Download or read book A Unified Approach to Boundary Value Problems written by Athanassios S. Fokas and published by SIAM. This book was released on 2008-11-06 with total page 327 pages. Available in PDF, EPUB and Kindle. Book excerpt: A novel approach to analysing initial-boundary value problems for integrable partial differential equations (PDEs) in two dimensions, based on ideas of the inverse scattering transform that the author introduced in 1997. This method is unique in also yielding novel integral representations for linear PDEs. Several new developments are addressed in the book, including a new transform method for linear evolution equations on the half-line and on the finite interval; analytical inversion of certain integrals such as the attenuated Radon transform and the Dirichlet-to-Neumann map for a moving boundary; integral representations for linear boundary value problems; analytical and numerical methods for elliptic PDEs in a convex polygon; and integrable nonlinear PDEs. An epilogue provides a list of problems on which the author's new approach has been used, offers open problems, and gives a glimpse into how the method might be applied to problems in three dimensions.

Download or read book Unified Transform for Boundary Value Problems written by Athanasios S. Fokas and published by SIAM. This book was released on 2015-01-01 with total page 290 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book A Unified Approach to Boundary Value Problems at Resonance with the First Eigenvalue written by E. Serra and published by . This book was released on 1996 with total page 19 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Numerical Solution of Two Point Boundary Value Problems written by Herbert B. Keller and published by SIAM. This book was released on 1976-01-01 with total page 69 pages. Available in PDF, EPUB and Kindle. Book excerpt: Lectures on a unified theory of and practical procedures for the numerical solution of very general classes of linear and nonlinear two point boundary-value problems.

Download or read book Initial boundary Value Problems and the Navier Stokes Equations written by Heinz-Otto Kreiss and published by SIAM. This book was released on 1989-01-01 with total page 408 pages. Available in PDF, EPUB and Kindle. Book excerpt: Annotation This book provides an introduction to the vast subject of initial and initial-boundary value problems for PDEs, with an emphasis on applications to parabolic and hyperbolic systems. The Navier-Stokes equations for compressible and incompressible flows are taken as an example to illustrate the results. Researchers and graduate students in applied mathematics and engineering will find Initial-Boundary Value Problems and the Navier-Stokes Equations invaluable. The subjects addressed in the book, such as the well-posedness of initial-boundary value problems, are of frequent interest when PDEs are used in modeling or when they are solved numerically. The reader will learn what well-posedness or ill-posedness means and how it can be demonstrated for concrete problems. There are many new results, in particular on the Navier-Stokes equations. The direct approach to the subject still gives a valuable introduction to an important area of applied analysis.

Download or read book Boundary Value Problems written by Chi Yeung Lo and published by World Scientific. This book was released on 2000 with total page 282 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book has been designed for a one-year graduate course on boundary value problems for students of mathematics, engineering, and the physical sciences. It deals mainly with the three fundamental equations of mathematical physics, namely the heat equation, the wave equation, and Laplace's equation. The goal of the book is to obtain a formal solution to a given problem either by the method of separation of variables or by the method of general solutions and to verify that the formal solution possesses all the required properties. To provide the mathematical justification for this approach, the theory of Sturm-Liouville problems, the Fourier series, and the Fourier transform are fully developed. The book assumes a knowledge of advanced calculus and elementary differential equations.

Download or read book Dynamics of Structure and Foundation A Unified Approach written by Indrajit Chowdhury and published by CRC Press. This book was released on 2008-12-17 with total page 564 pages. Available in PDF, EPUB and Kindle. Book excerpt: Designed to provide engineers with quick access to current and practical information on the dynamics of structure and foundation, this unique work, consisting of two separately available volumes, serves as a complete reference, especially for those involved with earthquake or dynamic analysis, or the design of machine foundations in the oil, gas, a

Download or read book Elliptic Boundary Value Problems with Indefinite Weights Variational Formulations of the Principal Eigenvalue and Applications written by Fethi Belgacem and published by CRC Press. This book was released on 1997-05-05 with total page 260 pages. Available in PDF, EPUB and Kindle. Book excerpt: Elliptic Boundary Value Problems With Indefinite Weights presents a unified approach to the methodologies dealing with eigenvalue problems involving indefinite weights. The principal eigenvalue for such problems is characterized for various boundary conditions. Such characterizations are used, in particular, to formulate criteria for the persistence and extinctions of populations, and applications of the formulations obtained can be quite extensive.

Download or read book Boundary Value Problems written by F. D. Gakhov and published by Elsevier. This book was released on 2014-07-10 with total page 585 pages. Available in PDF, EPUB and Kindle. Book excerpt: Boundary Value Problems is a translation from the Russian of lectures given at Kazan and Rostov Universities, dealing with the theory of boundary value problems for analytic functions. The emphasis of the book is on the solution of singular integral equations with Cauchy and Hilbert kernels. Although the book treats the theory of boundary value problems, emphasis is on linear problems with one unknown function. The definition of the Cauchy type integral, examples, limiting values, behavior, and its principal value are explained. The Riemann boundary value problem is emphasized in considering the theory of boundary value problems of analytic functions. The book then analyzes the application of the Riemann boundary value problem as applied to singular integral equations with Cauchy kernel. A second fundamental boundary value problem of analytic functions is the Hilbert problem with a Hilbert kernel; the application of the Hilbert problem is also evaluated. The use of Sokhotski's formulas for certain integral analysis is explained and equations with logarithmic kernels and kernels with a weak power singularity are solved. The chapters in the book all end with some historical briefs, to give a background of the problem(s) discussed. The book will be very valuable to mathematicians, students, and professors in advanced mathematics and geometrical functions.

