Lecture Notes in Physics

With Applications in Mechanics and Plasma Physics

By: N. Kh. Ibragimov, Yurii N. Grigoriev, Vladimir F. Kovalev, Sergey V. Meleshko

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Paperback | 12 July 2010 | Edition Number 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.

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"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)

"The reviewed collective monograph is very useful to both physicists and mathematicians interested in general methods for investigation of nonlinear problems using symmetries. It can also serve as a textbook on symmetries of I-DEs, stochastic and delay equations for graduate students in applied mathematics, physics and engineering." (Boris V. Loginov, Zentralblatt MATH, Vol. 1203, 2011)

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Introduction to Group Analysis of Differential Equations	p. 1
One-Parameter Groups	p. 1
Definition of a Transformation Group	p. 1
Generator of a One-Parameter Group	p. 2
Construction of a Group with a Given Generator	p. 3
Introduction of Canonical Variables	p. 5
Invariants (Invariant Functions)	p. 5
Invariant Equations (Manifolds)	p. 7
Representation of Regular Invariant Manifolds via Invariants	p. 9
Symmetries and Integration of Ordinary Differential Equations	p. 10
The Frame of Differential Equations	p. 10
Prolongation of Group Transformations and Their Generators	p. 12
Group Admitted by Differential Equations	p. 13
Determining Equation for Infinitesimal Symmetries	p. 14
An Example on Calculation of Symmetries	p. 14
Lie Algebras. Specific Property of Determining Equations	p. 16
Integration of First-Order Equations: Lie's Integrating Factor	p. 17
Integration of First-Order Equations: Method of Canonical Variables	p. 18
Standard Forms of Two-Dimensional Lie Algebras	p. 21
Lie's Method of Integration for Second-Order Equations	p. 22
Symmetries and Invariant Solutions of Partial Differential Equations	p. 24
Discussion of Symmetries for Evolution Equations	p. 24
Calculation of Symmetries for Burgers' Equation	p. 27
Invariant Solutions and Their Calculation	p. 29
Group Transformations of Solutions	p. 30
Optimal Systems of Subalgebras	p. 31
All Invariant Solutions of the Burgers Equation	p. 36
General Definitions of Symmetry Groups	p. 38
Differential Variables and Function	p. 38
Frame and Extended Frame	p. 39
Definition Using Solutions	p. 40
Definition Using the Frame	p. 40
Definition Using the Extended Frame	p. 41
Lie-Bäcklund Transformation Groups	p. 42
Lie-Bäcklund Operators	p. 42
Integration of Lie-Bäcklund Equations	p. 45
Lie-Bäcklund Symmetries	p. 48
Approximate Transformation Groups	p. 49
Approximate Transformations and Generators	p. 50
Approximate Lie Equations	p. 52
Approximate Symmetries	p. 53
References	p. 54
Introduction to Group Analysis and Invariant Solutions of Integro-Differential Equations	p. 57
Integro-Differential Equations in Mathematics and in Applications	p. 57
Survey of Various Approaches or Finding Invariant Solutions	p. 58
Methods Using a Presentation of a Solution or an Admitted Lie Group	p. 59
Methods of Moments	p. 73
Methods Using a Transition to Equivalent Differential Equations	p. 78
A Regular Method for Calculating Symmetries of Equations with Nonlocal Operators	p. 89
Admitted Lie Group of Partial Differential Equations	p. 90
The Approach for Equations with Nonlocal Operators	p. 92
Illustrative Examples	p. 94
The Fourier-Image of the Spatially Homogeneous Isotropic Boltzmann Equation	p. 95
Equations of One-Dimensional Viscoelastic Continuum Motion	p. 102
References	p. 108
The Boltzmann Kinetic Equation and Various Models	p. 113
Studies of Invariant Solutions of the Boltzmann Equation	p. 113
Introduction to the Boltzmann Equation	p. 114
Group Analysis of the Full Boltzmann Equation	p. 116
Admitted Lie Algebras	p. 116
Isomorphism of Algebras	p. 121
Invariant Solutions of the Full Boltzmann Equation	p. 123
Classification of Invariant Solutions of the Fourier Transformation of the Full Boltzmann Equation	p. 126
Complete Group Analysis of Some Kinetic Equations	p. 127
The Boltzmann Kinetic Equation with an Approximate Asymptotic Collision Integral	p. 128
The Smolukhovsky Kinetic Equation	p. 130
A System of the Space Homogeneous Boltzmann Kinetic Equations	p. 131
Homogeneous Relaxation of a Binary Model Gas	p. 139
References	p. 142
Plasma Kinetic Theory: Vlasov-Maxwell and Related Equations	p. 145
Mathematical Model	p. 145
