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Symmetries of Integro-Differential Equations : With Applications in Mechanics and Plasma Physics - Yurii N. Grigoriev

Symmetries of Integro-Differential Equations

With Applications in Mechanics and Plasma Physics

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Published: 11th September 2010
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The present book is an introduction to a new eld in applied group analysis. The book deals with symmetries of integro-differential, stochastic and delay equations that form the basis of a large variety of mathematical models, used to describe va- ous phenomena in uid mechanics and plasma physics and other elds of nonlinear science. Because of its baf ing complexity the mathematical study of nonlocal equations is far from completion, although the equations have been intensively studied in - merous applications over more than fty last years using both numerical and anal- ical methods. The principal aim of analytical approaches is to obtain exact solutions, admitted symmetries, conservation laws and other mathematical properties, which allow one to make sound decisions in more detailed applied investigations. Classical Lie group theory provides a universal tool for calculating symmetry groups for systems of differential equations. Consequently, group theoretical me- ods appear ef cient in analyzing different phenomena using mathematical models that employ differential equations. However Lie's methods cannot be directly - plied to integro-differential equations, in nite systems of differential equations, - lay equations, etc. Hence it is natural to extend the ideas of modern group analysis to these mathematical objects that up to recently were not in mainstream of classical group theoretical approaches.

From the reviews: "The present book deals with a large number of topics concerning symmetries of integro-differential, stochastic and delay equations that are met in many fields, from fluid mechanics and plasma physics to biology and so on. ! In conclusion, the book is rich in results and is of interest not only to physicists and applied scientists, but also to mathematicians working with (nonlinear) integrodifferential equations." (Alfredo Lorenzi, Mathematical Reviews, Issue 2011 i)

