A Basic Guide to Uniqueness Problems for Evolutionary Differential Equations

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ISBN 13 : 9783031347979
Total Pages : 0 pages
Book Rating : 4.3/5 (479 download)

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Book Synopsis A Basic Guide to Uniqueness Problems for Evolutionary Differential Equations by : Mi-Ho Giga

Download or read book A Basic Guide to Uniqueness Problems for Evolutionary Differential Equations written by Mi-Ho Giga and published by . This book was released on 2023 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book addresses the issue of uniqueness of a solution to a problem - a very important topic in science and technology, particularly in the field of partial differential equations, where uniqueness guarantees that certain partial differential equations are sufficient to model a given phenomenon. This book is intended to be a short introduction to uniqueness questions for initial value problems. One often weakens the notion of a solution to include non-differentiable solutions. Such a solution is called a weak solution. It is easier to find a weak solution, but it is more difficult to establish its uniqueness. This book examines three very fundamental equations: ordinary differential equations, scalar conservation laws, and Hamilton-Jacobi equations. Starting from the standard Gronwall inequality, this book discusses less regular ordinary differential equations. It includes an introduction of advanced topics like the theory of maximal monotone operators as well as what is called DiPerna-Lions theory, which is still an active research area. For conservation laws, the uniqueness of entropy solution, a special (discontinuous) weak solution is explained. For Hamilton-Jacobi equations, several uniqueness results are established for a viscosity solution, a kind of a non-differentiable weak solution. The uniqueness of discontinuous viscosity solution is also discussed. A detailed proof is given for each uniqueness statement.The reader is expected to learn various fundamental ideas and techniques in mathematical analysis for partial differential equations by establishing uniqueness. No prerequisite other than simple calculus and linear algebra is necessary. For the reader's convenience, a list of basic terminology is given at the end of this book.

A Basic Guide to Uniqueness Problems for Evolutionary Differential Equations

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Publisher : Springer Nature
ISBN 13 : 303134796X
Total Pages : 163 pages
Book Rating : 4.0/5 (313 download)

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Book Synopsis A Basic Guide to Uniqueness Problems for Evolutionary Differential Equations by : Mi-Ho Giga

Download or read book A Basic Guide to Uniqueness Problems for Evolutionary Differential Equations written by Mi-Ho Giga and published by Springer Nature. This book was released on 2023-10-16 with total page 163 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book addresses the issue of uniqueness of a solution to a problem – a very important topic in science and technology, particularly in the field of partial differential equations, where uniqueness guarantees that certain partial differential equations are sufficient to model a given phenomenon. This book is intended to be a short introduction to uniqueness questions for initial value problems. One often weakens the notion of a solution to include non-differentiable solutions. Such a solution is called a weak solution. It is easier to find a weak solution, but it is more difficult to establish its uniqueness. This book examines three very fundamental equations: ordinary differential equations, scalar conservation laws, and Hamilton-Jacobi equations. Starting from the standard Gronwall inequality, this book discusses less regular ordinary differential equations. It includes an introduction of advanced topics like the theory of maximal monotone operators as well as what is called DiPerna-Lions theory, which is still an active research area. For conservation laws, the uniqueness of entropy solution, a special (discontinuous) weak solution is explained. For Hamilton-Jacobi equations, several uniqueness results are established for a viscosity solution, a kind of a non-differentiable weak solution. The uniqueness of discontinuous viscosity solution is also discussed. A detailed proof is given for each uniqueness statement. The reader is expected to learn various fundamental ideas and techniques in mathematical analysis for partial differential equations by establishing uniqueness. No prerequisite other than simple calculus and linear algebra is necessary. For the reader’s convenience, a list of basic terminology is given at the end of this book.

