Download or read book Virtual Turning Points and Bifurcation of Stokes Curves for Higher Order Ordinary Differential Equations written by Takashi Aoki and published by . This book was released on 2004 with total page 7 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: "For a higher order linear ordinary differential operator P, its Stokes curve bifurcates in general when it hits another turning point of P. This phenomenon is most neatly understandable by taking into account Stokes curves emanating from virtual turning points, together with those from ordinary turning points. This understanding of the bifurcation of a Stokes curve plays an important role in resolving a paradox recently found in the Noumi-Yamada system, a system of linear differential equations associated with the fourth Painlevé equation."
Download or read book Virtual Turning Points written by Naofumi Honda and published by Springer. This book was released on 2015-07-07 with total page 133 pages. Available in PDF, EPUB and Kindle. Book excerpt: The discovery of a virtual turning point truly is a breakthrough in WKB analysis of higher order differential equations. This monograph expounds the core part of its theory together with its application to the analysis of higher order Painlevé equations of the Noumi–Yamada type and to the analysis of non-adiabatic transition probability problems in three levels. As M.V. Fedoryuk once lamented, global asymptotic analysis of higher order differential equations had been thought to be impossible to construct. In 1982, however, H.L. Berk, W.M. Nevins, and K.V. Roberts published a remarkable paper in the Journal of Mathematical Physics indicating that the traditional Stokes geometry cannot globally describe the Stokes phenomena of solutions of higher order equations; a new Stokes curve is necessary.
Download or read book Algebraic Analysis of Differential Equations written by T. Aoki and published by Springer Science & Business Media. This book was released on 2009-03-15 with total page 349 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains 23 articles on algebraic analysis of differential equations and related topics, most of which were presented as papers at the conference "Algebraic Analysis of Differential Equations – from Microlocal Analysis to Exponential Asymptotics" at Kyoto University in 2005. This volume is dedicated to Professor Takahiro Kawai, who is one of the creators of microlocal analysis and who introduced the technique of microlocal analysis into exponential asymptotics.
Download or read book On a Complete Description of the Stokes Geometry for Higher Order Ordinary Differential Equations with a Large Parameter Via Integral Representations written by Takashi Aoki and published by . This book was released on 1999 with total page 6 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: "We make a detailed study of particular third order linear ordinary differential equations with a large parameter of Laplace type through integral representations of their solutions, hoping that it will give us a better understanding of the redundant new turning points. In the particular examples we discuss, the redundancy is tied up with a simple function-theoretic property of a real one-dimensional curve determined by a Hamiltonian system associated with the third order differential equations."
Download or read book Algebraic Analysis of Singular Perturbation Theory written by Takahiro Kawai and published by American Mathematical Soc.. This book was released on 2005 with total page 148 pages. Available in PDF, EPUB and Kindle. Book excerpt: The topic of this book is the study of singular perturbations of ordinary differential equations, i.e., perturbations that represent solutions as asymptotic series rather than as analytic functions in a perturbation parameter. The main method used is the so-called WKB (Wentzel-Kramers-Brillouin) method, originally invented for the study of quantum-mechanical systems. The authors describe in detail the WKB method and its applications to the study of monodromy problems for Fuchsian differential equations and to the analysis of Painleve functions. This volume is suitable for graduate students and researchers interested in differential equations and special functions.
Download or read book On the Complete Description of the Stokes Geometry for the First Painleve Hierarchy written by and published by . This book was released on 2004 with total page 28 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: "One of the central issues of this article is the introduction of the notion of virtual turning points for higher order Painlevé equations, and two of the authors (Kawai and Takei), together with T. Aoki, fondly remember the stimulating and comfortable conference (Algebraic analysis of singular perturbations, 1991), which Professor Boutet de Monvel, together with Professor M. Sato, organized, and where the notion of a virtual turning point for linear ordinary differential equations was first made public (under the modest name 'a new turning point'). The notion of virtual turning points is one of the most important gifts to the exact WKB analysis from microlocal analysis, and hence we believe this article to be most appropriate to dedicate to Professor Boutet de Monvel, who has made substantial contributions to the development of microlocal analysis and asymptotic analysis."
