Aissa Boukarou

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 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.

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.

This book is devoted to multiplicative analytic geometry. The book reflects recent investigations into the topic. The reader can use the main formulae for investigations of multiplicative differential equations, multiplicative integral equations and multiplicative geometry. The authors summarize the most recent contributions in this area. The goal of the authors is to bring the most recent research on the topic to capable senior undergraduate students, beginning graduate students of engineering and science and researchers in a form to advance further study. The book contains eight chapters. The chapters in the book are pedagogically organized. Each chapter concludes with a section with practical problems. 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. In the period 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. Multiplicative calculus can especially be useful as a mathematical tool for economics and finance. Multiplicative Analytic Geometry builds upon multiplicative calculus and advances the theory to the topics of analytic and differential geometry.