The present textbook on ordinary and partial differential equations is tailored to develop analytic skills and "working knowledge'' in both classical and Lie's methods for solving linear and nonlinear equations. It is based on lectures delivered by the author at Moscow Institute of Physics and Technology, and Blekinge Institute of Technology. The prerequisites are linear algebra and mathematical analysis with one and several variables including elements of ordinary differential equations.
An overview is given of lie group analysis and symmetry methods applied to transport phenomena
The purpose of this paper is to develop a theory of approximate representations of the de Sitter group considered as a perturbation of the Poincare group. This approach simplifies investigation of relativistic effects pertaining to the mechanics in the de Sitter universe. Utility of the approximate approach is manifest if one compares the transformations of the de Sitter group with their approximate representations.
These notes provide an easy to follow introduction to the topic and are based on my talks at various conferences, in particular on the plenary lecture at the International Workshop on ``Differential equations and chaos" (University of Witwatersrand, Johannesburg, South Africa, January 1996). The book is prepared for the new graduate course ``Approximate transformation groups" that will be given at Blekinge Institute of Technology during January-March, 2009.
Volume 1 contains 3 articles: Nail H. Ibragimov, Equivalence groups and invariants of linear and non-linear equations; Nail H. Ibragimov and Sergey V. Meleshko, Linearization of third-order ordinary differential equations; Nail H. Ibragimov, Gazanfer Ünal and Claes Jogreús, Group analysis of stochastic differential systems: Approximate symmetries and conservation laws.
Volume 2 contains 3 articles: Ilir Berisha, Translation of Bäcklunds paper ”Surfaces of constant negative curvature”; Johan Erlandsson, "Survey of mathematical models in biology from point of view of Lie group analysis"; Niklas Säfström, "Group analysis of a tumour growth model"
Volume 4 contains articles: Nail H. Ibragimov, Ewald J.H. Wessels and George F.R. Ellis, Group classication of the Sachs equations for a radiating axisymmetric, non-rotating, vacuum space-time; Nail H. Ibragimov A discussion of conservation laws for over-determined systems with application to the Maxwell equations in vacuum; Nail H. Ibragimov, Quasi-self-adjoint differential equations; Nail H. Ibragimov and Salavat V. Khabirov, Existence of integrating factors for higher-order ordinary differential equations; Nail H. Ibragimov, Integration of second-order linear equations via linearization of Riccati's equations; Nail H. Ibragimov and Sergey V. Meleshko Linearization of second-order ordinary differential equations by changing the order; Nail H. Ibragimov and Sergey V. Meleshko Second-order ordinary differential equations equivalent to y''= H(y); Nail H. Ibragimov, Sergey V. Meleshko and Supaporn Suksern, Linearization of fourth-order ordinary differential equations by point transformations; Nail H. Ibragimov and Emrullah Yasar, Non-local conservation laws in fluid dynamics
Volume 7/8 contains 3 articles by N.H. Ibragimov, an article by N.H. Ibragimov, E. D. Avdonina and an article by N. H. Ibragimov , R. Khamitova.
Volume 9 contains articles of different authors.
The recent paper mentioned in the title contains a confusing statement on computing conservation laws corresponding to symmetries of nonlinearly self-adjoint differential equations. The present brief article contains clarifying comments.
Differentialekvationer och matematisk modellering" är avsedd för högskolestuderande på nybörjarnivå, exempelvis inom ingenjörs- och naturvetarprogram. Den kombinerar klassiska lösningsmetoder med moderna metoder från gruppanalys. Texten är generös med exempel och övningsuppgifter, vissa med fullständiga lösningar. Många vanliga läroböcker är begränsade till ett antal speciella differentialekvationer som man råkar kunna lösa med enkla recept. De flesta matematiska modeller inom naturvetenskap och teknik leder dock till ickelinjära differentialekvationer som man inte kan lösa med traditionella metoder. Med de redskap som ges i denna bok ligger många intressanta tillämpningar inom räckhåll.
Recently I developed a systematic method for determining invariants of families of equations. The method is based on the infinitesimal approach and is applicable to algebraic and differential equations possessing finite or infinite equivalence groups. Moreover, it does not depend on the assumption of linearity of equations. The method was applied to variety of ordinary and partial differential equations. The present paper is aimed at discussing the main principles of the method and its applications with emphasis on the use of infinite Lie groups.
