Khaled Zennir

This book presents recent investigations on nonlinear evolution partial differential equations with time delay, with particular focus on hyperbolic PDEs and their modern applications. Hyperbolic PDEs describe wave propagation and signals traveling at finite speeds and arise in many areas of science and engineering, including acoustics, elasticity, fluid dynamics, electromagnetism, traffic flow, and quantum mechanics. The book also examines dissipative hyperbolic systems, where energy decreases over time due to damping or internal resistance, as well as models involving time delay, in which the future state of a system depends on its past states. Such features appear naturally in systems with memory, feedback mechanisms, and nonlinear interactions, but they also present significant mathematical challenges concerning the existence, stability, and long-term behavior of solutions. The book consists of eight chapters. Chapter 1 introduces basic definitions and foundational results related to delay terms and damping mechanisms. The subsequent chapters study several important models, including the nonlinear Euler-Bernoulli beam with neutral delay, delayed transmission problems with infinite memory, plate equations with variable delay, wave equations with dynamical boundary conditions, wave equations with variable-exponent nonlinearities, thermoelastic laminated beams with time-varying delay, and delayed Timoshenko beam systems. Written with scholarly rigor, this book is intended for mathematicians, physicists, engineers, and researchers in applied sciences. It can serve both as a graduate-level text and as a reference for researchers working on partial differential equations and related fields.

This book is devoted to a systematic and comprehensive study of existence, stability, boundary value problems, oscillatory behavior, and linear systems associated with generalized quantum impulsive differential equations. In recent decades, the rapid development of quantum calculus—calculus without limits—together with the theory of impulsive differential equations has opened new and fertile directions for research. The synthesis of these two domains has led to the emergence of generalized quantum impulsive differential equations, a field that captures both discrete–continuous hybrid behavior and sudden state changes within a non-classical calculus setting. This book aims to provide both a rigorous theoretical foundation and practical analytical tools for researchers, graduate students, and specialists working in nonlinear analysis, dynamic equations, and applied mathematics.

The exposition of this Book begins with essential elements of stochastic analysis, stochastic calculus, and elements of functional analysis. We then progress to detailed discussions on existence, uniqueness, and stability of solutions, as well as qualitative behaviors under varying conditions. The text also incorporates selected applications, illustrating how stochastic models naturally arise in diverse scientific and engineering problems. This work is intended for graduate students, researchers, and professionals who wish to deepen their understanding of stochastic systems. It may serve as both a book for beginner researchers and a reference for specialists pursuing further studies in the field. The presentation balances rigor with accessibility, combining mathematical depth with an emphasis on clarity. We are indebted to the contributions of many mathematicians whose pioneering work laid the foundations of this subject. Our gratitude extends to colleagues and students whose questions and insights have helped shape the material presented here. In the first two chapters, the book introduces selected topics from probability the ory: Brownian motion and the Wiener process, the stochastic integral in Hilbert spaces, and fractional Brownian motion. It explains in detail the essential properties of functional analysis, such as generalized metrics and Banach spaces, compactness criteria, measures of non-compactness (MNC), fixed point theory, some properties of set-valued maps, fixed point results, and semi-group theory. The question of the quantitative study of impulsive stochastic differential equations/ systems is treated with particular attention in Chapter 3 and Chapter 4. With fixed moments and multiple delays, the existence of solutions with fixed moments and multiple delays is addressed through the application of Schaefer and Perov fixed point theorems in generalized Banach spaces, driven by standard Brownian motion. WhereasinChapter5, sufficient conditions for the local and global existence and exponential stability of mild solutions of semi-linear systems of stochastic differential equations with infinite fractional Brownian motions and impulses are established with the Hurst index H > 1/2. In Chapter 6, we discuss some results on the existence and uniqueness of mild solutions for systems of semilinear impulsive differential equations with infinite fractional Brownian motions and Wiener processes. The approach is based on a new ver sion of the fixed point theorem due to Krasnoselskii in generalized Banach spaces. Chapter 7 deals with impulsive neutral stochastic functional differential equations driven by fBm with a noncompact semigroup. In Chapter 8, we prove some existence results based on a nonlinear alternative of the Leray-Schauder type theorem in generalized Banach spaces for the convex case; we establish a multi-valued version of Perov’s fixed point theorem in a non-convex setting. In Chapter 9, we provide sufficient conditions for the existence of solutions for a class of second-order systems of stochastic impulsive differential inclusions. In Chapter 10, it is devoted to the study of a convex stochastic sweeping process with fractional Brownian motion and time delay. The approach is based on discretizing stochastic functional differential inclusions. This book is addressed to a wide audience of specialists such as mathematicians, physicists, engineers and biologists. It can be used as a textbook at the graduate level and as a reference book for several disciplines. 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.