Download or read book Finite Difference Methods for Ordinary and Partial Differential Equations written by Randall J. LeVeque and published by SIAM. This book was released on 2007-01-01 with total page 356 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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. A unified view of stability theory for ODEs and PDEs is presented, and the interplay between ODE and PDE analysis is stressed. The text emphasizes standard classical methods, but several newer approaches also are introduced and are described in the context of simple motivating examples.

Download or read book Solving Ordinary and Partial Boundary Value Problems in Science and Engineering written by Karel Rektorys and published by CRC Press. This book was released on 1998-10-20 with total page 218 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an elementary, accessible introduction for engineers and scientists to the concepts of ordinary and partial boundary value problems, acquainting readers with fundamental properties and with efficient methods of constructing solutions or satisfactory approximations. Discussions include: ordinary differential equations classical theory of partial differential equations Laplace and Poisson equations heat equation variational methods of solution of corresponding boundary value problems methods of solution for evolution partial differential equations The author presents special remarks for the mathematical reader, demonstrating the possibility of generalizations of obtained results and showing connections between them. For the non-mathematician, the author provides profound functional-analytical results without proofs and refers the reader to the literature when necessary. Solving Ordinary and Partial Boundary Value Problems in Science and Engineering contains essential functional analytical concepts, explaining its subject without excessive abstraction.

Download or read book Multiple Solutions of Boundary Value Problems written by John R. Graef and published by Trends in Abstract and Applied Analysis. This book was released on 2016 with total page 279 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Initial and Boundary Value Problems in Two and Three Dimensions written by Konstantinos Kalimeris and published by . This book was released on 2010 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis: (a) presents the solution of several boundary value problems (BVPs) for the Laplace and the modified Helmholtz equations in the interior of an equilateral triangle; (b) presents the solution of the heat equation in the interior of an equilateral triangle; (c) computes the eigenvalues and eigenfunctions of the Laplace operator in the interior of an equilateral triangle for a variety of boundary conditions; (d) discusses the solution of several BVPs for the non-linear Schrödinger equation on the half line. In 1967 the Inverse Scattering Transform method was introduced; this method can be used for the solution of the initial value problem of certain integrable equations including the celebrated Korteweg-de Vries and nonlinear Schrödinger equations. The extension of this method from initial value problems to BVPs was achieved by Fokas in 1997, when a unified method for solving BVPs for both integrable nonlinear PDEs, as well as linear PDEs was introduced. This thesis applies "the Fokas method" to obtain the results mentioned earlier. For linear PDEs, the new method yields a novel integral representation of the solution in the spectral (transform) space; this representation is not yet effective because it contains certain unknown boundary values. However, the new method also yields a relation, known as "the global relation", which couples the unknown boundary values and the given boundary conditions. By manipulating the global relation and the integral representation, it is possible to eliminate the unknown boundary values and hence to obtain an effective solution involving only the given boundary conditions. This approach is used to solve several BVPs for elliptic equations in two dimensions, as well as the heat equation in the interior of an equilateral triangle. The implementation of this approach: (a) provides an alternative way for obtaining classical solutions; (b) for problems that can be solved by classical methods, it yields novel alternative integral representations which have both analytical and computational advantages over the classical solutions; (c) yields solutions of BVPs that apparently cannot be solved by classical methods. In addition, a novel analysis of the global relation for the Helmholtz equation provides a method for computing the eigenvalues and the eigenfunctions of the Laplace operator in the interior of an equilateral triangle for a variety of boundary conditions. Finally, for the nonlinear Schrödinger on the half line, although the global relation is in general rather complicated, it is still possible to obtain explicit results for certain boundary conditions, known as "linearizable boundary conditions". Several such explicit results are obtained and their significance regarding the asymptotic behavior of the solution is discussed.

Download or read book Boundary Value Problems on Time Scales Volume I written by Svetlin G. Georgiev and published by CRC Press. This book was released on 2021-10-15 with total page 324 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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. AUTHORS Svetlin 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.

Download or read book An Introduction to Nonlinear Boundary Value Problems written by Lakshmikantham and published by Academic Press. This book was released on 1974-05-31 with total page 399 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Chaos Fractals and Complexity written by Tassos Bountis and published by Springer Nature. This book was released on 2023-10-23 with total page 381 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume of proceedings contains research results within the framework of the fields of Chaos, Fractals and Complexity, written by experienced professors, young researchers, and applied scientists. It includes reviews of the fields, which are presented in an educational way for the widest possible audience, analytical results, computer simulations and experimental evidence, focusing on mathematical modelling. The papers presented here are selected from lectures given at the 28th Summer School “Dynamical Systems and Complexity”, July 18 – 27, 2022. Topics cover applications of complex systems in Neuroscience, Biology, Photonics, Seismology, Meteorology, and more broadly Physical and Engineering systems. The summer school has a long history, which began at the University of Patras in 1987 and continues with great success to this day. The original main purpose was to introduce young students and researchers of Greece to a new science that emerged several decades ago and continues to grow internationally at an ever increasing rate around the world.

Download or read book Initial Boundary Value Problems in Mathematical Physics written by Rolf Leis and published by Courier Corporation. This book was released on 2013-07-17 with total page 274 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introduction to classical scattering theory and time-dependent theory of linear equations in mathematical physics. Topics include wave operators, exterior boundary value problems, radiation conditions, limiting absorption principles, and more. 1986 edition.