Handbook of Differential Equations: Evolutionary Equations

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Publisher : Elsevier
ISBN 13 : 9780080932590
Total Pages : 534 pages
Book Rating : 4.9/5 (325 download)

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Book Synopsis Handbook of Differential Equations: Evolutionary Equations by : C.M. Dafermos

Download or read book Handbook of Differential Equations: Evolutionary Equations written by C.M. Dafermos and published by Elsevier. This book was released on 2009-04-29 with total page 534 pages. Available in PDF, EPUB and Kindle. Book excerpt: Handbook of Differential Equations: Evolutionary Equations is the last text of a five-volume reference in mathematics and methodology. This volume follows the format set by the preceding volumes, presenting numerous contributions that reflect the nature of the area of evolutionary partial differential equations. The book is comprised of five chapters that feature the following: • A thorough discussion of the shallow-equations theory, which is used as a model for water waves in rivers, lakes and oceans. It covers the issues of modeling, analysis and applications. • Evaluation of the singular limits of reaction-diffusion systems, where the reaction is fast compared to the other processes; and applications that range from the theory of the evolution of certain biological processes to the phenomena of Turing and cross-diffusion instability • Detailed discussion of numerous problems arising from nonlinear optics, at the high-frequency and high-intensity regime • Geometric and diffractive optics, including wave interactions • Presentation of the issues of existence, blow-up and asymptotic stability of solutions, from the equations of solutions to the equations of linear and non-linear thermoelasticity • Answers to questions about unique space, such as continuation and backward uniqueness for linear second-order parabolic equations. Research mathematicians, mathematics lecturers and instructors, and academic students will find this book invaluable. - Review of new results in the area - Continuation of previous volumes in the handbook series covering evolutionary PDEs - New content coverage of DE applications

A Concise Guide To Semigroups And Evolution Equations

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Publisher : World Scientific
ISBN 13 : 9813104570
Total Pages : 180 pages
Book Rating : 4.8/5 (131 download)

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Book Synopsis A Concise Guide To Semigroups And Evolution Equations by : Belleni-morante Aldo

Download or read book A Concise Guide To Semigroups And Evolution Equations written by Belleni-morante Aldo and published by World Scientific. This book was released on 1994-05-18 with total page 180 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a simple and concise introduction to the theory of semigroups and evolution equations, both in the linear and in the semilinear case. The subject is presented by a discussion of two standard boundary value problems (from particle transport theory and from population theory), and by showing how such problems can be rewritten as evolution problems in suitable Banach spaces.Each section of the book is completed by some notes, where the relevant notions of functional analysis are explained. Some other definitions and theorems of functional analysis are discussed in the Appendices (so that the only prerequisites to read the book are classical differential and integral calculus).

Evolutionary Equations

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Publisher : Springer Nature
ISBN 13 : 3030893979
Total Pages : 321 pages
Book Rating : 4.0/5 (38 download)

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Book Synopsis Evolutionary Equations by : Christian Seifert

Download or read book Evolutionary Equations written by Christian Seifert and published by Springer Nature. This book was released on 2022 with total page 321 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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. Nevertheless, key properties of solutions can be recovered in an elegant manner. Moreover, the strength of this method is demonstrated by a large variety of examples, showing the applicability of the approach of evolutionary equations in various fields. Additionally, a quantitative theory for evolutionary equations is developed. The text is self-contained, providing an excellent source for a first study on evolutionary equations and a decent guide to the available literature on this subject, thus bridging the gap to state-of-the-art mathematical research.

Numerical Methods for Evolutionary Differential Equations

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Publisher : SIAM
ISBN 13 : 0898718910
Total Pages : 404 pages
Book Rating : 4.8/5 (987 download)

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Book Synopsis Numerical Methods for Evolutionary Differential Equations by : Uri M. Ascher

Download or read book Numerical Methods for Evolutionary Differential Equations written by Uri M. Ascher and published by SIAM. This book was released on 2008-01-01 with total page 404 pages. Available in PDF, EPUB and Kindle. Book excerpt: Methods for the numerical simulation of dynamic mathematical models have been the focus of intensive research for well over 60 years, and the demand for better and more efficient methods has grown as the range of applications has increased. Mathematical models involving evolutionary partial differential equations (PDEs) as well as ordinary differential equations (ODEs) arise in diverse applications such as fluid flow, image processing and computer vision, physics-based animation, mechanical systems, relativity, earth sciences, and mathematical finance. This textbook develops, analyzes, and applies numerical methods for evolutionary, or time-dependent, differential problems. Both PDEs and ODEs are discussed from a unified viewpoint. The author emphasizes finite difference and finite volume methods, specifically their principled derivation, stability, accuracy, efficient implementation, and practical performance in various fields of science and engineering. Smooth and nonsmooth solutions for hyperbolic PDEs, parabolic-type PDEs, and initial value ODEs are treated, and a practical introduction to geometric integration methods is included as well. Audience: suitable for researchers and graduate students from a variety of fields including computer science, applied mathematics, physics, earth and ocean sciences, and various engineering disciplines. Researchers who simulate processes that are modeled by evolutionary differential equations will find material on the principles underlying the appropriate method to use and the pitfalls that accompany each method.

Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations

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Publisher : World Scientific Publishing Company
ISBN 13 : 9813104619
Total Pages : 324 pages
Book Rating : 4.8/5 (131 download)

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Book Synopsis Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations by : R P Agarwal

Download or read book Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations written by R P Agarwal and published by World Scientific Publishing Company. This book was released on 1993-03-31 with total page 324 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph aims to fill a void by making available a source book which first systematically describes all the available uniqueness and nonuniqueness criteria for ordinary differential equations, and compares and contrasts the merits of these criteria, and second, discusses open problems and offers some directions towards possible solutions. Contents:First Order Differential EquationsFirst Order Differential SystemsHigher Order Differential EquationsDifferential Equations in Abstract SpacesComplex Differential EquationsFunctional Differential EquationsImpulsive Differential EquationsDifferential Equations with HysteresisGeneralized Differential Equations Readership: Applied mathematicians, mathematicians and mathematical physicists.

A Concise Guide to Semigroups and Evolution Equations

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Publisher : World Scientific
ISBN 13 : 9789810212940
Total Pages : 188 pages
Book Rating : 4.2/5 (129 download)

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Book Synopsis A Concise Guide to Semigroups and Evolution Equations by : Aldo Belleni-Morante

Download or read book A Concise Guide to Semigroups and Evolution Equations written by Aldo Belleni-Morante and published by World Scientific. This book was released on 1994 with total page 188 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a simple and concise introduction to the theory of semigroups and evolution equations, both in the linear and in the semilinear case. The subject is presented by a discussion of two standard boundary value problems (from particle transport theory and from population theory), and by showing how such problems can be rewritten as evolution problems in suitable Banach spaces.Each section of the book is completed by some notes, where the relevant notions of functional analysis are explained. Some other definitions and theorems of functional analysis are discussed in the Appendices (so that the only prerequisites to read the book are classical differential and integral calculus).

Evolution PDEs with Nonstandard Growth Conditions

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Publisher : Springer
ISBN 13 : 9462391122
Total Pages : 409 pages
Book Rating : 4.4/5 (623 download)

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Book Synopsis Evolution PDEs with Nonstandard Growth Conditions by : Stanislav Antontsev

Download or read book Evolution PDEs with Nonstandard Growth Conditions written by Stanislav Antontsev and published by Springer. This book was released on 2015-04-01 with total page 409 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph offers the reader a treatment of the theory of evolution PDEs with nonstandard growth conditions. This class includes parabolic and hyperbolic equations with variable or anisotropic nonlinear structure. We develop methods for the study of such equations and present a detailed account of recent results. An overview of other approaches to the study of PDEs of this kind is provided. The presentation is focused on the issues of existence and uniqueness of solutions in appropriate function spaces and on the study of the specific qualitative properties of solutions, such as localization in space and time, extinction in a finite time and blow-up, or nonexistence of global in time solutions. Special attention is paid to the study of the properties intrinsic to solutions of equations with nonstandard growth.

Guide To Mathematical Methods For Physicists, A: Advanced Topics And Applications

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Publisher : World Scientific
ISBN 13 : 1786345501
Total Pages : 308 pages
Book Rating : 4.7/5 (863 download)

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Book Synopsis Guide To Mathematical Methods For Physicists, A: Advanced Topics And Applications by : Petrini Michela

Download or read book Guide To Mathematical Methods For Physicists, A: Advanced Topics And Applications written by Petrini Michela and published by World Scientific. This book was released on 2018-08-29 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a self-contained and rigorous presentation of the main mathematical tools needed to approach many courses at the last year of undergraduate in Physics and MSc programs, from Electromagnetism to Quantum Mechanics. It complements A Guide to Mathematical Methods for Physicists with advanced topics and physical applications. The different arguments are organised in three main sections: Complex Analysis, Differential Equations and Hilbert Spaces, covering most of the standard mathematical method tools in modern physics.One of the purposes of the book is to show how seemingly different mathematical tools like, for instance, Fourier transforms, eigenvalue problems, special functions and so on, are all deeply interconnected. It contains a large number of examples, problems and detailed solutions, emphasising the main purpose of relating concrete physical examples with more formal mathematical aspects. remove