Euler generalized d'Alembert's solution to a wide class of linear hyperbolic equations with two independent variables. He introduced in 1769 the quantities that were rediscovered by Laplace in 1773 and became known as the Laplace invariants. The present paper is devoted to an extension of Euler's method to linear parabolic equations with two independent variables. The new method allows one to derive an explicit formula for the general solution of a wide class of parabolic equations. In particular, the general solution of the Black-Scholes equation is obtained. (c) 2008 Elsevier B.V. All rights reserved.
Översättning av Louise Petréns doktorsavhandling vid Lunds universitet
Integrating factors and adjoint equations are determined for linear and non-linear differential equations of an arbitrary order. The new concept of an adjoint equation is used for construction of a Lagrangian for an arbitrary differential equation and for any system of differential equations where the number of equations is equal to the number of dependent variables. The method is illustrated by considering several equations traditionally regarded as equations without Lagrangians. Noether's theorem is applied to the Maxwell equations.
Integrating factors are determined for ordinary differential equations of an arbitrary order. In the case of first-order equations the new definition coincides with the usual definition of integrating factor introduced by A. Clairaut in 1739. It is shown that higher-order equations, unlike the first-order equations, can have an integrating factor even when they do not admit Lie point symmetries. The method is illustrated by several examples.
The test for linearization of the Riccati equations by a change of the dependent variable, proved by the author in 1989, is utilized for integration of second-order linear equations by quadratures.
A method for solving the inverse variational problem for differential equations admitting a Lie group is presented. The method is used for determining invariant Lagrangians and integration of second-order nonlinear differential equations admitting two-dimensional non-commutative Lie algebras. The method of integration suggested here is different from Lie's classical method of integration of second-order ordinary differential equations based on canonical forms of two-dimensional Lie algebras. The new method reveals existence and significance of one-parameter families of singular solutions to nonlinear equations of second order.
We investigate invariants of equations u_t=f(x,u)u_{xx}+g(x,u,u_x). We show that these equations have two functionally independent invariants of the second order.
The renown Laplace invariants were used by P.S. Laplace in 1773 in his integration theory of linear hyperbolic differential equations in two variables. In 1960, L.V.Ovsyannikov, tackling the problem of group classification of hyperbolic equations, came across two proper invariants that do not change under the most general equivalence transformation of hyperbolic equations. The question on existence of other invariants remained open. The present paper is dedicated to solution of Laplace's problem which consists in finding all invariants for hyperbolic equations and constructing a basis of invariants. Three new invariants of first and second order as well as operators of invariant differentiations are constructed. It is shown that the new invariants, together with Ovsyannikov's invariants, provide a basis of invariants, so that any invariant of higher order is a function of the basic invariants and their invariant derivatives.
The Laplace invariants pertain to linear hyperbolic differential equations with two independent variables. They were discovered by Laplace in 1773 and used in his integration theory of hyperbolic equations. Cotton extended the Laplace invariants to elliptic equations in 1900. Cotton's invariants can be obtained from the Laplace invariants merely by the complex change of variables relating the elliptic and hyperbolic equations. To the best of my knowledge, the invariants for parabolic equations were not found thus far. The purpose of this paper is to fill this gap by considering what will be called Laplace type invariants (or seminvariants), i.e. the quantities that remain unaltered under the linear transformation of the dependent variable. Laplace type invariants are calculated here for all hyperbolic, elliptic and parabolic equations using the unified infinitesimal method. A new invariant is found for parabolic equations.
The paper is devoted to the Lie group analysis of a nonlinear equation arising in metallurgical applications of Magnetohydrodynamics. Self-adjointness of the basic equations is investigated. The analysis reveals two exceptional values of the exponent playing a significant role in the model.
In the present paper, a new method is proposed for constructing exact solutions for systems of nonlinear partial differential equations. It is called the method of conservation laws. Application of the method to the Chaplygin gas allowed to construct new solutions containing several arbitrary parameters. It is shown that these solutions cannot be obtained, in general, as group invariant solutions.
Översättning av E. Bessel Hagens artikel om elektrodynamik i Mathematische Annalen 1921
An optimal system of invariant solution is derived for the Burger´s equation.
The concept of self-adjoint equations is useful for calculating conservation laws. The aim of the present paper is extend the property of self-adjoint equations to a wider class of equations called quasi-self-adjoint.
Volume III contains Dr.Sci thesis (1972) and papers written during 1987-2004. The main topics in this volume include Lie groups in mathematical physics and mathematical modelling, fluid dynamics, Huygens’ principle and approximate symmetries of equations with small parameter.
