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Generalized Quantum Calculus with Applications is devoted to the qualitative theory of general quantum calculus and its applications to general quantum differential equations and inequalities. The book is aimed at upper-level undergraduate students and beginning graduate students in a range of interdisciplinary courses including physical sciences and engineering, from quantum mechanics to differential equations, with pedagogically organized chapters that each concludes with a section of practical problems. Generalized quantum calculus includes a generalization of the q-quantum calculus and the time scale calculus. There are many open problems and difficulties in q-quantum calculus and time-scale calculus, and this book explores how to use the generalized quantum operators to solve difficulties arising in q-quantum calculus and time-scale calculus, including but not limited to generalized quantum integration, generalized quantum chain rules, and generalized quantum Taylor formula.Since generalized quantum calculus includes the q-quantum and time-scale calculus, this book can be utilized by a wide audience of researchers and students. This text is one of few foundational books on generalized quantum calculus and can be used for future discoveries in the area of integral transforms, variational calculus, integral equations, and inequalities in the language of generalized quantum calculus. This book also offers detailed proofs, exercises, and examples to aid instructors, researchers, and users in their studies.Explores cutting-edge research trends in quantum calculusProvides practical information and techniques for building fundamental knowledge and applying contemporary quantum calculus in upper-undergraduate and graduate-level studiesServes as a front-line book for budding researchers and experts of mathematics, along with students from several interdisciplinary fieldsOffers additional resources such as detailed proofs, exercises, and examples to aid instructors and students in their work

This book presents the theory of dynamic equations on time scales and applications, providing an overview of recent developments in the foundations of the field as well as its applications. It discusses the recent results related to the qualitative properties of solutions like existence and uniqueness, stability, continuous dependence, controllability, oscillations, etc.• Presents cutting-edge research trends of dynamic equations and recent advances in contemporary research on the topic of time scales• Connects several new areas of dynamic equations on time scales with applications in different fields• Includes mathematical explanation from the perspective of existing knowledge of dynamic equations on time scales• Offers several new recently developed results, which are useful for the mathematical modeling of various phenomena• Useful for several interdisciplinary fields like economics, biology, and population dynamics from the perspective of new trendsThe text is for postgraduate students, professionals, and academic researchers working in the fields of Applied Mathematics.

This book presents contemporary mathematical concepts and techniques including theories of summability, fixed point and non-absolute integration and applications, providing an overview of recent developments in the foundations of the field as well as its applications. It discusses the recent results of double sequence spaces as the four-dimensional forward difference matrix in double sequence spaces, several new fixed point on Hadamard type fractional integral and differential operator related to the qualitative properties of solutions like, existence and uniqueness, stability, continuous dependence, controllability, oscillations, etc. It also includes several new areas of nonabsolute integration theory are introduced and their applications to other fields. This reference text is for researchers, academics, and professionals in the field of pure and applied mathematics.Covers recent research breakthroughs in this field offering new approaches and methods for both theoretical exploration and practical applicationPresents insights into functional analytic methods in summability, absolute and strong summability, direct theorems on summability, special and general summability methods, and their applicationsHighlights fixed-point theory’s application to real-world problems and offers solutions to various complex challengesIntroduces new areas of non-absolute integration theory, such as the Henstock-Kurzweil integral and generalized Riemann integralDiscusses sequence spaces and functional analysis, including the exploration of double sequence spaces and the four-dimensional forward difference matrix, offering valuable contributions to ongoing research

This book presents advanced research on quantum calculus, functional analysis, and their interdisciplinary applications. It focuses on solving complex challenges in areas such as quantum transport, carbon nanotubes, approximation theory, and renewable natural resources by providing cutting-edge developments in these specialised fields.

This book is about the β-quantum inequalities. This book covers a range of generalized quantum inequalities in detail. It begins with the basic calculus of the β-quantum operator and then explores various types of inequalities. The content is presented in a straightforward manner and is aimed at senior undergraduate and postgraduate students in engineering and science programs. The book consists of six chapters, and almost all of the material is new and represents the authors' original work. The chapters are organized in a way that is easy to follow, and each chapter concludes with a section containing practical problems.

The Stieltjes derivative is a modification of the usual derivative through a nondecreasing and left-continuous map. This change in the definition allows us to study several differential problems under the same framework.This monograph is the first published book that offers a comprehensive view of the fundamentals of Stieltjes calculus and its applications, making it approachable to newcomers and experts. It aims to provide an integrated approach to the foundations and recent developments in the area of the Stieltjes derivatives and the qualitative theory of the Stieltjes differential equations. Through 10 pedagogically organized chapters, the authors examine a wide scope of the concept of the Stieltjes derivative and its applications. Each chapter focuses on theory, and proofs, and contains sufficient examples to enrich the reader's understanding.The Stieltjes derivative contains the Hilger delta derivative on time scales. Thus, offering a new unification and extension of continuous and discrete calculus. Further, a study of differential equations in the sense of the Stieltjes derivative allows the study of many classical problems in a unique framework. This theory has the advantage that ordinary differential equations, ordinary difference equations, quantum difference equations, impulsive differential equations, dynamic equations on time scales, and generalized differential equations can be treated as particular instances of the Stieltjes differential equations. Hence, this book serves as a basic reference for researchers to harness this powerful technique further to unlock new insights and embrace the intricacies of natural processes. Researchers and graduate students at various levels interested in learning about the Stieltjes differential calculus and related fields will find this text a valuable resource of both introductory and advanced material.

The process of authoring this book is inspired by the recent increased activity of research on dynamic equations on time scales and other closely related areas. This monograph is the first published book that attempts to provide a comprehensive view of the theory and applications of set dynamic equations on time scales. The main focus of the book is the qualitative theory of set dynamic equations and their applications to fuzzy dynamic equations. The key topics include the solvability of set dynamic equations, stability of set dynamic equations, and applications to certain types of fuzzy dynamic equations.There are five chapters in the book, through which the authors examine a wide scope of the concept of set dynamic equations and their applications. Each chapter focuses on theory and proofs to enrich the reader's understanding of the topic.This book will be particularly useful to those experts who work in applied analysis, in general. It will also be a good reference for computer scientists since it covers fuzzy dynamic equations. Researchers and graduate students at various levels interested in learning about set dynamic equations and related fields will find this text a valuable resource of both introductory and advanced material.