Definition and Infinitesimal Test	p. 148
Definition of Symmetry Group	p. 149
Variational Derivative for Functionals	p. 149
Infinitesimal Criterion	p. 150
Prolongation on Nonlocal Variables	p. 151
Symmetry of Plasma Kinetic Equations in One-Dimensional Approximation	p. 152
Non-relativistic Electron Gas	p. 152
Relativistic Electron Gas	p. 160
Collisionless Non-relativistic Electron-Ion Plasma	p. 161
Collisionless Relativistic Electron-Ion Plasma	p. 162
Non-relativistic Electron Plasma Kinetics with a Moving and Stationary Ion Background	p. 164
Relativistic Electron Plasma Kinetics with a Moving Ion Background	p. 165
Non-relativistic Electron-Ion Plasma in Quasi-neutral Approximation	p. 166
Group Analysis of Three Dimensional Collisionless Plasma Kinetic Equations	p. 168
Relativistic Electron Gas Kinetics	p. 168
Relativistic Electron-Ion Plasma Kinetic Equations	p. 170
Symmetry of Vlasov-Maxwell Equations in Lagrangian Variables	p. 172
Vlasov-Type Equations: Symmetries of the Benney Equations	p. 175
Different Forms of the Benney Equations	p. 175
Lie Subgroup and Lie-Bäcklund Group: Statement of the Problem	p. 176
Incompleteness of the Point Group: Statement of the Problem	p. 177
Determining Equations and Their Solution	p. 178
Discussion of the Solution of the Determining Equations	p. 180
Illustrative Example for Matrix Elements	p. 181
Symmetries in Application to Plasma Kinetic Theory. Renormalization Group Symmetries for Boundary Value Problems and Solution Functionals	p. 184
Introduction to Renormgroup Symmetries	p. 185
RG Symmetry: An Idea of Construction and Its Simple Realization	p. 186
Renormgroup Algorithm	p. 190
Examples of RG Symmetries in Plasma Theory	p. 193
References	p. 206
Symmetries of Stochastic Differential Equations	p. 209
Stochastic Integration of Processes	p. 210
Stochastic Processes	p. 210
The Itô Integral	p. 211
The Itô Formula	p. 213
Time Change in Stochastic Integrals	p. 214
Stochastic ODEs	p. 215
Itô Stochastic Differential Equations	p. 215
Stratonovich Stochastic Differential Equations	p. 216
Kolmogorov Equations	p. 217
Linearization of First-Order SODE	p. 218
Weak Linearization Problem	p. 219
Strong Linearization of First-Order SODE	p. 220
Linearization of Second-Order SODE	p. 225
Linearization Conditions	p. 225
Transformations of Autonomous SODEs	p. 226
Admitted Transformations	p. 226
Autonomous Systems of Stochastic First-Order ODEs	p. 231
Transformations of Stochastic ODEs	p. 233
Short Historical Review	p. 233
Admitted Lie Group and Determining Equations	p. 235
Symmetries of Stochastic ODEs	p. 239
Determining Equations	p. 240
Admitted Lie Group of Geometric Brownian Motion	p. 241
Lie Groups of Transformations of Some Stochastic ODEs	p. 244
Geometric Brownian Motion	p. 244
Narrow-Sense Linear System	p. 245
Black and Scholes Market	p. 247
Nonlinear Itô System	p. 248
References	p. 249
Delay Differential Equations	p. 251
Delay Differential Equations in Mathematical Modeling	p. 251
Mathematical Background of Delay Ordinary Differential Equations	p. 252
Definitions and Theorems	p. 253
Definition of Symmetries of DDEs	p. 255
Example	p. 255
Admitted Lie Group of DODE	p. 255
Symmetries of a Model Equation	p. 257
Differential-Difference Equations	p. 258
Group Classification of Second-Order DODEs	p. 259
Introduction into the Problem	p. 259
Determining Equations	p. 260
Properties of Admitted Generators	p. 261
Strategy for Obtaining a Complete Classification of DODEs	p. 264
Illustrative Examples	p. 264
List of Invariant DODEs	p. 270
Equivalence Lie Group for DDE	p. 271
Lie-Bäcklund Representation of Determining Equations for the Equivalence Lie Group of PDEs	p. 271
Potential Equivalence Lie Group of Delay Differential Equations	p. 274
The Reaction-Diffusion Equation with a Delay	p. 274
The Cauchy Problem	p. 275
The Equivalence Lie Group	p. 275
Admitted Lie Group of Equation	p. 277
Case ≠ 0	p. 279
Case = 0	p. 283
Summary of the Group Classification	p. 288
Invariant Solutions	p. 289
References	p. 291
Appendix A	p. 293
Optimal Systems of Subalgebras	p. 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 B	p. 297
Realizations of Lie Algebras on the Real Plane	p. 297
Group Classification of Second-Order DODEs	p. 299
Index	p. 303
Table of Contents provided by Ingram. All Rights Reserved.

Lecture Notes in Physics

With Applications in Mechanics and Plasma Physics

At a Glance

Paperback

Industry Reviews

Shipping

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