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This book provides a unified analysis and scheme for the existence
and uniqueness of strong and mild solutions to certain fractional
kinetic equations. This class of equations is characterized by the
presence of a nonlinear time-dependent source, generally of
arbitrary growth in the unknown function, a time derivative in the
sense of Caputo and the presence of a large class of diffusion
operators. The global regularity problem is then treated separately
and the analysis is extended to some systems of fractional kinetic
equations, including prey-predator models of Volterra-Lotka type
and chemical reactions models, all of them possibly containing some
fractional kinetics. Besides classical examples involving the
Laplace operator, subject to standard (namely, Dirichlet, Neumann,
Robin, dynamic/Wentzell and Steklov) boundary conditions, the
framework also includes non-standard diffusion operators of
"fractional" type, subject to appropriate boundary conditions. This
book is aimed at graduate students and researchers in mathematics,
physics, mathematical engineering and mathematical biology, whose
research involves partial differential equations.
This book investigates several classes of partial differential
equations of real time variable and complex spatial variables,
including the heat, Laplace, wave, telegraph, Burgers,
Black-Merton-Scholes, Schroedinger and Korteweg-de Vries
equations.The complexification of the spatial variable is done by
two different methods. The first method is that of complexifying
the spatial variable in the corresponding semigroups of operators.
In this case, the solutions are studied within the context of the
theory of semigroups of linear operators. It is also interesting to
observe that these solutions preserve some geometric properties of
the boundary function, like the univalence, starlikeness, convexity
and spirallikeness. The second method is that of complexifying the
spatial variable directly in the corresponding evolution equation
from the real case. More precisely, the real spatial variable is
replaced by a complex spatial variable in the corresponding
evolution equation and then analytic and non-analytic solutions are
sought.For the first time in the book literature, we aim to give a
comprehensive study of the most important evolution equations of
real time variable and complex spatial variables. In some cases,
potential physical interpretations are presented. The generality of
the methods used allows the study of evolution equations of spatial
variables in general domains of the complex plane.
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