Differential Equations And Group Methods For Scientists And Engineers

Differential Equations and Group Methods for Scientists and Engineers presents a basic introduction to the technically complex area of invariant one-parameter Lie group methods and their use in solving differential equations. The book features discussions on ordinary differential equations (first, second, and higher order) in addition to partial differential equations (linear and nonlinear). Each chapter contains worked examples with several problems at the end; answers to these problems and hints on how to solve them are found at the back of the book. Students and professionals in mathematics, science, and engineering will find this book indispensable for developing a fundamental understanding of how to use invariant one-parameter group methods to solve differential equations.

Differential equations, especially nonlinear, present the most effective way for describing complex physical processes. Methods for constructing exact solutions of differential equations play an important role in applied mathematics and mechanics. This book aims to provide scientists, engineers and students with an easy-to-follow, but comprehensive, description of the methods for constructing exact solutions of differential equations.

This expanded and revised second edition is a comprehensive and systematic treatment of linear and nonlinear partial differential equations and their varied applications. Building upon the successful material of the first book, this edition contains updated modern examples and applications from diverse fields. Methods and properties of solutions, along with their physical significance, help make the book more useful for a diverse readership. The book is an exceptionally complete text/reference for graduates, researchers, and professionals in mathematics, physics, and engineering.

This significantly expanded fourth edition is designed as an introduction to the theory and applications of linear PDEs. The authors provide fundamental concepts, underlying principles, a wide range of applications, and various methods of solutions to PDEs. In addition to essential standard material on the subject, the book contains new material that is not usually covered in similar texts and reference books. It also contains a large number of worked examples and exercises dealing with problems in fluid mechanics, gas dynamics, optics, plasma physics, elasticity, biology, and chemistry; solutions are provided.

It is the first text that in addition to standard convergence theory treats other necessary ingredients for successful numerical simulations of physical systems encountered by every practitioner. The book is aimed at users with interests ranging from application modeling to numerical analysis and scientific software development. It is strongly influenced by the authors research in in space physics, electrical and optical engineering, applied mathematics, numerical analysis and professional software development. The material is based on a year-long graduate course taught at the University of Arizona since 1989. The book covers the first two-semesters of a three semester series. The second semester is based on a semester-long project, while the third semester requirement consists of a particular methods course in specific disciplines like computational fluid dynamics, finite element method in mechanical engineering, computational physics, biology, chemistry, photonics, etc. The first three chapters focus on basic properties of partial differential equations, including analysis of the dispersion relation, symmetries, particular solutions and instabilities of the PDEs; methods of discretization and convergence theory for initial value problems. The goal is to progress from observations of simple numerical artifacts like diffusion, damping, dispersion, and anisotropies to their analysis and management technique, as it is not always possible to completely eliminate them. In the second part of the book we cover topics for which there are only sporadic theoretical results, while they are an integral part and often the most important part for successful numerical simulation. We adopt a more heuristic and practical approach using numerical methods of investigation and validation. The aim is teach students subtle key issues in order to separate physics from numerics. The following topics are addressed: Implementation of transparent and absorbing boundary conditions; Practical stability analysis in the presence of the boundaries and interfaces; Treatment of problems with different temporal/spatial scales either explicit or implicit; preservation of symmetries and additional constraints; physical regularization of singularities; resolution enhancement using adaptive mesh refinement and moving meshes. Self contained presentation of key issues in successful numerical simulation Accessible to scientists and engineers with diverse background Provides analysis of the dispersion relation, symmetries, particular solutions and instabilities of the partial differential equations

Group analysis of differential equations has applications to various problems in nonlinear mechanics and physics. For the first time, this book gives the systematic group analysis of main postulates of classical and relativistic mechanics. The consistent presentation of Lie group theory is illustrated by plentiful examples. Symmetries and conservation laws of differential equations are studied. Specific equations and problems of mechanics and physics are considered, and exact solutions are given for the following equations: dynamics of rigid body, heat transfer, wave, hydrodynamics, Thomas-Fermi and more. The author pays particular attention to the application of group analysis to developing asymptotic methods of applied mathematics in problems with small parameter. The methods are used to solve basic equations (Van Der Pol's equation, Duffing equation, etc.) encountered in the theory of nonlinear oscillations. This book is intended for a wide range of scientists, engineers and students in the fields of applied mathematics, mechanics and physics.

Volume 2 offers a unique blend of classical results of Sophus Lie with new, modern developments and numerous applications which span a period of more than 100 years. As a result, this reference is up to date, with the latest information on the group theoretic methods used frequently in mathematical physics and engineering. Volume 2 is divided into three parts. Part A focuses on relevant definitions, main algorithms, group classification schemes for partial differential equations, and multifaceted possibilities offered by Lie group theoretic philosophy. Part B contains the group analysis of a variety of mathematical models for diverse natural phenomena. It tabulates symmetry groups and solutions for linear equations of mathematical physics, classical field theory, viscous and non-Newtonian fluids, boundary layer problems, Earth sciences, elasticity, plasticity, plasma theory (Vlasov-Maxwell equations), and nonlinear optics and acoustics. Part C offers an English translation of Sophus Lie's fundamental paper on the group classification and invariant solutions of linear second-order equations with two independent variables. This will serve as a concise, practical guide to the group analysis of partial differential equations.

REFEREN CES . 156 9 Transforma.tion of a Boundary Value Problem to an Initial Value Problem . 157 9.0 Introduction . 157 9.1 Blasius Equation in Boundary Layer Flow . 157 9.2 Longitudinal Impact of Nonlinear Viscoplastic Rods . 163 9.3 Summary . 168 REFERENCES . . . . . . . . . . . . . . . . . . 168 . 10 From Nonlinear to Linear Differential Equa.tions Using Transformation Groups. . . . . . . . . . . . . . 169 . 10.1 From Nonlinear to Linear Differential Equations . 170 10.2 Application to Ordinary Differential Equations -Bernoulli's Equation . . . . . . . . . . . 173 10.3 Application to Partial Differential Equations -A Nonlinear Chemical Exchange Process . 178 10.4 Limitations of the Inspectional Group Method . 187 10.5 Summary . 188 REFERENCES . . . . 188 11 Miscellaneous Topics . 190 11.1 Reduction of Differential Equations to Algebraic Equations 190 11.2 Reduction of Order of an Ordinary Differential Equation . 191 11.3 Transformat.ion From Ordinary to Partial Differential Equations-Search for First Integrals . . . . . . " 193 . 11.4 Reduction of Number of Variables by Multiparameter Groups of Transformations . . . . . . . . .. . . . 194 11.5 Self-Similar Solutions of the First and Second Kind . . 202 11.6 Normalized Representation and Dimensional Consideration 204 REFERENCES .206 Problems . 208 .220 Index .. Chapter 1 INTRODUCTION AND GENERAL OUTLINE Physical problems in engineering science are often described by dif ferential models either linear or nonlinear. There is also an abundance of transformations of various types that appear in the literature of engineer ing and mathematics that are generally aimed at obtaining some sort of simplification of a differential model.