This textbook is a real contribution to mathematical analysis, a discipline that requires a great deal of study and attention and finds interesting applications in many disciplines of sciences and engineering. This book is designed for undergraduate students in mathematics and engineering. From the first chapters, the book introduces real and complex numbers, the cornerstones of analysis. It explains in detail the properties of real numbers, such as the square root, absolute value. The analysis of sequences and series occupies a central place in this textbook. The reader discovers, step by step, the convergence through the study of real sequences, drawing on the concepts of Cauchy sequences and monotone sequences. The book also covers numerical series, presenting a variety of convergence criteria. The field of functions and asymptotic study of functions is treated with particular attention. The book contains proofs, numerous examples, and exercises with hints and solutions. Our proposal differs from all similar books in that it is sequential, simplified concepts, and includes numerous solved examples and exercises that serve the reader, enabling them to understand the content well.

This book presents the main tools for investigations of the existence and uniqueness of as well as the existence of multiple solutions for initial- and boundary-value problems for fuzzy impulsive dynamic equations on time scales. Time-scale theory is relatively new. The basic theory attempts to unify both approaches of dynamic modeling: difference and differential equations. Similar ideas have been used before and go back in the introduction of the Riemann-Stieltjes integral, which unifies sums and integrals. Many results in differential equations easily carry over to the corresponding results for difference equations, while other results seem to be totally different in nature. For these reasons, the theory of dynamic equations is an active area of research. The time scale calculus can be applied to any fields in which dynamic processes are described by discrete or continuous time models. The calculus of time scales has various applications involving noncontinuous domains such as certain insect populations, phytoremediation of metals, wound healing, maximization problems in economics, and traffic problems. This book is intended for researchers and students in engineering and science. There are eight chapters in this book. The chapters in the book are organized in a way that is pedagogically accessible. Each chapter concludes with a section on practical problems to develop further understanding.

The book consists of eight chapters, each focusing on different aspects of multiple integrals and related topics in mathematical analysis. In Chapter 1, multiple integrals are defined and developed. The Jordan measure in n-dimensional unit balls is introduced, along with the definition and criteria for multiple integrals, as well as their properties. Chapter 2 delves into advanced techniques for computing multiple integrals. It introduces the Taylor formula, discusses linear maps on measurable sets, and explores the metric properties of differentiable maps. In Chapter 3, we focus on improper multiple integrals and their properties. The chapter deduces criteria for the integrability of functions of several variables and develops concepts such as improper integrals of nonnegative functions, comparison criteria, and absolute convergence. Chapter 4 investigates the Stieltjes integral and its properties. Topics covered include the differentiation of monotone functions of finite variation and the Helly principle of choice, as well as continuous functions of finite variation. Chapter 5 addresses curvilinear integrals, defining line integrals of both the first and second kinds. It also discusses the independence of line integrals from the path of integration. In Chapter 6, surface integrals of the first and second kinds are introduced. The chapter presents the Gauss-Ostrogradsky theorem and Stokes’ formulas, along with advanced practical problems to practice these concepts.

