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 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-BÃ¤cklund Transformation Groups | p. 42 |
Lie-BÃ¤cklund Operators | p. 42 |
Integration of Lie-BÃ¤cklund Equations | p. 45 |
Lie-BÃ¤cklund 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-BÃ¤cklund 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 ItÃ´ Integral | p. 211 |
The ItÃ´ Formula | p. 213 |
Time Change in Stochastic Integrals | p. 214 |
Stochastic ODEs | p. 215 |
ItÃ´ 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 ItÃ´ 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-BÃ¤cklund 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 L_{11} (Y) | p. 294 |
Six-Dimensional Subalgebras of the Lie Algebra L_{12}(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. |
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