Development Of New Methods For The Solution Of Nonlinear Differential Equations By The Method Of Lie Series And Extension To New Fields PDF Download
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Author | : W. Groebner |
Publisher | : |
Total Pages | : 120 |
Release | : 1970 |
Genre | : |
ISBN | : |
Download Development of New Methods for the Solution of Nonlinear Differential Equations by the Method of Lie Series and Extension to New Fields Book in PDF, ePub and Kindle
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).
Author | : Wolfgang Gröbner |
Publisher | : |
Total Pages | : 116 |
Release | : 1970 |
Genre | : |
ISBN | : |
Download Development of New Methods for the Solution of Nonlinear Differential Equations by the Method of Lie Series and Extension to New Fields Book in PDF, ePub and Kindle
Author | : |
Publisher | : |
Total Pages | : 922 |
Release | : 1970 |
Genre | : Science |
ISBN | : |
Download Bibliography of Scientific and Industrial Reports Book in PDF, ePub and Kindle
Author | : |
Publisher | : |
Total Pages | : 1368 |
Release | : 1971 |
Genre | : Science |
ISBN | : |
Download U.S. Government Research & Development Reports Book in PDF, ePub and Kindle
Author | : W. Groebner |
Publisher | : |
Total Pages | : 164 |
Release | : 1969 |
Genre | : |
ISBN | : |
Download Development of New Methods for the Solution of Differential Equations by the Method of Lie Series Book in PDF, ePub and Kindle
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.
Author | : Bruno Carpentieri |
Publisher | : BoD – Books on Demand |
Total Pages | : 374 |
Release | : 2021-09-08 |
Genre | : Mathematics |
ISBN | : 1839686561 |
Download Recent Developments in the Solution of Nonlinear Differential Equations Book in PDF, ePub and Kindle
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.
Author | : |
Publisher | : |
Total Pages | : 1014 |
Release | : 1966-11 |
Genre | : Science |
ISBN | : |
Download U.S. Government Research & Development Reports Book in PDF, ePub and Kindle
Author | : |
Publisher | : |
Total Pages | : 238 |
Release | : 1971-02-25 |
Genre | : Technology |
ISBN | : |
Download Government Reports Announcements Book in PDF, ePub and Kindle
Author | : Victor A. Galaktionov |
Publisher | : CRC Press |
Total Pages | : 538 |
Release | : 2006-11-02 |
Genre | : Mathematics |
ISBN | : 9781584886631 |
Download Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics Book in PDF, ePub and Kindle
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.
Author | : Ellis Robert Kolchin |
Publisher | : American Mathematical Soc. |
Total Pages | : 660 |
Release | : 1999 |
Genre | : Mathematics |
ISBN | : 9780821805428 |
Download Selected Works of Ellis Kolchin with Commentary Book in PDF, ePub and Kindle
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.