Differentialregning og differentialligninger

This book provides a comprehensive survey of the Sharkovsky ordering, its different aspects and its role in dynamical systems theory and applications. It addresses the coexistence of cycles for continuous interval maps and one-dimensional spaces, combinatorial dynamics on the interval and multidimensional dynamical systems. Also featured is a short chapter of personal remarks by O.M. Sharkovsky on the history of the Sharkovsky ordering, the discovery of which almost 60 years ago led to the inception of combinatorial dynamics. Now one of cornerstones of dynamics, bifurcation theory and chaos theory, the Sharkovsky ordering is an important tool for the investigation of dynamical processes in nature. Assuming only a basic mathematical background, the book will appeal to students, researchers and anyone who is interested in the subject.

This book develops a class of graded finite element methods to solve singular elliptic boundary value problems in two- and three-dimensional domains. It provides an approachable and self-contained presentation of the topic, including both the mathematical theory and numerical tools necessary to address the major challenges imposed by the singular solution. Moreover, by focusing upon second-order equations with constant coefficients, it manages to derive explicit results that are accessible to the broader computation community. Although written with mathematics graduate students and researchers in mind, this book is also relevant to applied and computational mathematicians, scientists, and engineers in numerical methods who may encounter singular problems.

This book is an attempt to make available to the student a coherent modern view of the theory of partial differential equations.

The modelling of systems by differential equations usually requires that the parameters involved be completely known. Such models often originate from problems in physics or economics where we have insufficient information on parameter values. One important class of stochastic mathematical models is stochastic partial differential equations (SPDEs), which can be seen as deterministic partial differential equations (PDEs) with finite or infinite dimensional stochastic processes - either with colour noise or white noise. Though white noise is a purely mathematical construction, it can be a good model for rapid random fluctuations.Stochastic Integral and Differential Equations in Mathematical Modelling concerns the analysis of discrete-time approximations for stochastic differential equations (SDEs) driven by Wiener processes. It also provides a theoretical basis for working with SDEs and stochastic processes.This book is written in a simple and clear mathematical logical language, with basic definitions and theorems on stochastic calculus provided from the outset. Each chapter contains illustrated examples via figures and tables. The reader can also construct new wavelets by using the procedure presented in the book. Stochastic Integral and Differential Equations in Mathematical Modelling fulfils the existing gap in the literature for a comprehensive account of this subject area.

This CIME Series book provides mathematical and simulation tools to help resolve environmental hazard and security-related issues.The contributions reflect five major topics identified by the SIES (Strategic Initiatives for the Environment and Security) as having significant societal impact: optimal control in waste management, in particular the degradation of organic waste by an aerobic biomass, by means of a mathematical model; recent developments in the mathematical analysis of subwave resonators; conservation laws in continuum mechanics, including an elaboration on the notion of weak solutions and issues related to entropy criteria; the applications of variational methods to 1-dimensional boundary value problems, in particular to light ray-tracing in ionospheric physics; and the mathematical modelling of potential electromagnetic co-seismic events associated to large earthquakes.This material will provide a sound foundation for those who intend to approach similar problems from a multidisciplinary perspective.

This study guide is designed for students taking courses in differential equations. The textbook includes examples, questions, and exercises that will help engineering students to review and sharpen their knowledge of the subject and enhance their performance in the classroom. Offering detailed solutions, multiple methods for solving problems, and clear explanations of concepts, this hands-on guide will improve student's problem-solving skills and basic and advanced understanding of the topics covered in electric circuit analysis courses. 

Functional analysis is an important branch of mathematical analysis which deals with the transformations of functions and their algebraic and topological properties. Motivated by their large applicability to real life problems, applications of functional analysis have been the aim of an intensive study effort in the last decades, yielding significant progress in the theory of functions and functional spaces, differential and difference equations and boundary value problems, differential and integral operators and spectral theory, and mathematical methods in physical and engineering sciences. The present volume is devoted to these investigations. The publication of this collection of papers is based on the materials of the mini-symposium "Functional Analysis in Interdisciplinary Applications" organized in the framework of the Fourth International Conference on Analysis and Applied Mathematics (ICAAM 2018, September 6¿9, 2018). Presenting a widerange of topics and results, this book will appeal to anyone working in the subject area, including researchers and students interested to learn more about different aspects and applications of functional analysis. Many articles are written by experts from around the world, strengthening international integration in the fields covered. The contributions to the volume, all peer reviewed, contain numerous new results.This volume contains four different chapters. The first chapter contains the contributed papers focusing on various aspects of the theory of functions and functional spaces. The second chapter is devoted to the research on difference and differential equations and boundary value problems. The third chapter contains the results of studies on differential and integral operators and on the spectral theory. The fourth chapter is focused on the simulation of problems arising in real-world applications of applied sciences.

