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Author: Ilpo Laine
Publisher: Walter de Gruyter
ISBN: 3110863146
Category : Mathematics
Languages : en
Pages : 353
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Book Description
The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level. The series de Gruyter Studies in Mathematics was founded ca. 30 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics. While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community. Please submit any book proposals to Niels Jacob.
Author: Ilpo Laine
Publisher: Walter de Gruyter
ISBN: 3110863146
Category : Mathematics
Languages : en
Pages : 353
Get Book Here
Book Description
The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level. The series de Gruyter Studies in Mathematics was founded ca. 30 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics. While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community. Please submit any book proposals to Niels Jacob.
Author: Katsuya Ishizaki
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Author: Chengfa Wu
Publisher: Open Dissertation Press
ISBN: 9781361348758
Category :
Languages : en
Pages :
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Book Description
This dissertation, "Meromorphic Solutions of Complex Differential Equations" by Chengfa, Wu, 吳成發, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: The objective of this thesis is to study meromorphic solutions of complex algebraic ordinary differential equations (ODEs). The thesis consists of two main themes. One of them is to find explicitly all meromorphic solutions of certain class of complex algebraic ODEs. Since constructing explicit solutions of complex ODEs in general is very difficult, the other theme (motivated by the classical conjecture proposed by Hayman in 1996) is to establish estimations on the growth of meromorphic solutions in terms of Nevanlinna characteristic function. The tools from complex analysis that will be used have been collected in Chapter 1. Chapter 2 is devoted to introducing a method, which was first used by Eremenko and later refined by Conte and Ng, to give a classification of some complex algebraic autonomous ODEs. Under certain assumptions, based on local singularity analysis and Nevanlinna theory, this method shows that all meromorphic solutions of these ODEs if exist, must belong to 'class W', which consists of elliptic functions and their degenerations. Combined with knowledge from function theory, as shown by Demina and Kudryashov, it further allows us to find all of them explicitly and the details of the method will be illustrated by constructing new real meromorphic solutions of the stationary case of cubic-quintic Swift-Hohenberg equation. In Chapter 3, the same method is used to construct on R DEGREESn, n >= 2 some explicit Bryant solitons and on R DEGREESn\{0}, n >= 2 some Ricci solitons, and one of them turns out to be a new Ricci soliton on R DEGREES5\{0}. In addition, the completeness of corresponding metrics on the Ricci solitons that we have constructed are also discussed. In 1996, Hayman conjectured an upper bound on the growth, in terms of Nevanlinna characteristic function, of meromorphic solutions of complex algebraic ODEs. Related work in the literature towards this so-called classical conjecture is first reviewed in Chapter 4. The classical conjecture for three types of second order complex algebraic ODEs will then be verified by either giving a classification of the meromorphic solutions or obtaining them explicitly in Chapter 4. As the classical conjecture seems to be out of reach at present, we proposed in Chapter 5 to study a particular class of complex algebraic ODEs which can be factorized into certain form. On one hand, for these factorizable ODEs, it has been proven for the generic case that all their meromorphic solutions must be elliptic functions or their degenerations. On the other hand, the second order factorizable ODEs have been carefully studied so that their meromorphic solutions have been obtained explicitly except one case. This will allow the classical conjecture for most of the second order factorizable ODEs to be verified by employing Nevanlinna theory and certain qualitative results from complex differential equations. Finally, the classical conjecture has been shown to be sharp in certain cases. DOI: 10.5353/th_b5317034 Subjects: Differential equations
Author: 吳成發
Publisher:
ISBN:
Category : Differential equations
Languages : en
Pages : 224
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Author: Janne Heittokangas
Publisher:
ISBN:
Category : Differential equations
Languages : en
Pages : 62
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Author: Kwok-Kin Wong
Publisher: Open Dissertation Press
ISBN: 9781361281901
Category :
Languages : en
Pages :
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This dissertation, "Exact Meromorphic Solutions of Complex Algebraic Differential Equations" by Kwok-kin, Wong, 黃國堅, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: For any given complex algebraic ordinary differential equation (ODE), one major task of both pure and applied mathematicians is to find explicit meromorphic solutions due to their extensive applications in science. In 2010, Conte and Ng in [12] proposed a new technique for solving complex algebraic ODEs. The method consists of an idea due to Eremenko in [20] and the subequation method of Conte and Musette, which was first proposed in [9]. Eremenko's idea is to make use of the Nevanlinna theory to analyze the value distribution and growth rate of the solutions, from which one would be able to show that in some cases, all the meromorphic solutions of the studied differential equation are in a class of functions called "class W," which consists of elliptic functions and their degenerates. The establishment of solutions is then achieved by the subequation method. The main idea is to build subequations which have solutions that also satisfy the original differential equation, hoping that the subequations will be easier to solve. As in [12], the technique has been proven to be very successful in obtaining explicit particular meromorphic solutions as well as giving complete classification of meromorphic solutions. In this thesis, the necessary theoretical background, including the Nevanlinna theory and the subequation method, will be developed. The technique will then be applied to obtain all meromorphic stationary wave solutions of the real cubic Swift-Hohenberg equation (RCSH). This last part is joint work with Conte and Ng and will appear in Studies in Applied Mathematics [13]. RCSH is important in several studies in physics and engineering problems. For instance, RCSH is used as modeling equation for Rayleigh- B?nard convection in hydrodynamics [43] as well as in pattern formation [16]. Among the explicit stationary wave solutions obtained by the technique used in this thesis, one of them appears to be new and could be written down as a rational function composite with Weierstrass elliptic function. DOI: 10.5353/th_b4833021 Subjects: Differential-algebraic equations
Author: Kai Liu
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 3110560569
Category : Mathematics
Languages : en
Pages : 302
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Book Description
This book presents developments and new results on complex differential-difference equations, an area with important and interesting applications, which also gathers increasing attention. Key problems, methods, and results related to complex differential-difference equations are collected to offer an up-to-date overview of the field.
Author: Valerii I. Gromak
Publisher: Walter de Gruyter
ISBN: 3110198096
Category : Mathematics
Languages : en
Pages : 313
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Book Description
This book is the first comprehensive treatment of Painlevé differential equations in the complex plane. Starting with a rigorous presentation for the meromorphic nature of their solutions, the Nevanlinna theory will be applied to offer a detailed exposition of growth aspects and value distribution of Painlevé transcendents. The subsequent main part of the book is devoted to topics of classical background such as representations and expansions of solutions, solutions of special type like rational and special transcendental solutions, Bäcklund transformations and higher order analogues, treated separately for each of these six equations. The final chapter offers a short overview of applications of Painlevé equations, including an introduction to their discrete counterparts. Due to the present important role of Painlevé equations in physical applications, this monograph should be of interest to researchers in both mathematics and physics and to graduate students interested in mathematical physics and the theory of differential equations.
Author: Kwok-kin Wong
Publisher:
ISBN:
Category : Differential-algebraic equations
Languages : en
Pages : 128
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Author: Kwok-kin Wong
Publisher:
ISBN:
Category : Differential-algebraic equations
Languages : en
Pages : 0
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