The book contains six chapters. In Chapter 1, the set Rn is defined. n-dimensional ball, sphere, and rectangular neighborhood of a point are defined. Sequences in Rn are defined by their properties. Open, closed and compact sets in Rn are investigated. The Heine-Borel theorem is formulated and proved. In Chapter 2, functions of several variables are introduced. Limits of functions of several variables are defined with their properties are deducted. Continuous functions of several variables are investigated. Uniform continuity is introduced and explored. In Chapter 3, partial derivatives of higher order and differentials for functions of several variables and their properties are introduced. Criteria for the differentiability of functions of several variables are deducted. Gradient of a function explored. Directional derivatives are defined and investigated. In Chapter 4, higher order partial derivatives of functions of several variables are investigated. The minimum and maximum of functions of several variables are introduced and investigated. Implicit functions are defined and explored. The method of Lagrange multipliers is introduced. The book contains proofs, numerous examples, and exercises with solutions in the two last chapters.

The book discusses the stability, observability, and controllability of nonlinear systems of PDEs (such as Wave, Heat, Euler-Bernoulli beam, Petrovsky, Kirchhoff, equations, and more). Methods based on the theory of classical weak functions analysis and movements in Sobolev spaces are used to analyze nonlinear systems of evolutionary partial differential equations. With the unifying theme of evolutionary dynamic equations, both linear and nonlinear, in more complex environments with different approaches, the book presents a multidisciplinary blend of topics, spanning the fields of PDEs applied to various models coming from theoretical physics, biology, engineering, and natural sciences. This comprehensive book is prepared for a diverse audience interested in applied mathematics. With its broad applicability, this book aims to foster interdisciplinary collaboration and facilitate a deeper understanding of complex phenomenon concepts, practically in electromagnetic waves, the acoustic model for seismic waves, waves in blood vessels, wind drag on space, the linear shallow water equations, sound waves in liquids and gases, non-elastic effects in the string.

Partial Differential Equations (PDEs) are fundamental in fields such as physics and engineering, underpinning our understanding of sound, heat, diffusion, electrostatics, electrodynamics, thermodynamics, fluid dynamics, elasticity, general relativity, and quantum mechanics. They also arise in areas like differential geometry and the calculus of variations. This book focuses on recent investigations of PDEs in Sobolev and analytic spaces. It consists of twelve chapters, starting with foundational definitions and results on linear, metric, normed, and Banach spaces, which are essential for introducing weak solutions to PDEs. Subsequent chapters cover topics such as Lebesgue integration, Lp spaces, distributions, Fourier transforms, Sobolev and Bourgain spaces, and various types of KdV equations. Advanced topics include higher order dispersive equations, local and global well-posedness, and specific classes of Kadomtsev-Petviashvili equations. This book is intended for specialists like mathematicians, physicists, engineers, and biologists. It can serve as a graduate-level textbook and a reference for multiple disciplines.

The book contains eleven chapters introduced by an introductory description. Qualitative properties for the semilinear dissipative wave equations are discussed in Chapter 2 and Chapter 3 based on the solutions with compactly supported initial data. The purpose of Chapter 4 is to present results according to the well-possednes and behavior f solutions the nonlinear viscoelastic wave equations in weighted spaces. Elements of theory of Kirchhoff problem is introduced in Chapter 5. It is introduced same decay rate of second order evolution equations with density. Chapter 6 is devoted on the original method for Well posedness and general decay for wave equation with logarithmic nonlinearities. In Chapter 7, it is investigated the uniform stabilization of the Petrovsky-Wave nonlinear coupled system. The question of well-posedness and general energy decay of solutions for a system of three wave equations with a nonlinear strong dissipation are investigated in chapter 8 using the weighied. In sofar as Chapter 9 and chapter 10 are concerned with damped nonlinear wave problems in Fourier spaces. The last Chapter 11 analysis the existence/ nonexistence of solutions for structural damped wave equations with nonlinear memory terms in Rn.