Volume IV contains papers written during 1996-2007. The main topics in this volume include Equivalence groups and invariants of differential equations, Extension of Eulers’ method of integration of hyperbolic equations to parabolic equations, Invariant and formal Lagrangians, Conservation laws.
These notes are based on my lectures delivered in Russia (Novosibirsk and Moscow State Universities, 1972-1973 and 1988-1990), France (Collège de France, 1980), South Africa (University of the Witwatersrand, 1995-1997) and Sweden (Blekinge Institute of Technology, during 2004-2008).
In 1973, during a discussion of the pioneering works on soliton theory at ``Theoretical seminar of the Institute of Hydrodynamics" in Novosibirsk, Professor Ovsyannikov asked me if the infinite number of conservation laws for the Korteweg-de Vries equation can be obtained from its symmetries. The answer was by no means evident because the KdV equation did not have the usual Lagrangian, and hence the Noether theorem was not applicable. In my talk I give the affirmative answer to Ovsyannikov's question by proving a general theorem on conservation laws for arbitrary differential equations. The new conservation theorem does not require existence of a Lagrangian and is based on a concept of adjoint equations for non-linear equations. For derivation of the infinites series of conservation laws for the KdV equation, I modify the notion of self-adjoint equations and extend it to non-linear equations.
Time-dependent exact solutions of the Kompaneets photon diffusion equation are obtained for several approximations of this equation. One of the approximations describes the case when the induced scattering is dominant. In this case, the Kompaneets equation has an additional symmetry which is used for constructing some exact solutions as group invariant solutions.
These notes are designed for the graduate course on Transformation groups and Lie algebras that I have been teaching at Blekinge Institute of Technology since 2002. The course is aimed at augmenting a preliminary knowledge in this field obtained by students during the course on Differential equations based on my book "A practical course in differential equations and mathematical modelling".
This book is based on the extensive experience of teaching for mathematics, physics and engineering students in Russia, USA, South Africa and Sweden. The author provides students and teachers with an easy to follow textbook spanning a variety of topics. The methods of local Lie groups discussed in the book provide universal and effective method for solving nonlinear differential equations analytically. Introduction to approximate transformation groups also contained in the book helps to develop skills in constructing approximate solutions for differential equations with a small parameter. Readership: Professional mathematics researchers and graduated students.
This book is based on lectures in differential equations delivered by the authors, namely, by A. Al-Hammadi at the University of Bahrain and by N. Ibragimov at the Blekinge Institute of Technology, Sweden. It is designed for a one semester course in differential equations. We assume that the reader has taken standard courses in mathematical analysis and has encountered basic concepts from linear algebra such as vectors, matrices and algebraic linear systems. Our aim is to provide students with an easy to follow introduction to the subject containing classical devices enriched by fundamentals of Lie group methods given in Chapter 9.
Conservation laws and exact solutions of nonlinear differential equations describing diffusion phenomena in anisotropic media with external sources are constructed. The construction is based on the method of nonlinear self-adjointness. Numerous exact solutions are obtained by using the recent method of conservation laws. These solutions are different from group invariant solutions and can be useful for investigating diffusion phenomena in complex media, e.g. in oil industry.
Different approximations of the Kompaneets equation are studied using approximate symmetries, which allows consideration of the contributions of all terms of this equation previously neglected in the analysis of the limiting cases.
Propagation of nonlinear baroclinic Kelvin waves in a rotating column of uniformly stratified fluid under the Boussinesq approximation is investigated. The model is constrained. by the Kelvin's conjecture saying that the velocity component normal to the interface between rotating fluid and surrounding medium (e.g. a seashore) is possibly zero everywhere in the domain of fluid motion, not only at the boundary. Three classes of distinctly different exact solutions for the nonlinear model are obtained. The obtained solutions are associated with symmetries of the Boussinesq model. It is shown that one class of the obtained solutions can be visualized as rotating whirlpools along which the pressure deviation from the mean state is zero, is positive inside and negative outside of the whirlpools. The angular velocity is zero at the center of the whirlpools and it is monotonically increasing function of radius of the whirlpools.
Many mathematical models formulated in terms of non-linear differential equations can successfully be treated and solved by Lie group methods. Lie group analysis is especially valuable in investigating non-linear differential equations, for its algorithms act here as reliably as for linear cases. The aim of this article is to provide the group theoretical modeling of internal waves in the ocean. The approach is based on a new concept of conservation laws that is utilized to systematically derive the conservation laws of non-linear equations describing propagation of internal waves in the ocean. It was shown in our previous publication that uni-directional internal wave beams can be obtained as invariant solutions of non-linear equations of motion. The main goal of the present publication is to thoroughly analyze another physically significant exact solution, namely the rotationally symmetric solution and the energy carried by this solution. It is shown that the rotationally symmetric solution and its energy are presented by means of a bounded oscillating function.
