Differential Equations

This book is devoted to impulsive differential equations and inclusions. Initial and boundary value problems for both impulsive differential equations and inclusions, as well as for each of impulsive functional differential equations or inclusions, and neutral functional differential equations, are studied. Moreover, results on impulsive semilinear ordinary and functional differential inclusions satisfying nonlocal boundary conditions are given careful attention. Positive solutions and multiple positive solutions for impulsive ordinary and functional differential equations, as well as impulsive differential inclusions satisfying periodic boundary conditions, are also studied. Some of these results are extended to functional differential equations and inclusions for operators that are nondensely defined on a Branch Scope, to impulsive hyperbolic differential inclusions, and to impulsive dynamic equations on time scales.

The book provides a comprehensive introduction to the mathematical theory of nonlinear problems described by elliptic partial differential equations. These equations can be seen as nonlinear versions of the classical Laplace equation, and they appear as mathematical models in different branches of physics, chemistry, biology, genetics and engineering and are also relevant in differential geometry and relativistic physics. Much of the modern theory of such equations is based on the calculus of variations and functional analysis.

The topic of singular boundary value problems has been of substantial and rapidly growing interest for many scientists and engineers. This book is devoted to singular boundary value problems for ordinary differential equations. It presents existence theory for a variety of problems having unbounded nonlinearities in regions where their solutions are searched for. The importance of thorough investigation of analytical solvability is emphasized by the fact that numerical simulations of solutions to such problems usually break down near singular points. The book provides both general existence principles and effective existence criteria which give a theoretical framework for investigation of variety singular boundary value problems, such as Dirichlet, periodic, mixed, focal, conjugate, Sturm-Liouville, Lidstone, or nonlocal problems.

Nonlocal problems concerning the conditions of the existence of different classes of solutions play an important role in the qualitative theory of differential equations. Here belong the problem of boundedness, periodicity, almost periodicity, stability in the sense of Poisson, and the problem of the existence of limit regimes of different types, convergence, dissipativity, and so on.

This highly visual introductory textbook presents an in-depth treatment suitable for undergraduates in mathematics and physics, gradually introducing abstraction while always keeping the link to physical motivation. Designed for lecturers as well as students, downloadable files for all figures, exercises, and practice problems are available online, as are solutions.

Written in a clear and accurate language that students can understand, this book minimizes the number of explicitly stated theorems and definitions. It deals with concepts in a conversational style that engages students. It includes more than 250 illustrated, worked examples for easy reading and comprehension.

This text has been written in clear and accurate language that students can read and comprehend. The author has minimized the number of explicitly state theorems and definitions, in favor of dealing with concepts in a more conversational manner. This is illustrated by over 250 worked out examples. The problems are extremely high quality and are regarded as one of the text's many strengths.

In this book, there are five chapters: The Laplace Transform, Systems of Homogenous Linear Differential Equations (HLDE), Methods of First and Higher Orders Differential Equations, Extended Methods of First and Higher Orders Differential Equations, and Applications of Differential Equations.

An introductory course on differential equations aimed at engineers. The book covers first order ODEs, higher order linear ODEs, systems of ODEs, Fourier series and PDEs, eigenvalue problems, the Laplace transform, and power series methods. The book originated as class notes for Math 286 at the University of Illinois at Urbana-Champaign in the Fall 2008 and Spring 2009 semesters. It has since been successfully used in many university classrooms as the main textbook.

This open access book provides a solution theory for time-dependent partial differential equations, which classically have not been accessible by a unified method. Instead of using sophisticated techniques and methods, the approach is elementary in the sense that only Hilbert space methods and some basic theory of complex analysis are required.

This book introduces Finite Difference Methods for both ordinary differential equations (ODEs) and partial differential equations (PDEs) and discusses the similarities and differences between algorithm design and stability analysis for different types of equations.

This text provides a very simple, initial introduction to the complete scientific computing pipeline: models, discretization, algorithms, programming, verification, and visualization.

This textbook teaches Finite Element Methods from a computational point of view. It focuses on how to develop flexible computer programs with Python, a programming language in which a combination of symbolic and numerical tools is used to achieve an explicit and practical derivation of finite element algorithms.

The FEniCS Project is an open-source Finite Element library. It gives its users easy access to powerful and scalable numerical tools for solving a great number of problems based on partial differential equations (PDEs).

The theory of difference equations, the methods used in their solutions, and their wide applications have advanced beyond their adolescent stage to occupy a central position in applicable analysis. In fact, in the last ten years, the proliferation of the subject is witnessed by numerous research articles and several monographs, annual international conferences, and new journals devoted to the study of difference equations and their applications.