Quantum calculus is the modern name for the investigation of calculus without limits. Quantum calculus, or q-calculus, began with F. H. Jackson in the early twentieth century, but this kind of calculus had already been worked out by renowned mathematicians Euler and Jacobi. Lately, quantum calculus has aroused a great amount of interest due to the high demand of mathematics that model quantum computing. The q-calculus appeared as a connection between mathematics and physics. It has a lot of applications in different mathematical areas such as number theory, combinatorics, orthogonal polynomials, basic hypergeometric functions and other quantum theory sciences, mechanics, and the theory of relativity. Recently, the concept of general quantum difference operators that generalize quantum calculus has been defined. General Quantum Variational Calculus is specially designed for those who wish to understand this important mathematical concept, as the text encompasses recent developments of general quantum variational calculus. The material is presented in a highly readable, mathematically solid format. Many practical problems are illustrated, displaying a wide variety of solution techniques. This book is addressed to a wide audience of specialists such as mathematicians, physicists, engineers, and biologists. It can be used as a textbook at the graduate level and as a reference for several disciplines.

Quantum calculus is the modern name for the investigation of calculus without limits. Quantum calculus, or q-calculus, began with F. H. Jackson in the early twentieth century, but this kind of calculus had already been worked out by renowned mathematicians Euler and Jacobi. Lately, quantum calculus has aroused a great amount of interest due to the high demand of mathematics that model quantum computing. The q-calculus appeared as a connection between mathematics and physics. It has a lot of applications in different mathematical areas such as number theory, combinatorics, orthogonal polynomials, basic hypergeometric functions and other quantum theory sciences, mechanics, and the theory of relativity. Recently, the concept of general quantum difference operators that generalize quantum calculus has been defined. General Quantum Variational Calculus is specially designed for those who wish to understand this important mathematical concept, as the text encompasses recent developments of general quantum variational calculus. The material is presented in a highly readable, mathematically solid format. Many practical problems are illustrated, displaying a wide variety of solution techniques. This book is addressed to a wide audience of specialists such as mathematicians, physicists, engineers, and biologists. It can be used as a textbook at the graduate level and as a reference for several disciplines.

Boundary Value Problems on Time Scales, Volume I is devoted to the qualitative theory of boundary value problems on time scales. Summarizing the most recent contributions in this area, it addresses a wide audience of specialists such as mathematicians, physicists, engineers and biologists. It can be used as a textbook at the graduate level and as a reference book for several disciplines. The text contains two volumes, both published by Chapman & Hall/CRC Press. Volume I presents boundary value problems for first- and second-order dynamic equations on time scales. Volume II investigates boundary value problems for three, four, and higher-order dynamic equations on time scales. Many results to differential equations carry over easily to corresponding results for difference equations, while other results seem to be totally different in nature. Because of these reasons, the theory of dynamic equations is an active area of research. The time-scale calculus can be applied to any field in which dynamic processes are described by discrete or continuous time models. The calculus of time scales has various applications involving noncontinuous domains such as certain bug populations, phytoremediation of metals, wound healing, maximization problems in economics, and traffic problems. Boundary value problems on time scales have been extensively investigated in simulating processes and the phenomena subject to short-time perturbations during their evolution. The material in this book is presented in highly readable, mathematically solid format. Many practical problems are illustrated displaying a wide variety of solution techniques. AUTHORSSvetlin 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. Khaled Zennir earned his PhD in mathematics in 2013 from Sidi Bel Abbès University, Algeria. In 2015, he received 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.

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. AuthorsSvetlin 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 Abbès 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.

