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E-grāmata: Nonlinear Reaction-Diffusion-Convection Equations: Lie and Conditional Symmetry, Exact Solutions and Their Applications

(Department of Mathematics, National Technical University, Ukraine), (Department of Mathematics, National Technical University, Ukraine),
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Nonlinear Reaction-Diffusion-Convection Equations: Lie and Conditional Symmetry, Exact Solutions and Their ApplicationsPalielināt
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It is well known that symmetry-based methods are very powerful tools for investigating nonlinear partial differential equations (PDEs), notably for their reduction to those of lower dimensionality (e.g. to ODEs) and constructing exact solutions. This book is devoted to (1) search Lie and conditional (non-classical) symmetries of nonlinear RDC equations, (2) constructing exact solutions using the symmetries obtained, and (3) their applications for solving some biologically and physically motivated problems. The book summarises the results derived by the authors during the last 10 years and those obtained by some other authors.

Recenzijas

"This work is devoted to an in-depth study of Lie and nonclassical (called Q-conditional) symmetries for nonlinear Reaction-Di usion-Convection (RDC) equations. This type of PDEs is ubiquitous in many fields of applications, and many models for describing natural phenomena such as heat transfer, filtration of liquids, di usion in chemical reactions, or bio-medical processes lead to equations of this type. Clearly, the determination of exact solutions is an integral part of gaining an understanding of such models, and symmetry methods are a key technique in this endeavour. Since a number of nonlinear RDC equations have a quite small Lie transformation group of symmetries, more refined methods, like Q-conditional symmetries, are typically the method of choice. The authors are among the leading researchers in this field and have collected in this book many results they have obtained over the past decades. Chapter 1 gives a general introduction to nonlinear RDC equations as well as a brief recapitulation of the determination of Lie symmetries for PDEs. This is applied in Chapter 2 to the complete description of Lie symmetries of RDC equations. Particular emphasis is put on equivalence and form-preserving transformations, which are consistently employed to enhance the classical algorithm for determining Lie symmetries. In Chapter 3, after a brief introduction to the topic of conditional symmetries, Q-conditional symmetries are studied and are determined explicitly for a number of relevant classes (in particular, power-law and exponential-law di usivity). Chapter 4 is devoted to determining exact solutions of RDC equations of the general form

ut = [ A(u)ux]x +B(u)ux +C(u):

Explicit solutions are derived for relevant types of such PDEs (e.g., Fisher and Murray and FitzHugh-Nagumo), and the cases of power-law and exponential di usivity are studied in detail. The final Chapter 5 introduces the method of additional generating conditions for constructing exact solutions. This carefully written book collects many results that had previously only been available in the journal literature in a unified and applicable manner. As such it is a most welcome addition to the literature on symmetry methods for di erential equations."

- Michael Kunzinger - Mathematical Reviews Clippings February 2019

Preface ix
List of Figures
xv
List of Tables
xvii
1 Introduction
1(18)
1.1 Nonlinear reaction-diffusion-convection equations in mathematical modeling
1(2)
1.2 Main methods for exact solving nonlinear reaction-diffusion-convection equations
3(5)
1.3 Lie symmetry of differential equations: historical review, definitions and properties
8(11)
2 Lie symmetries of reaction-diffusion-convection equations
19(58)
2.1 Symmetry of the linear diffusion equation
19(2)
2.2 Symmetry of the nonlinear diffusion equation
21(8)
2.3 Equivalence transformations and form-preserving transformations
29(7)
2.3.1 The group of equivalence transformations
30(2)
2.3.2 Form-preserving transformations
32(4)
2.4 Determining equations for reaction-diffusion-convection equations
36(3)
2.5 Complete description of Lie symmetries of reaction-diffusion-convection equations
39(30)
2.5.1 Principal algebra of invariance
39(1)
2.5.2 Necessary conditions for nontrivial Lie symmetry
39(10)
2.5.3 Lie symmetry classification via the Lie-Ovsiannikov algorithm
49(11)
2.5.4 Application of form-preserving transformation
60(9)
2.6 Nonlinear equations arising in applications and their Lie symmetry
69(8)
2.6.1 Heat (diffusion) equations with power-law nonlinearity
69(2)
2.6.2 Diffusion equations with a convective term
71(2)
2.6.3 Nonlinear equations describing three types of transport mechanisms
73(4)
3 Conditional symmetries of reaction-diffusion-convection equations
77(58)
3.1 Conditional symmetry of differential equations: historical review, definitions and properties
77(6)
3.2 Q-conditional symmetry of the nonlinear heat equation
83(4)
3.3 Determining equations for finding Q-conditional symmetry of reaction-diffusion-convection equations
87(5)
3.4 Q-conditional symmetry of reaction-diffusion-convection equations with constant diffusivity
92(8)
3.5 Q-conditional symmetry of reaction-diffusion-convection equations with power-law diffusivity
100(11)
3.5.1 The case of proportional diffusion and convection coefficients
101(6)
3.5.2 The case of different diffusion and convection coefficients
107(4)
3.6 Q-conditional symmetry of reaction-diffusion-convection equations with exponential diffusivity
111(18)
3.6.1 Solving the nonlinear system (3.166)
118(7)
3.6.2 Solving the nonlinear system (3.169)
125(4)
3.7 Nonlinear equations arising in applications and their conditional symmetry
129(6)
4 Exact solutions of reaction-diffusion-convection equations and their applications
135(56)
4.1 Classification of exact solutions from the symmetry point of view
135(3)
4.2 Examples of exact solutions for some well-known nonlinear equations
138(5)
4.3 Solutions of some reaction-diffusion-convection equations arising in biomedical applications
143(13)
4.3.1 The Fisher and Murray equations
143(3)
4.3.2 The Fitzhugh-Nagumo equation and its generalizations
146(10)
4.4 Solutions of reaction-diffusion-convection equations with power-law diffusivity
156(20)
4.4.1 Lie's solutions of an equation with power-law diffusion and convection
156(3)
4.4.2 Non-Lie solutions of some equations with power-law diffusion and convection
159(17)
4.5 Solutions of reaction-diffusion-convection equations with exponential diffusivity
176(15)
4.5.1 Lie's solutions of an equation with exponential diffusion and convection
176(4)
4.5.2 Non-Lie solutions of an equation with exponential diffusion and convection
180(8)
4.5.3 Application of the solutions obtained for population dynamics
188(3)
5 The method of additional generating conditions for constructing exact solutions
191(28)
5.1 Description of the method and the general scheme of implementation
191(4)
5.2 Application of the method for solving nonlinear reaction-diffusion-convection equations
195(21)
5.2.1 Reduction of the nonlinear equations (5.10) and (5.11) to ODE systems
196(5)
5.2.2 Exact solutions of the nonlinear equations (5.10) and (5.11)
201(10)
5.2.3 Application of the solutions obtained for solving boundary-value problems
211(5)
5.3 Analysis of the solutions obtained and comparison with the known results
216(3)
References 219(20)
Index 239
Roman Cherniha is a professor at the Institute of Mathematics, National Academy of Sciences, Ukraine. His main areas of interest are Non-linear PDEs: Lie and conditional symmetries, exact solutions and their properties and the application of modern methods for analytical solving nonlinear boundary value problems. He is the author of over 100 scientific papers and has acted as the referee for several international scientific journals.

Mykola I. Serov is a professor at the National Technical University, Ukraine. His main areas of interest at Lie symmetries of partial differential equations (PDEs), Conditional symmetries of PDEs and nonlocal symmetries of PDEs. He has authored over 60 scientific papers and published 6 Monographs (in Ukrainian).

Oleksii H. Pliukhin is an associate professor at the National Technical University, Ukraine. His main areas of interest are Lie symmetries of partial differential equations (PDEs), Conditional symmetries of PDEs and exact soutions and their properties of PDEs. He has participated in many scientific conferences and workshops, and published 13 scientific papers.
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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