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

"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