Boundary Value Problems on Time Scales, Volume II is devoted to the qualitative theory of boundary value problems on time scales. Summarizing themost recent contributions in this area, it addresses a wide audience of specialists such as mathematicians, physicists, engineers and biologists. It can be used asa textbook at the graduate level and as a reference book for several disciplines. The text contains two volumes, both published by Chapman & Hall/CRC Press. Volume I presents boundary value problems for first- and second-order dynamic equations on time scales. Volume II investigates boundary value problems forthree, four, and higher-order dynamic equations on time scales. Many results to differential equations carry over easily to corresponding resultsfor difference equations, while other results seem to be totally different in nature. Because of these reasons, the theory of dynamic equations is an active area ofresearch. The time-scale calculus can be applied to any field in which dynamic processes are described by discrete or continuous time models. The calculus of time scales has various applications involving noncontinuous domains such as certain bug populations, phytoremediation of metals, woundhealing, maximization problems in economics, and traffic problems. Boundary value problems on time scales have been extensively investigated in simulatingprocesses and the phenomena subject to short-time perturbations during their evolution. The material in this book is presented in highly readable, mathematically solid format. Many practical problems are illustrated displaying a wide variety ofsolution techniques. AUTHORSSvetlin G. Georgiev is a mathematician who has worked in various areas of the study. He currently focuses on harmonic analysis, functional analysis, partialdifferential equations, ordinary differential equations, Clifford and quaternion analysis, integral equations, and dynamic calculus on time scales. Khaled Zennir earned his PhD in mathematics in 2013 from Sidi Bel Abbès University, Algeria. In 2015, he received his highest diploma in Habilitation inmathematics from Constantine University, Algeria. He is currently assistant professor at Qassim University in the Kingdom of Saudi Arabia. His researchinterests lie in the subjects of nonlinear hyperbolic partial differential equations: global existence, blowup, and long-time behavior.

The main subject of our book is to use the (p, p(x) and p(x))-bi-Laplacian operator in some partial differential systems, where we developed and obtained many results in quantitative and qualitative point of view.

This book is focused on the qualitative theory of general quantum calculus, the modern name for the investigation of calculus without limits. It centers on designing, analysing and applying computational techniques for general quantum differential equations. The quantum calculus or q-calculus began with F. H. Jackson in the early twentieth century, but this kind of calculus had already been worked out by Euler and Jacobi. Recently, it has aroused interest due to high demand of mathematics that models quantum computing and the connection between mathematics and physics. Quantum calculus has many applications in different mathematical areas such as number theory, combinatorics, orthogonal polynomials, basic hyper-geometric functions and other sciences such as quantum theory, mechanics and the theory of relativity. The authors summarize the most recent contributions in this area. General Quantum Numerical Analysis is intended for senior undergraduate students and beginning graduate students of engineering and science courses. The twelve chapters in this book are pedagogically organized, each concluding with a section of practical problems.

This book presents a projector analysis of dynamic systems on time scales. The dynamic systems are classified as first, second, third and fourth kinds. For each classes of dynamic systems the basic matrix chains are constructed. The proposed theory is applied for decoupling of dynamic equations on time scales. Properly involved derivatives, constraints and consistent initial values for the considered equations are defined. A linearization for nonlinear dynamic systems is introduced and the total derivative for regular linearized equations with tractability index one is investigated.

Multiplicative Partial Differential Equations presents an introduction to the theory of multiplicative partial differential equations (MPDEs). It is suitable for all types of basic courses on MPDEs. The authors' aim is to present a clear and well-organized treatment of the concepts behind the development of mathematics and solution techniques. The text is presented in a highly readable, mathematically solid format. Many practical problems are illustrated, displaying a wide variety of solution techniques. FeaturesIncludes new classification and canonical forms of second-order MPDEsProposes the latest techniques in solving the multiplicative wave equation such as the method of separation of variables and the energy methodUseful in allowing for the basic properties of multiplicative elliptic problems, fundamental solutions, multiplicative integral representation of multiplicative harmonic functions, meant-value formulas, strong principle of maximum, multiplicative Poisson equation, multiplicative Green functions, method of separation of variables, and theorems of Liouville and Harnack

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. 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.