A Practical Course in Differential Equations and Mathematical Modeling: Classical and New Methods, Nonlinear Mathematical Models, Symmetry and Invariance Principles
Description
Елена Авдонина has uploaded on YouTube 19 video lectures from the MOGRAN (Modern Group Analysis) 18 conference in 2015. (cf. course on Canvas by Ibragimov, Авдонина, and Гайнетдинова)
has been translated into Swedish (2009), Chinese (2010), Russian (2012) and recently also appeared in German (2018)!
cited in Ibragimov's ch. of Symmetries of Integro-Differential Equations
See the ALGA publications.
The PDF_OLD format is the non-rasterized version of chs. 1 & 5 and the index of that (2006) version (pagination slightly different than this 2010 Higher Education Press / World Scientific version).
- ch. 1 is a good, concise overview of basic mathematics
- PDF p. 122-3 is on Jean Bernoulli's ingenious variation of parameters method of solving non-homogeneous linear equations.
- PDF p. 128 is on Euler's ansatz for homogeneous equations of constant coefficients for reducing DE→algebraic eqn. (characterstic equation & polynomial)
- p. 152 is on self-adjoint linear operators (PDF p. 167)
- Lie theory begins on ch. 6, p. 179 (PDF p. 194).
- §6.5 p. 201 (PDF p. 216) is on 2nd-order nonlinear ODEs.
- ch. 8 "Generalized functions or distributions" pp. 291ff. (PDF pp. 306ff.) ("Distribution" in this sense means "generalized function.")
- pp. 327ff. (PDF pp. 342ff.) contains answers to the problems
This book addresses modern symmetry analysis of differential equations as a universal tool and an effective method for solving differential equations. In traditional courses on ordinary differential equations there are numerous classical methods to integrate differential equations but they are rather mysterious: they do not explain the interplay between the form of an equation and the method for integrating this equation in the framework of some universal theory. This textbook presents a new conceptual framework, leading to an effective structured method, for a main course on differential equations and mathematical modeling. The program is carried out by the systematic detailed exposition of training materials on the basis of Lie group analysis. In explaining this concept, the author also provides comments on their historical development, beginning with L. Euler, E. Goursat, S. Lie and others.
This text was developed by the author through his teaching of a partial differential equations course at Moscow Institute of Physics and Technology, a special course on Lie group analysis of differential equations at Novosibirsk State University and similar lectures in South Africa and Sweden.
This self-contained textbook begins with the foundational materials in the first three chapters, which cover selected topics from analysis, basic mathematical models arising in physics, biology and engineering sciences, and a comprehensive traditional course on ordinary differential equations. The next two chapters are devoted to the basic classical methods of integration of both first-order partial differential equations and linear partial differential equations of the second order. The final, and basic, four chapters consists of a modern course on differential equations that introduces students to elements of the theory of group transformations, Lie's infinitesimal technique and Lie algebras which are most important in practice. It is also demonstrated how the classical methods of integration of differential equations are incorporated into Lie group techniques. Chapters 6 and 7 make use of such main topics as calculation of symmetries, Lie's integration method, integrating factor, nonlinear superposition, symmetries and conservation laws, and group invariant solutions. A modern approach for finding fundamental solutions based on the invariance principle and an extension of Lie's infinitesimal technique to the space of distributions is exploited in Chapter 9. Examples and exercises are given throughout the text in order to motivate and illustrate the theory and to teach students to do mathematics independently. Reviewed by Vladimir Grebenev