A Stability Technique for Evolution Partial Differential Equations

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Publisher : Springer Science & Business Media
ISBN 13 : 1461220505
Total Pages : 388 pages
Book Rating : 4.4/5 (612 download)

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Book Synopsis A Stability Technique for Evolution Partial Differential Equations by : Victor A. Galaktionov

Download or read book A Stability Technique for Evolution Partial Differential Equations written by Victor A. Galaktionov and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 388 pages. Available in PDF, EPUB and Kindle. Book excerpt: * Introduces a state-of-the-art method for the study of the asymptotic behavior of solutions to evolution partial differential equations. * Written by established mathematicians at the forefront of their field, this blend of delicate analysis and broad application is ideal for a course or seminar in asymptotic analysis and nonlinear PDEs. * Well-organized text with detailed index and bibliography, suitable as a course text or reference volume.

Handbook of differential equations

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Publisher :
ISBN 13 : 9780444517432
Total Pages : 616 pages
Book Rating : 4.5/5 (174 download)

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Book Synopsis Handbook of differential equations by : M. Chipot

Download or read book Handbook of differential equations written by M. Chipot and published by . This book was released on 2006 with total page 616 pages. Available in PDF, EPUB and Kindle. Book excerpt: This handbook is volume III in a series devoted to stationary partial differential quations. Similarly as volumes I and II, it is a collection of self contained state-of-the-art surveys written by well known experts in the field. The topics covered by this handbook include singular and higher order equations, problems near critically, problems with anisotropic nonlinearities, dam problem, T-convergence and Schauder-type estimates. These surveys will be useful for both beginners and experts and speed up the progress of corresponding (rapidly developing and fascinating) areas of mathematics. Key features: - Written by well-known experts in the field - Self-contained volume in series covering one of the most rapid developing topics in mathematics - Written by well-known experts in the field - Self-contained volume in series covering one of the most rapid developing topics in mathematics.

Basic Theory Of Fractional Differential Equations (Third Edition)

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Publisher : World Scientific
ISBN 13 : 9811271704
Total Pages : 516 pages
Book Rating : 4.8/5 (112 download)

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Book Synopsis Basic Theory Of Fractional Differential Equations (Third Edition) by : Yong Zhou

Download or read book Basic Theory Of Fractional Differential Equations (Third Edition) written by Yong Zhou and published by World Scientific. This book was released on 2023-10-06 with total page 516 pages. Available in PDF, EPUB and Kindle. Book excerpt: This accessible monograph is devoted to a rapidly developing area on the research of qualitative theory of fractional ordinary differential equations and evolution equations. It is self-contained and unified in presentation, and provides the readers the necessary background material required to go further into the subject and explore the rich research literature. The tools used include many classical and modern nonlinear analysis methods such as fixed point theory, measure of noncompactness method, topological degree method, Picard operators technique, critical point theory and semigroups theory. This book is based on the research work done so far by the author and other experts, and contains comprehensive up-to-date materials on the topic.In this third edition, four new topics have been added: Hilfer fractional evolution equations and infinite interval problems, oscillations and nonoscillations, fractional Hamiltonian systems, fractional Rayleigh-Stokes equations, and wave equations. The bibliography has also been updated and expanded.This book is useful to researchers, graduate or PhD students dealing with fractional calculus and applied analysis, differential equations, and related areas of research.

Evolution Equations, Semigroups and Functional Analysis

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Publisher : Birkhäuser
ISBN 13 : 3034882211
Total Pages : 404 pages
Book Rating : 4.0/5 (348 download)

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Book Synopsis Evolution Equations, Semigroups and Functional Analysis by : Alfredo Lorenzi

