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This book is devoted to the multiplicative differential calculus.
Its seven pedagogically organized chapters summarize the most
recent contributions in this area, concluding with a section of
practical problems to be assigned or for self-study. Two
operations, differentiation and integration, are basic in calculus
and analysis. In fact, they are the infinitesimal versions of the
subtraction and addition operations on numbers, respectively. From
1967 till 1970, Michael Grossman and Robert Katz gave definitions
of a new kind of derivative and integral, moving the roles of
subtraction and addition to division and multiplication, and thus
established a new calculus, called multiplicative calculus. It is
also called an alternative or non-Newtonian calculus.
Multiplicative calculus can especially be useful as a mathematical
tool for economics, finance, biology, and engineering.
Multiplicative Differential Calculus is written to be of interest
to a wide audience of specialists such as mathematicians,
physicists, engineers, and biologists. It is primarily a textbook
at the senior undergraduate and beginning graduate level and may be
used for a course on differential calculus. It is also for students
studying engineering and science. Authors Svetlin G. Georgiev is a
mathematician who has worked in various areas of the study. He
currently focuses on harmonic analysis, functional analysis,
partial differential equations, ordinary differential equations,
Clifford and quaternion analysis, integral equations, and dynamic
calculus on time scales. He is also the author of Dynamic Geometry
of Time Scales (CRC Press). He is a co-author of Conformable
Dynamic Equations on Time Scales, with Douglas R. Anderson (CRC
Press). Khaled Zennir earned his PhD in mathematics from Sidi Bel
Abbes University, Algeria. He earned his highest diploma in
Habilitation in Mathematics from Constantine University, Algeria.
He is currently Assistant Professor at Qassim University in the
Kingdom of Saudi Arabia. His research interests lie in the subjects
of nonlinear hyperbolic partial differential equations: global
existence, blowup, and long-time behavior. The authors have also
published: Multiple Fixed-Point Theorems and Applications in the
Theory of ODEs, FDEs and PDE; Boundary Value Problems on Time
Scales, Volume 1 and Volume II, all with CRC Press.
This book presents an introduction to the theory of multiplicative
partial differential equations (MPDEs). It is suitable for all
types of basic courses on MPDEs. The author’s aim is to present a
clear and well-organized treatment of the concept behind the
development of mathematics and solution techniques. The text
material of this book is presented in highly readable,
mathematically solid format. Many practical problems are
illustrated displaying a wide variety of solution techniques. The
book features: - The book includes new classification and canonical
forms of Second order MPDEs - Proposes a new technique to solve the
multiplicative wave equation such as method of separation of
variables, energy method. - The proposed technique in the book can
be used to give the basic properties of multiplicative elliptic
problems, the fundamental solutions, multiplicative integral
representation of multiplicative harmonic functions, mean-value
formulas, strong principle of maximum, the multiplicative Poisson
equation, multiplicative Green functions, method of separation of
variables, theorems of Liouville and Harnack.
Multiplicative Differential Equations: Volume 2 is the second part
of a comprehensive approach to the subject. It continues a series
of books written by the authors on multiplicative, geometric
approaches to key mathematical topics. This volume is devoted to
the theory of multiplicative differential systems. The asymptotic
behavior of the solutions of such systems is studied. Stability
theory for multiplicative linear and nonlinear systems is
introduced and boundary value problems for second order
multiplicative linear and nonlinear equations are explored. The
authors also present first order multiplicative partial
differential equations. Each chapter ends with a section of
practical problems. The book is accessible to graduate students and
researchers in mathematics, physics, engineering and biology.
Multiplicative Differential Equations: Volume I is the first part
of a comprehensive approach to the subject. It continues a series
of books written by the authors on multiplicative, geometric
approaches to key mathematical topics. This volume begins with a
basic introduction to multiplicative differential equations and
then moves on to first and second order equations, as well as the
question of existence and unique of solutions. Each chapter ends
with a section of practical problems. The book is accessible to
graduate students and researchers in mathematics, physics,
engineering and biology.
The book is a follow-up to the first book on the topic published
here. The book can be used for teaching and research purposes. The
book offers different techniques for investigations of Ordinary and
Partial Differential Equations and should promote interest in
functional analysis.
This is the second book in a two-volume set. Boundary value
problems are of interest to mathematicians, engineers, scientists
and the technique of investigating these problems for time scales
is unique. The key topics here are BVDs, ordinary and partial
differential equations, difference equations, and integral
equations and so has broad appeal. The techniques presented here
are applicable to these topics and the teaching and research. This
book is a different take on the topic than the competitors, most
offered at a higher level. This book will be accessible to advanced
undergraduates, graduate students, and appeal to researchers as
well.
Boundary value problems are of interest to mathematicians,
engineers, scientists and the technique of investigating these
problems for time scales is unique. The key topics here are BVDs,
ordinary and partial differential equations, difference equations,
and integral equations and so has broad appeal. The techniques
presented here are applicable to these topics and the teaching and
research. This book is a different take on the topic than the
competitors, most offered at a higher level. This book will be
accessible to advanced undergraduates, graduate students, and
appeal to researchers as well.
Multiple Fixed-Point Theorems and Applications in the Theory of
ODEs, FDEs and PDEs covers all the basics of the subject of
fixed-point theory and its applications with a strong focus on
examples, proofs and practical problems, thus making it ideal as
course material but also as a reference for self-study. Many
problems in science lead to nonlinear equations T x + F x = x posed
in some closed convex subset of a Banach space. In particular,
ordinary, fractional, partial differential equations and integral
equations can be formulated like these abstract equations. It is
desirable to develop fixed-point theorems for such equations. In
this book, the authors investigate the existence of multiple fixed
points for some operators that are of the form T + F, where T is an
expansive operator and F is a k-set contraction. This book offers
the reader an overview of recent developments of multiple
fixed-point theorems and their applications. About the Authors
Svetlin G. Georgiev is a mathematician who has worked in various
areas of mathematics. He currently focuses on harmonic analysis,
functional analysis, partial differential equations, ordinary
differential equations, Clifford and quaternion analysis, integral
equations and dynamic calculus on time scales. Khaled Zennir is
assistant professor at Qassim University, KSA. He received his PhD
in mathematics in 2013 from Sidi Bel Abbes University, Algeria. He
obtained his Habilitation in mathematics from Constantine
University, Algeria in 2015. His research interests lie in
nonlinear hyperbolic partial differential equations: global
existence, blow up and long-time behavior.
This book is written as a textbook and includes examples and
exercises. This is a companion volume to the author's other books
published here on Multiplicative Geometry. There are no similar
books on this topic.
This book on functional analysis covers all the basics of the
subject (normed, Banach and Hilbert spaces, Lebesgue integration
and spaces, linear operators and functionals, compact and
self-adjoint operators, small parameters, fixed point theory) with
a strong focus on examples, exercises and practical problems, thus
making it ideal as course material but also as a reference for
self-study.
This book is devoted on recent developments of linear and nonlinear
fractional Riemann-Liouville and Caputo integral inequalities on
time scales. The book is intended for the use in the field of
fractional dynamic calculus on time scales and fractional dynamic
equations on time scales. It is also suitable for graduate courses
in the above fields, and contains ten chapters. The aim of this
book is to present a clear and well-organized treatment of the
concept behind the development of mathematics as well as solution
techniques. The text material of this book is presented in a
readable and mathematically solid format.
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