The Qualitative Theory of Ordinary Differential Equations (ODEs) occupies a rather special position both in Applied and Theoretical Mathematics. On the one hand, it is a continuation of the standard course on ODEs. On the other hand, it is an introduction to Dynamical Systems, one of the main mathematical disciplines in recent decades. Moreover, it turns out to be very useful for graduates when they encounter differential equations in their work; usually those equations are very complicated and cannot be solved by standard methods.The main idea of the qualitative analysis of differential equations is to be able to say something about the behavior of solutions of the equations, without solving them explicitly. Therefore, in the first place such properties like the stability of solutions stand out. It is the stability with respect to changes in the initial conditions of the problem. Note that, even with the numerical approach to differential equations, all calculations are subject to a certain inevitable error. Therefore, it is desirable that the asymptotic behavior of the solutions is insensitive to perturbations of the initial state.Each chapter contains a series of problems (with varying degrees of difficulty) and a self-respecting student should solve them. This book is based on Raul Murillo's translation of Henryk ¿ö¿dek's lecture notes, which were in Polish and edited in the portal Matematyka Stosowana (Applied Mathematics) in the University of Warsaw.

Physics is expressed in the language of mathematics; it is deeply ingrained in how physics is taught and how it's practiced. A study of the mathematics used in science is thus asound intellectual investment for training as scientists and engineers. This first volume of two is centered on methods of solving partial differential equations (PDEs) and the special functions introduced. Solving PDEs can't be done, however, outside of the context in which they apply to physical systems. The solutions to PDEs must conform to boundary conditions, a set of additional constraints in space or time to be satisfied at the boundaries of the system, that small part of the universe under study. The first volume is devoted to homogeneous boundary-value problems (BVPs), homogeneous implying a system lacking a forcing function, or source function. The second volume takes up (in addition to other topics) inhomogeneous problems where, in addition to the intrinsic PDE governing a physical field, source functions are an essential part of the system. This text is based on a course offered at the Naval Postgraduate School (NPS) and while produced for NPS needs, it will serve other universities well. It is based on the assumption that it follows a math review course, and was designed to coincide with the second quarter of student study, which is dominated by BVPs but also requires an understanding of special functions and Fourier analysis.

This book brings together eleven topics on different aspects of fractional calculus in a single volume. It provides readers the basic knowledge of fractional calculus and introduces advanced topics and applications. The information in the book is presented in four parts:1.Fractional Diffusion Equations: (i) solutions of fractional diffusion equations using wavelet methods, (ii) the maximum principle for time fractional diffusion equations, (iii) nonlinear sub-diffusion equations. 2.Mathematical Analysis: (i) shifted Jacobi polynomials for solving and identifying coupled fractional delay differential equations, (ii) the monotone iteration principle in the theory of Hadamard fractional delay differential equations, (iii) dynamics of fractional order modified Bhalekar-Gejji System, (iv) Fractional Derivative. 3.Computational Techniques: GPU computing of special mathematical functions used in fractional calculus.4.Reviews: (i) the popular iterative method NIM, (ii) fractional derivative with non-singular kernels, (iii) some open problems in fractional order nonlinear system This is a useful reference for researchers and graduate level mathematics students seeking knowledge about of fractional calculus and applied mathematics.

This volume! aims at introducing some basic ideas for studying approxima- tion processes and, more generally, discrete processes. The study of discrete processes, which has grown together with the study of infinitesimal calcu- lus, has become more and more relevant with the use of computers. The volume is suitably divided in two parts. In the first part we illustrate the numerical systems of reals, of integers as a subset of the reals, and of complex numbers. In this context we intro- duce, in Chapter 2, the notion of sequence which invites also a rethinking of the notions of limit and continuity2 in terms of discrete processes; then, in Chapter 3, we discuss some elements of combinatorial calculus and the mathematical notion of infinity. In Chapter 4 we introduce complex num- bers and illustrate some of their applications to elementary geometry; in Chapter 5 we prove the fundamental theorem of algebra and present some of the elementary properties of polynomials and rational functions, and of finite sums of harmonic motions. In the second part we deal with discrete processes, first with the process of infinite summation, in the numerical case, i.e., in the case of numerical series in Chapter 6, and in the case of power series in Chapter 7. The last chapter provides an introduction to discrete dynamical systems; it should be regarded as an invitation to further study.

Inverse limits with set-valued functions are quickly becoming a popular topic of research due to their potential applications in dynamical systems and economics. This brief provides a concise introduction dedicated specifically to such inverse limits. The theory is presented along with detailed examples which form the distinguishing feature of this work. The major differences between the theory of inverse limits with mappings and the theory with set-valued functions are featured prominently in this book in a positive light.   The reader is assumed to have taken a senior level course in analysis and a basic course in topology. Advanced undergraduate and graduate students, and researchers working in this area will find this brief useful.  ¿

"Hypernumbers and Extrafunctions" presents a rigorous mathematical approach to operate with infinite values. First, concepts of real and complex numbers are expanded to include a new universe of numbers called hypernumbers which includes infinite quantities. This brief extends classical calculus based on real functions by introducing extrafunctions, which generalize not only the concept of a conventional function but also the concept of a distribution. Extrafucntions have been also efficiently used for a rigorous mathematical definition of the Feynman path integral, as well as for solving some problems in probability theory, which is also important for contemporary physics. This book introduces a new theory that includes the theory of distributions as a subtheory, providing more powerful tools for mathematics and its applications. Specifically, it makes it possible to solve PDE for which it is proved that they do not have solutions  in distributions. Also illustrated in this text is how this new theory allows the differentiation and integration of any real function. This text can be used for enhancing traditional courses of calculus for undergraduates, as well as for teaching a separate course for graduate students.