Download or read book Evolution Equations, Semigroups and Functional Analysis written by Alfredo Lorenzi and published by Birkhäuser. This book was released on 2012-12-06 with total page 404 pages. Available in PDF, EPUB and Kindle. Book excerpt: Brunello Terreni (1953-2000) was a researcher and teacher with vision and dedication. The present volume is dedicated to the memory of Brunello Terreni. His mathematical interests are reflected in 20 expository articles written by distinguished mathematicians. The unifying theme of the articles is "evolution equations and functional analysis", which is presented in various and diverse forms: parabolic equations, semigroups, stochastic evolution, optimal control, existence, uniqueness and regularity of solutions, inverse problems as well as applications. Contributors: P. Acquistapace, V. Barbu, A. Briani, L. Boccardo, P. Colli Franzone, G. Da Prato, D. Donatelli, A. Favini, M. Fuhrmann, M. Grasselli, R. Illner, H. Koch, R. Labbas, H. Lange, I. Lasiecka, A. Lorenzi, A. Lunardi, P. Marcati, R. Nagel, G. Nickel, V. Pata, M. M. Porzio, B. Ruf, G. Savaré, R. Schnaubelt, E. Sinestrari, H. Tanabe, H. Teismann, E. Terraneo, R. Triggiani, A. Yagi.

Inverse Problems and Nonlinear Evolution Equations

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Publisher : Walter de Gruyter
ISBN 13 : 3110258617
Total Pages : 354 pages
Book Rating : 4.1/5 (12 download)

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Book Synopsis Inverse Problems and Nonlinear Evolution Equations by : Alexander L. Sakhnovich

Download or read book Inverse Problems and Nonlinear Evolution Equations written by Alexander L. Sakhnovich and published by Walter de Gruyter. This book was released on 2013-07-31 with total page 354 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is based on the method of operator identities and related theory of S-nodes, both developed by Lev Sakhnovich. The notion of the transfer matrix function generated by the S-node plays an essential role. The authors present fundamental solutions of various important systems of differential equations using the transfer matrix function, that is, either directly in the form of the transfer matrix function or via the representation in this form of the corresponding Darboux matrix, when Bäcklund–Darboux transformations and explicit solutions are considered. The transfer matrix function representation of the fundamental solution yields solution of an inverse problem, namely, the problem to recover system from its Weyl function. Weyl theories of selfadjoint and skew-selfadjoint Dirac systems, related canonical systems, discrete Dirac systems, system auxiliary to the N-wave equation and a system rationally depending on the spectral parameter are obtained in this way. The results on direct and inverse problems are applied in turn to the study of the initial-boundary value problems for integrable (nonlinear) wave equations via inverse spectral transformation method. Evolution of the Weyl function and solution of the initial-boundary value problem in a semi-strip are derived for many important nonlinear equations. Some uniqueness and global existence results are also proved in detail using evolution formulas. The reading of the book requires only some basic knowledge of linear algebra, calculus and operator theory from the standard university courses.

Surface Evolution Equations

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Publisher : Springer Science & Business Media
ISBN 13 : 3764373911
Total Pages : 264 pages
Book Rating : 4.7/5 (643 download)

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Book Synopsis Surface Evolution Equations by : Yoshikazu Giga

Download or read book Surface Evolution Equations written by Yoshikazu Giga and published by Springer Science & Business Media. This book was released on 2006-03-30 with total page 264 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a self-contained introduction to the analytic foundation of a level set approach for various surface evolution equations including curvature flow equations. These equations are important in many applications, such as material sciences, image processing and differential geometry. The goal is to introduce a generalized notion of solutions allowing singularities, and to solve the initial-value problem globally-in-time in a generalized sense. Various equivalent definitions of solutions are studied. Several new results on equivalence are also presented. Moreover, structures of level set equations are studied in detail. Further, a rather complete introduction to the theory of viscosity solutions is contained, which is a key tool for the level set approach. Although most of the results in this book are more or less known, they are scattered in several references, sometimes without proofs. This book presents these results in a synthetic way with full proofs. The intended audience are graduate students and researchers in various disciplines who would like to know the applicability and detail of the theory as well as its flavour. No familiarity with differential geometry or the theory of viscosity solutions is required. Only prerequisites are calculus, linear algebra and some basic knowledge about semicontinuous functions.

Existence and Uniqueness Theorem for Differential Equations

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Publisher :
ISBN 13 : 9780335011230
Total Pages : 42 pages
Book Rating : 4.0/5 (112 download)

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Book Synopsis Existence and Uniqueness Theorem for Differential Equations by :

Download or read book Existence and Uniqueness Theorem for Differential Equations written by and published by . This book was released on 1972 with total page 42 pages. Available in PDF, EPUB and Kindle. Book excerpt: