Nonlinear Reaction-Diffusion-Convection Equations

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.

• 1. Introduction. 2. Lie symmetries of reaction-diffusion-convection equations. 3. Conditional symmetries of reaction-diffusion-convection equations. 4. Exact solutions of reaction-diffusion-convection equations. 5. Method additional generating conditions for constructing exact solutions. 6. Concluding remarks.

"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