteration, we obtain very accurate analytical solutions which are very easy to be verified. Moreover, no simplifying hypothesis or assumptions are made. The book contains a large amount of practical models from various fields of engineering such as classical and fluid mechanics, thermodynamics, nonlinear oscillations, electrical machines, and many more. The book is a continuation of our previous books “Nonlinear Dynamical Systems in Engineering. Some Approximate Approaches”, Springer-2011 and “The Optimal Homotopy Asymptotic Method. Engineering Applications”, Springer-2015.
Download or read book Applied Mechanics Reviews written by and published by . This book was released on 1970 with total page 772 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book An Index and Other Useful Information written by A. Dold and published by Springer. This book was released on 2013-12-11 with total page 82 pages. Available in PDF, EPUB and Kindle. Book excerpt:
