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Numerical analysis forms a cornerstone of numeric computing and
optimization, in particular recently, interval numerical
computations play an important role in these topics. The interest
of researchers in computations involving uncertain data, namely
interval data opens new avenues in coping with real-world problems
and deliver innovative and efficient solutions. This book provides
the basic theoretical foundations of numerical methods, discusses
key technique classes, explains improvements and improvements, and
provides insights into recent developments and challenges. The
theoretical parts of numerical methods, including the concept of
interval approximation theory, are introduced and explained in
detail. In general, the key features of the book include an
up-to-date and focused treatise on error analysis in calculations,
in particular the comprehensive and systematic treatment of error
propagation mechanisms, considerations on the quality of data
involved in numerical calculations, and a thorough discussion of
interval approximation theory. Moreover, this book focuses on
approximation theory and its development from the perspective of
linear algebra, and new and regular representations of numerical
integration and their solutions are enhanced by error analysis as
well. The book is unique in the sense that its content and
organization will cater to several audiences, in particular
graduate students, researchers, and practitioners.
Soft Numerical Computing in Uncertain Dynamic Systems is intended
for system specialists interested in dynamic systems that operate
at different time scales. The book discusses several types of
errors and their propagation, covering numerical methods-including
convergence and consistence properties and characteristics-and
proving of related theorems within the setting of soft computing.
Several types of uncertainty representation like interval, fuzzy,
type 2 fuzzy, granular, and combined uncertain sets are discussed
in detail. The book can be used by engineering students in control
and finite element fields, as well as all engineering, applied
mathematics, economics, and computer science students. One of the
important topics in applied science is dynamic systems and their
applications. The authors develop these models and deliver
solutions with the aid of numerical methods. Since they are
inherently uncertain, soft computations are of high relevance here.
This is the reason behind investigating soft numerical computing in
dynamic systems. If these systems are involved with
complex-uncertain data, they will be more practical and important.
Real-life problems work with this type of data and most of them
cannot be solved exactly and easily-sometimes they are impossible
to solve. Clearly, all the numerical methods need to consider error
of approximation. Other important applied topics involving
uncertain dynamic systems include image processing and pattern
recognition, which can benefit from uncertain dynamic systems as
well. In fact, the main objective is to determine the coefficients
of a matrix that acts as the frame in the image. One of the
effective methods exhibiting high accuracy is to use finite
differences to fill the cells of the matrix.
This book contains new and useful materials concerning fuzzy
fractional differential and integral operators and their
relationship. As the title of the book suggests, the fuzzy subject
matter is one of the most important tools discussed. Therefore, it
begins by providing a brief but important and new description of
fuzzy sets and the computational calculus they require. Fuzzy
fractals and fractional operators have a broad range of
applications in the engineering, medical and economic sciences.
Although these operators have been addressed briefly in previous
papers, this book represents the first comprehensive collection of
all relevant explanations. Most of the real problems in the
biological and engineering sciences involve dynamic models, which
are defined by fuzzy fractional operators in the form of fuzzy
fractional initial value problems. Another important goal of this
book is to solve these systems and analyze their solutions both
theoretically and numerically. Given the content covered, the book
will benefit all researchers and students in the mathematical and
computer sciences, but also the engineering sciences.
This book contains the topics of artificial intelligence and deep
learning that do have much application in real-life problems. The
concept of uncertainty has long been used in applied science,
especially decision making and a logical decision must be made in
the field of uncertainty or in the real-life environment that is
formed and combined with vague concepts and data. The chapters of
this book are connected to the new concepts and aspects of decision
making with uncertainty. Besides, other chapters are involved with
the concept of data mining and decision making under uncertain
computations.
This book identifies the important uncertainties to use in
real-world problem modeling. Having information about several types
of ambiguities, vagueness, and uncertainties is vital in modeling
problems that involve linguistic variables, parameters, and word
computing. Today, since most of our real-world problems are related
to decision-making at the right time, we need to apply intelligent
decision science. Clearly, in order to have an appropriate and
flexible mathematical model, every intelligent system requires real
data on our environment. Presenting problems that can be
represented using mathematical models to create a system of linear
equations, this book discusses the latest insights into uncertain
information.
This book contains new and useful materials concerning fuzzy
fractional differential and integral operators and their
relationship. As the title of the book suggests, the fuzzy subject
matter is one of the most important tools discussed. Therefore, it
begins by providing a brief but important and new description of
fuzzy sets and the computational calculus they require. Fuzzy
fractals and fractional operators have a broad range of
applications in the engineering, medical and economic sciences.
Although these operators have been addressed briefly in previous
papers, this book represents the first comprehensive collection of
all relevant explanations. Most of the real problems in the
biological and engineering sciences involve dynamic models, which
are defined by fuzzy fractional operators in the form of fuzzy
fractional initial value problems. Another important goal of this
book is to solve these systems and analyze their solutions both
theoretically and numerically. Given the content covered, the book
will benefit all researchers and students in the mathematical and
computer sciences, but also the engineering sciences.
This book identifies the important uncertainties to use in
real-world problem modeling. Having information about several types
of ambiguities, vagueness, and uncertainties is vital in modeling
problems that involve linguistic variables, parameters, and word
computing. Today, since most of our real-world problems are related
to decision-making at the right time, we need to apply intelligent
decision science. Clearly, in order to have an appropriate and
flexible mathematical model, every intelligent system requires real
data on our environment. Presenting problems that can be
represented using mathematical models to create a system of linear
equations, this book discusses the latest insights into uncertain
information.
This book presents the theory and application of the models
presented in this regard and establishes a meaningful relationship
between data envelopment analysis and multi-attribute decision
making. The issue of "choice" using the aggregation of voters'
votes is one of the most important group decision-making issues
that are always considered by decision makers in electoral systems.
Voting is a method of group decision making in a democratic society
that expresses the will of the majority. Voting is perhaps the
simplest way to gather the opinions of experts, and this ease of
application has made it a multi-attribute decision-making method in
group decisions. Preferential voting is a type of voting that may
refer to electoral systems or groups of the electoral system. In
preferential voting, voters vote for multiple candidates, and how
the candidates are arranged on the ballot is important. Researchers
have made many efforts to provide models of voter aggregation, and
one of the best results of these efforts is the aggregation of
votes based on the policy of data envelopment analysis. Thus, in
group decisions, the opinions of experts are obtained in a simple
structure and consolidated in an interactive and logical structure,
and the results can be a powerful tool for decision support. This
book provides a complete set of voting models based on data
envelopment analysis and expressing its various applications in
industry and society. However, most decision-making methods do not
use the opinions of experts or reduce the motivation of experts to
participate in complex interactions and time, while voting methods
do not have this shortcoming. This book is suitable for graduate
students in the fields of industrial management, business
management, industrial engineering, applied mathematics, and
economics. It can also be a good source for researchers in decision
science, decision support systems, data envelopment analysis,
supply chain management, healthcare management, and others. The
methods presented in this book can not only offer a comprehensive
framework for solving the problems of these areas but also can
inspire researchers to pursue new innovative hybrid methods.
This book suggests that the numerical analysis subjects' matter are
the important tools of the book topic, because numerical errors and
methods have important roles in solving integral equations.
Therefore, all needed topics including a brief description of
interpolation are explained in the book. The integral equations
have many applications in the engineering, medical, and economic
sciences, so the present book contains new and useful materials
about interval computations including interval interpolations that
are going to be used in interval integral equations. The concepts
of integral equations are going to be discussed in two directions,
analytical concepts, and numerical solutions which both are
necessary for these kinds of dynamic systems. The differences
between this book with the others are a full discussion of error
topics and also using interval interpolations concepts to obtain
interval integral equations. All researchers and students in the
field of mathematical, computer, and also engineering sciences can
benefit the subjects of the book.
This book delivers a concise and carefully structured introduction
to probability and random variables. It aims to build a linkage
between the theoretical conceptual topics and the practical
applications, especially in the undergraduate engineering area. The
book motivates the student to gain full understanding of the
fundamentals of probability theory and help acquire working
problem-solving skills and apply the theory to engineering
applications. Each chapter includes solved examples at varying
levels (both introductory and advanced) in addition to problems
that demonstrate the relevance of the probability and random
variables in engineering. As authors, we focused on to find out the
optimum ways in order to introduce the topics in probability and
random variables area.
As the title of the book suggests, the topics of this book are
organized into two parts. The first part points out the fuzzy
differential equations and the second one is related to the fuzzy
integral equations. The book contains nine chapters that six
chapters are about fuzzy differential equations and three of them
are about fuzzy integral equations. In each part, the chapters'
authors are going to discuss the topics theoretically and
numerically. All researchers and students in the field of
mathematical, computer, and also engineering sciences can benefit
from the subjects of the book.
This book suggests that the numerical analysis subjects' matter are
the important tools of the book topic, because numerical errors and
methods have important roles in solving integral equations.
Therefore, all needed topics including a brief description of
interpolation are explained in the book. The integral equations
have many applications in the engineering, medical, and economic
sciences, so the present book contains new and useful materials
about interval computations including interval interpolations that
are going to be used in interval integral equations. The concepts
of integral equations are going to be discussed in two directions,
analytical concepts, and numerical solutions which both are
necessary for these kinds of dynamic systems. The differences
between this book with the others are a full discussion of error
topics and also using interval interpolations concepts to obtain
interval integral equations. All researchers and students in the
field of mathematical, computer, and also engineering sciences can
benefit the subjects of the book.
This book delivers a concise and carefully structured introduction
to probability and random variables. It aims to build a linkage
between the theoretical conceptual topics and the practical
applications, especially in the undergraduate engineering area. The
book motivates the student to gain full understanding of the
fundamentals of probability theory and help acquire working
problem-solving skills and apply the theory to engineering
applications. Each chapter includes solved examples at varying
levels (both introductory and advanced) in addition to problems
that demonstrate the relevance of the probability and random
variables in engineering. As authors, we focused on to find out the
optimum ways in order to introduce the topics in probability and
random variables area.
As the title of the book suggests, the topics of this book are
organized into two parts. The first part points out the fuzzy
differential equations and the second one is related to the fuzzy
integral equations. The book contains nine chapters that six
chapters are about fuzzy differential equations and three of them
are about fuzzy integral equations. In each part, the chapters'
authors are going to discuss the topics theoretically and
numerically. All researchers and students in the field of
mathematical, computer, and also engineering sciences can benefit
from the subjects of the book.
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