Introduction to Group Analysis of Differential Equationsp. 1
One-Parameter Groupsp. 1
Definition of a Transformation Groupp. 1
Generator of a One-Parameter Groupp. 2
Construction of a Group with a Given Generatorp. 3
Introduction of Canonical Variablesp. 5
Invariants (Invariant Functions)p. 5
Invariant Equations (Manifolds)p. 7
Representation of Regular Invariant Manifolds via Invariantsp. 9
Symmetries and Integration of Ordinary Differential Equationsp. 10
The Frame of Differential Equationsp. 10
Prolongation of Group Transformations and Their Generatorsp. 12
Group Admitted by Differential Equationsp. 13
Determining Equation for Infinitesimal Symmetriesp. 14
An Example on Calculation of Symmetriesp. 14
Lie Algebras. Specific Property of Determining Equationsp. 16
Integration of First-Order Equations: Lie's Integrating Factorp. 17
Integration of First-Order Equations: Method of Canonical Variablesp. 18
Standard Forms of Two-Dimensional Lie Algebrasp. 21
Lie's Method of Integration for Second-Order Equationsp. 22
Symmetries and Invariant Solutions of Partial Differential Equationsp. 24
Discussion of Symmetries for Evolution Equationsp. 24
Calculation of Symmetries for Burgers' Equationp. 27
Invariant Solutions and Their Calculationp. 29
Group Transformations of Solutionsp. 30
Optimal Systems of Subalgebrasp. 31
All Invariant Solutions of the Burgers Equationp. 36
General Definitions of Symmetry Groupsp. 38
Differential Variables and Functionp. 38
Frame and Extended Framep. 39
Definition Using Solutionsp. 40
Definition Using the Framep. 40
Definition Using the Extended Framep. 41
Lie-Bäcklund Transformation Groupsp. 42
Lie-Bäcklund Operatorsp. 42
Integration of Lie-Bäcklund Equationsp. 45
Lie-Bäcklund Symmetriesp. 48
Approximate Transformation Groupsp. 49
Approximate Transformations and Generatorsp. 50
Approximate Lie Equationsp. 52
Approximate Symmetriesp. 53
Referencesp. 54
Introduction to Group Analysis and Invariant Solutions of Integro-Differential Equationsp. 57
Integro-Differential Equations in Mathematics and in Applicationsp. 57
Survey of Various Approaches or Finding Invariant Solutionsp. 58
Methods Using a Presentation of a Solution or an Admitted Lie Groupp. 59
Methods of Momentsp. 73
Methods Using a Transition to Equivalent Differential Equationsp. 78
A Regular Method for Calculating Symmetries of Equations with Nonlocal Operatorsp. 89
Admitted Lie Group of Partial Differential Equationsp. 90
The Approach for Equations with Nonlocal Operatorsp. 92
Illustrative Examplesp. 94
The Fourier-Image of the Spatially Homogeneous Isotropic Boltzmann Equationp. 95
Equations of One-Dimensional Viscoelastic Continuum Motionp. 102
Referencesp. 108
The Boltzmann Kinetic Equation and Various Modelsp. 113
Studies of Invariant Solutions of the Boltzmann Equationp. 113
Introduction to the Boltzmann Equationp. 114
Group Analysis of the Full Boltzmann Equationp. 116
Admitted Lie Algebrasp. 116
Isomorphism of Algebrasp. 121
Invariant Solutions of the Full Boltzmann Equationp. 123
Classification of Invariant Solutions of the Fourier Transformation of the Full Boltzmann Equationp. 126
Complete Group Analysis of Some Kinetic Equationsp. 127
The Boltzmann Kinetic Equation with an Approximate Asymptotic Collision Integralp. 128
The Smolukhovsky Kinetic Equationp. 130
A System of the Space Homogeneous Boltzmann Kinetic Equationsp. 131
Homogeneous Relaxation of a Binary Model Gasp. 139
Referencesp. 142
Plasma Kinetic Theory: Vlasov-Maxwell and Related Equationsp. 145
Mathematical Modelp. 145
Definition and Infinitesimal Testp. 148
Definition of Symmetry Groupp. 149
Variational Derivative for Functionalsp. 149
Infinitesimal Criterionp. 150
Prolongation on Nonlocal Variablesp. 151
Symmetry of Plasma Kinetic Equations in One-Dimensional Approximationp. 152
Non-relativistic Electron Gasp. 152
Relativistic Electron Gasp. 160
Collisionless Non-relativistic Electron-Ion Plasmap. 161
Collisionless Relativistic Electron-Ion Plasmap. 162
Non-relativistic Electron Plasma Kinetics with a Moving and Stationary Ion Backgroundp. 164
Relativistic Electron Plasma Kinetics with a Moving Ion Backgroundp. 165
Non-relativistic Electron-Ion Plasma in Quasi-neutral Approximationp. 166
Group Analysis of Three Dimensional Collisionless Plasma Kinetic Equationsp. 168
Relativistic Electron Gas Kineticsp. 168
Relativistic Electron-Ion Plasma Kinetic Equationsp. 170
Symmetry of Vlasov-Maxwell Equations in Lagrangian Variablesp. 172
Vlasov-Type Equations: Symmetries of the Benney Equationsp. 175
Different Forms of the Benney Equationsp. 175
Lie Subgroup and Lie-Bäcklund Group: Statement of the Problemp. 176
Incompleteness of the Point Group: Statement of the Problemp. 177
Determining Equations and Their Solutionp. 178
Discussion of the Solution of the Determining Equationsp. 180
Illustrative Example for Matrix Elementsp. 181
Symmetries in Application to Plasma Kinetic Theory. Renormalization Group Symmetries for Boundary Value Problems and Solution Functionalsp. 184
Introduction to Renormgroup Symmetriesp. 185
RG Symmetry: An Idea of Construction and Its Simple Realizationp. 186
Renormgroup Algorithmp. 190
Examples of RG Symmetries in Plasma Theoryp. 193
Referencesp. 206
Symmetries of Stochastic Differential Equationsp. 209
Stochastic Integration of Processesp. 210
Stochastic Processesp. 210
The Itô Integralp. 211
The Itô Formulap. 213
Time Change in Stochastic Integralsp. 214
Stochastic ODEsp. 215
Itô Stochastic Differential Equationsp. 215
Stratonovich Stochastic Differential Equationsp. 216
Kolmogorov Equationsp. 217
Linearization of First-Order SODEp. 218
Weak Linearization Problemp. 219
Strong Linearization of First-Order SODEp. 220
Linearization of Second-Order SODEp. 225
Linearization Conditionsp. 225
Transformations of Autonomous SODEsp. 226
Admitted Transformationsp. 226
Autonomous Systems of Stochastic First-Order ODEsp. 231
Transformations of Stochastic ODEsp. 233
Short Historical Reviewp. 233
Admitted Lie Group and Determining Equationsp. 235
Symmetries of Stochastic ODEsp. 239
Determining Equationsp. 240
Admitted Lie Group of Geometric Brownian Motionp. 241
Lie Groups of Transformations of Some Stochastic ODEsp. 244
Geometric Brownian Motionp. 244
Narrow-Sense Linear Systemp. 245
Black and Scholes Marketp. 247
Nonlinear Itô Systemp. 248
Referencesp. 249
Delay Differential Equationsp. 251
Delay Differential Equations in Mathematical Modelingp. 251
Mathematical Background of Delay Ordinary Differential Equationsp. 252
Definitions and Theoremsp. 253
Definition of Symmetries of DDEsp. 255
Examplep. 255
Admitted Lie Group of DODEp. 255
Symmetries of a Model Equationp. 257
Differential-Difference Equationsp. 258
Group Classification of Second-Order DODEsp. 259
Introduction into the Problemp. 259
Determining Equationsp. 260
Properties of Admitted Generatorsp. 261
Strategy for Obtaining a Complete Classification of DODEsp. 264
Illustrative Examplesp. 264
List of Invariant DODEsp. 270
Equivalence Lie Group for DDEp. 271
Lie-Bäcklund Representation of Determining Equations for the Equivalence Lie Group of PDEsp. 271
Potential Equivalence Lie Group of Delay Differential Equationsp. 274
The Reaction-Diffusion Equation with a Delayp. 274
The Cauchy Problemp. 275
The Equivalence Lie Groupp. 275
Admitted Lie Group of Equationp. 277
Case ≠ 0p. 279
Case = 0p. 283
Summary of the Group Classificationp. 288
Invariant Solutionsp. 289
Referencesp. 291
Appendix Ap. 293
Optimal Systems of Subalgebrasp. 293
Six and Seven-Dimensional Subalgebras of the Lie Algebra L11 (Y)p. 294
Six-Dimensional Subalgebras of the Lie Algebra L12(Y)p. 295
Appendix Bp. 297
Realizations of Lie Algebras on the Real Planep. 297
Group Classification of Second-Order DODEsp. 299
Indexp. 303
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9789048137961
ISBN-10: 9048137969
Series: Lecture Notes in Physics
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 305
Published: 11th September 2010
Publisher: Springer
Country of Publication: NL
Dimensions (cm): 23.5 x 15.5  x 2.29
Weight (kg): 1.0