Nonlinear Theory and Modeling of the Free Boundary Problems Arising in Distributed Media

8vo (21.5 cm), 351 pp. Publisher's laminated wrappers. Mathematical Studies: Monograph Series, vol. 9. Edited by Michael Zarichnyi. A research monograph in nonlinear dynamics and mathematical physics by Vasyl Gafiychuk (Institute of Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv) and Ihor Lubashevsky (General Physics Institute of the Russian Academy of Sciences, Moscow), presenting a unified mathematical theory of free-boundary (Stefan) problems arising in a broad spectrum of physical, chemical, and biological systems exhibiting self-organization. The central methodological innovation is the systematic use of conformal mapping to transform moving interfaces into fixed boundaries, thereby reducing nonlinear free-boundary problems to field equations on stationary domains and making them amenable to the methods of modern field theory. The authors emphasize that the volume consolidates and substantially extends results previously scattered across journal articles and an earlier Russian-language monograph (Kyiv, 1990), making them accessible to a wider international audience. The work is divided into five parts. Chapter 1 introduces the mathematical structure of free-boundary problems through classical examples, including oscillatory zoning, dendritic crystal growth in undercooled melts, directional solidification, flame propagation, Saffman-Taylor viscous fingering, streamer formation, and electrodeposition. Part I (Chapters 2-3) develops the nonlinear theory of pattern formation in the one-dimensional Stefan problem, with particular attention to auto-oscillatory behaviour during directional solidification. Part II (Chapters 4-5) treats two-dimensional quasi-stationary Stefan problems for open and closed geometries using conformal mapping techniques, including numerical simulations and connections with the Shraiman-Bensimon equations. Part III (Chapters 6-8) extends the theory to fully non-stationary interfaces, introducing a variational formulation and generalized dissipative functionals applicable to dendritic growth, pattern formation, and interface evolution for both open and closed boundaries. Part IV (Chapters 9-13) applies the theoretical framework to a wide range of distributed-media phenomena, including combustion stability, systems with surface diffusion and singular boundaries, reaction-diffusion pattern formation, heat-diffusion-limited thermal coagulation in biological tissues (with applications to tumour necrosis and hyperthermia), and an order-parameter model for multilane traffic flow describing hysteresis, synchronized traffic, cluster formation, and jam transitions. Part V (Chapters 14-15) provides the mathematical foundations underlying the preceding analysis, covering conformal mappings of canonical domains--including the half-plane, strip, annulus, and related regions--and the operator theory of integral transformations such as convolution. A substantial contribution to the mathematical theory of nonlinear interface dynamics, combining rigorous analytical methods with applications ranging from phase transitions and combustion to biomedical physics and traffic modelling, and serving as an important reference for researchers in applied mathematics, nonlinear science, and mathematical physics.…