In this paper, we shall obtain the symmetries of the mathematical model describing spontaneous relaxation of eastward jets into a meandering state and use these symmetries for constructing the conservation laws. The basic eastward jet is a spectral parameter of the model, which is in geostrophic equilibrium with the basic density structure and which guarantees the existence of nontrivial conservation laws.
We investigate the concept of integrating factors for higher-order ordinary differential equations introduced recently by one of the authors (NHI). The integrating factors for equations of order higher than one are determined by overdetermined systems. Therefore one can expect that not all higher-order equations have integrating factors. We prove in this paper that in fact they have. Moreover, we demonstrate that every equation of order n has precisely n functionally independent integrating factors.
Lie group analysis provides a universal tool for tackling considerable numbers of differential equations even when other means of integration fail. In fact, group analysis is the only effective method for solving nonlinear differential equations analytically. However the philosophy of Lie groups in the theory of differential equations did not enjoy widespread acceptance in the past and the subject has been neglected in university programs. One of the main aims of ALGA is to improve the situation by developing courses based on group analysis. But the problem is that calculation of symmetries is a lengthy process and this can scare students. The calculation can be simplified by using computer algebra packages. We started to use one of them for teaching engineering students at BTH. This project was supported by STINT, Swedish Foundation for International Cooperation in Research and Higher Education.
Physically significant question on calculation of conservation laws of the Thomas equation is investigated. It is demonstrated that the Thomas equation is nonlinearly self-adjoint. Using this property and applying the theorem on nonlocal conservation laws the infinite set of conservation laws corresponding to the symmetries of the Thomas equation is computed. It is shown that the Noether theorem provide only one of these conservation laws.
In all areas of physics, conservation laws are essential since they allow us to draw conclusions of our physical system under study in an indirect but efficient way. Electrodynamics, in terms of the standard Maxwell electromagnetic equations for fields in vacuum, exhibit a rich set of symmetries to which conserved quantities are associated. We have derived conservation laws for Dirac's symmetric version of the Maxwell-Lorentz microscopic equations, allowing magnetic charges and magnetic currents, where the latter, just as electric currents, are assumed to be described by a linear relationship between the field and the current, i.e. an Ohm's law. We find that when we use the method of Ibragimov to construct the conservation laws, they will contain two new adjoint vector fields which fulfil Maxwell-like equations. In particular, we obtain conservation laws for the electromagnetic field which are nonlocal in time.
We show that one can apply a Lagrangian approach to certain evolution equations by considering them together with their associated equations. Consequently, one can employ Noether's theorem and derive conservation laws from symmetries of coupled systems of evolution equations. We discuss in detail the linear and non-linear heat equations as well as the Burgers equation and obtain new non-local conservation laws for the non-linear heat and the Burgers equations by extending their symmetries to the associated equations. We also provide Lagrangians for non-linear Schrödinger and Korteweg-de Vries type systems.
Approximate and Renormgroup Symmetries deals with approximate transformation groups, symmetries of integro-differential equations and renormgroup symmetries. It includes a concise and self contained introduction to basic concepts and methods of Lie group analysis, and provides an easy to follow introduction to the theory of approximate transformation groups and symmetries of integro-differential equations. The book is designed for specialists in nonlinear physics --mathematicians and nonmathematicians--interested in methods of applied group analysis for investigating nonlinear problems in physical, engineering and natural sciences.
The book deals with so-called renormalization group symmetries considered in the framework of approximate transformation groups. Renormgroup symmetries provide a basis for the renormgroup algorithm for improving solutions to boundary value problems by converting "less applicable solutions" into "more applicable solutions". the algorithm is particularly useful for improving approximate solutions given by the perturbation theory.
Classical Lie group theory provides a universal tool for calculating symmetry groups for systems of differential equations. However Lie's method is not as much effective in the case of integral or integro-differential equations as well as in the case of infinite systems of differential equations. This paper is aimed to survey the modern approaches to symmetries of integro-differential equations. As an illustration, an infinite symmetry Lie algebra is calculated for a system of integro-differential equations, namely the well-known Benny equations. The crucial idea is to look for symmetry generators in the form of canonical Lie-Backlund operators.