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| Author | : Elizabeth K. Leonard |
| Publisher | : |
| Total Pages | : 0 |
| Release | : 1977 |
| Genre | : Nonlinear boundary value problems |
| ISBN | : |
We show the existence of a solution, positive and continuous on 0 ^ t 00 , and tending to 0 as t -*• 00 , of the differential equation x + 2t - x + a(t)x k =0, 1
| Author | : John R Graef |
| Publisher | : World Scientific |
| Total Pages | : 290 |
| Release | : 2015-08-26 |
| Genre | : Mathematics |
| ISBN | : 9814696560 |
Variational methods and their generalizations have been verified to be useful tools in proving the existence of solutions to a variety of boundary value problems for ordinary, impulsive, and partial differential equations as well as for difference equations. In this monograph, we look at how variational methods can be used in all these settings. In our first chapter, we gather the basic notions and fundamental theorems that will be applied in the remainder of this monograph. While many of these items are easily available in the literature, we gather them here both for the convenience of the reader and for the purpose of making this volume somewhat self-contained. Subsequent chapters deal with the Sturm-Liouville problems, multi-point boundary value problems, problems with impulses, partial differential equations, and difference equations. An extensive bibliography is also included.
| Author | : Lakshmikantham |
| Publisher | : Academic Press |
| Total Pages | : 399 |
| Release | : 1974-05-31 |
| Genre | : Computers |
| ISBN | : 0080956181 |
A book on an advanced level that exposes the reader to the fascinating field of differential equations and provides a ready access to an up-to-date state of this art is of immense value. This book presents a variety of techniques that are employed in the theory of nonlinear boundary value problems. For example, the following are discussed: methods that involve differential inequalities; shooting and angular function techniques; functional analytic approaches; topological methods.
| Author | : Milan Kubicek |
| Publisher | : Courier Corporation |
| Total Pages | : 338 |
| Release | : 2008-01-01 |
| Genre | : Mathematics |
| ISBN | : 0486463001 |
A survey of the development, analysis, and application of numerical techniques in solving nonlinear boundary value problems, this text presents numerical analysis as a working tool for physicists and engineers. Starting with a survey of accomplishments in the field, it explores initial and boundary value problems for ordinary differential equations, linear boundary value problems, and the numerical realization of parametric studies in nonlinear boundary value problems. The authors--Milan Kubicek, Professor at the Prague Institute of Chemical Technology, and Vladimir Hlavacek, Professor at the University of Buffalo--emphasize the description and straightforward application of numerical techniques rather than underlying theory. This approach reflects their extensive experience with the application of diverse numerical algorithms.
| Author | : Marino Badiale |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 204 |
| Release | : 2010-12-07 |
| Genre | : Mathematics |
| ISBN | : 0857292277 |
Semilinear elliptic equations are of fundamental importance for the study of geometry, physics, mechanics, engineering and life sciences. The variational approach to these equations has experienced spectacular success in recent years, reaching a high level of complexity and refinement, with a multitude of applications. Additionally, some of the simplest variational methods are evolving as classical tools in the field of nonlinear differential equations. This book is an introduction to variational methods and their applications to semilinear elliptic problems. Providing a comprehensive overview on the subject, this book will support both student and teacher engaged in a first course in nonlinear elliptic equations. The material is introduced gradually, and in some cases redundancy is added to stress the fundamental steps in theory-building. Topics include differential calculus for functionals, linear theory, and existence theorems by minimization techniques and min-max procedures. Requiring a basic knowledge of Analysis, Functional Analysis and the most common function spaces, such as Lebesgue and Sobolev spaces, this book will be of primary use to graduate students based in the field of nonlinear partial differential equations. It will also serve as valuable reading for final year undergraduates seeking to learn about basic working tools from variational methods and the management of certain types of nonlinear problems.
| Author | : M. A. Lavrent’ev |
| Publisher | : Courier Dover Publications |
| Total Pages | : 164 |
| Release | : 2016-01-14 |
| Genre | : Mathematics |
| ISBN | : 0486160289 |
Famous monograph by a distinguished mathematician presents an innovative approach to classical boundary value problems. The treatment employs the basic scheme first suggested by Hilbert and developed by Tonnelli. 1963 edition.
| Author | : Seiro Omata |
| Publisher | : Springer Nature |
| Total Pages | : 99 |
| Release | : 2022-11-28 |
| Genre | : Mathematics |
| ISBN | : 9811967318 |
This volume is devoted to the study of hyperbolic free boundary problems possessing variational structure. Such problems can be used to model, among others, oscillatory motion of a droplet on a surface or bouncing of an elastic body against a rigid obstacle. In the case of the droplet, for example, the membrane surrounding the fluid in general forms a positive contact angle with the obstacle, and therefore the second derivative is only a measure at the contact free boundary set. We will show how to derive the mathematical problem for a few physical systems starting from the action functional, discuss the mathematical theory, and introduce methods for its numerical solution. The mathematical theory and numerical methods depart from the classical approaches in that they are based on semi-discretization in time, which facilitates the application of the modern theory of calculus of variations.
| Author | : Antonio Cañada |
| Publisher | : Springer |
| Total Pages | : 136 |
| Release | : 2015-11-24 |
| Genre | : Mathematics |
| ISBN | : 3319252895 |
This book highlights the current state of Lyapunov-type inequalities through a detailed analysis. Aimed toward researchers and students working in differential equations and those interested in the applications of stability theory and resonant systems, the book begins with an overview Lyapunov’s original results and moves forward to include prevalent results obtained in the past ten years. Detailed proofs and an emphasis on basic ideas are provided for different boundary conditions for ordinary differential equations, including Neumann, Dirichlet, periodic, and antiperiodic conditions. Novel results of higher eigenvalues, systems of equations, partial differential equations as well as variational approaches are presented. To this respect, a new and unified variational point of view is introduced for the treatment of such problems and a systematic discussion of different types of boundary conditions is featured. Various problems make the study of Lyapunov-type inequalities of interest to those in pure and applied mathematics. Originating with the study of the stability properties of the Hill equation, other questions arose for instance in systems at resonance, crystallography, isoperimetric problems, Rayleigh type quotients and oscillation and intervals of disconjugacy and it lead to the study of Lyapunov-type inequalities for differential equations. This classical area of mathematics is still of great interest and remains a source of inspiration.
| Author | : Athanasios S. Fokas |
| Publisher | : SIAM |
| Total Pages | : 290 |
| Release | : 2015-01-01 |
| Genre | : Mathematics |
| ISBN | : 1611973821 |
This book describes state-of-the-art advances and applications of the unified transform and its relation to the boundary element method. The authors present the solution of boundary value problems from several different perspectives, in particular the type of problems modeled by partial differential equations (PDEs). They discuss recent applications of the unified transform to the analysis and numerical modeling of boundary value problems for linear and integrable nonlinear PDEs and the closely related boundary element method, a well-established numerical approach for solving linear elliptic PDEs. The text is divided into three parts. Part I contains new theoretical results on linear and nonlinear evolutionary and elliptic problems. New explicit solution representations for several classes of boundary value problems are constructed and rigorously analyzed. Part II is a detailed overview of variational formulations for elliptic problems. It places the unified transform approach in a classic context alongside the boundary element method and stresses its novelty. Part III presents recent numerical applications based on the boundary element method and on the unified transform.