Download or read book Mathematical Reviews written by and published by . This book was released on 2005 with total page 1884 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book On the Stokes Geometry of Higher Order Painleve Equations written by Takahiro Kawai and published by . This book was released on 2004 with total page 67 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: "We show several basic properties concerning the relation between the Stokes geometry (i.e., configuration of Stokes curves and turning points) of a higher order Painlevé equation with a large parameter and the Stokes geometry of (one of) the underlying Lax pair. The higher-order Painlevé equation with a large parameter to be considered in this paper is one of the members of P[subscript J]-hierarchy with J=I, II-1 or II-2, which are concretely given in Section 1. Since we deal with higher order equations, the Stokes curves may cross; some anomaly called the Nishikawa phenomenon may occur at the crossing point, and in this paper we analyze the mechanism why and how the Nishikawa phenomenon occurs. Several examples of Stokes geometry are given in Section 5 to visualize the core part of our results."
Download or read book On a Complete Description of the Stokes Geometry for Higher Order Ordinary Differential Equations Wit a Large Parameter Via Integral Representations written by Takashi Aoki and published by . This book was released on 1999 with total page 6 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Exploring ODEs written by Lloyd N. Trefethen and published by SIAM. This book was released on 2017-12-21 with total page 343 pages. Available in PDF, EPUB and Kindle. Book excerpt: Exploring ODEs is a textbook of ordinary differential equations for advanced undergraduates, graduate students, scientists, and engineers. It is unlike other books in this field in that each concept is illustrated numerically via a few lines of Chebfun code. There are about 400 computer-generated figures in all, and Appendix B presents 100 more examples as templates for further exploration.?
Download or read book Nonlinear Dynamics and Chaos written by Steven H. Strogatz and published by CRC Press. This book was released on 2018-05-04 with total page 532 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook is aimed at newcomers to nonlinear dynamics and chaos, especially students taking a first course in the subject. The presentation stresses analytical methods, concrete examples, and geometric intuition. The theory is developed systematically, starting with first-order differential equations and their bifurcations, followed by phase plane analysis, limit cycles and their bifurcations, and culminating with the Lorenz equations, chaos, iterated maps, period doubling, renormalization, fractals, and strange attractors.
Download or read book Partial Differential Equations written by Walter A. Strauss and published by John Wiley & Sons. This book was released on 2007-12-21 with total page 467 pages. Available in PDF, EPUB and Kindle. Book excerpt: Our understanding of the fundamental processes of the natural world is based to a large extent on partial differential equations (PDEs). The second edition of Partial Differential Equations provides an introduction to the basic properties of PDEs and the ideas and techniques that have proven useful in analyzing them. It provides the student a broad perspective on the subject, illustrates the incredibly rich variety of phenomena encompassed by it, and imparts a working knowledge of the most important techniques of analysis of the solutions of the equations. In this book mathematical jargon is minimized. Our focus is on the three most classical PDEs: the wave, heat and Laplace equations. Advanced concepts are introduced frequently but with the least possible technicalities. The book is flexibly designed for juniors, seniors or beginning graduate students in science, engineering or mathematics.
Download or read book Ordinary Differential Equations with Applications written by Carmen Chicone and published by Springer Science & Business Media. This book was released on 2008-04-08 with total page 569 pages. Available in PDF, EPUB and Kindle. Book excerpt: Based on a one-year course taught by the author to graduates at the University of Missouri, this book provides a student-friendly account of some of the standard topics encountered in an introductory course of ordinary differential equations. In a second semester, these ideas can be expanded by introducing more advanced concepts and applications. A central theme in the book is the use of Implicit Function Theorem, while the latter sections of the book introduce the basic ideas of perturbation theory as applications of this Theorem. The book also contains material differing from standard treatments, for example, the Fiber Contraction Principle is used to prove the smoothness of functions that are obtained as fixed points of contractions. The ideas introduced in this section can be extended to infinite dimensions.
Download or read book Lectures on Analytic Differential Equations written by I︠U︡. S. Ilʹi︠a︡shenko and published by American Mathematical Soc.. This book was released on 2008 with total page 641 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book combines the features of a graduate-level textbook with those of a research monograph and survey of the recent results on analysis and geometry of differential equations in the real and complex domain. As a graduate textbook, it includes self-contained, sometimes considerably simplified demonstrations of several fundamental results, which previously appeared only in journal publications (desingularization of planar analytic vector fields, existence of analytic separatrices, positive and negative results on the Riemann-Hilbert problem, Ecalle-Voronin and Martinet-Ramis moduli, solution of the Poincare problem on the degree of an algebraic separatrix, etc.). As a research monograph, it explores in a systematic way the algebraic decidability of local classification problems, rigidity of holomorphic foliations, etc. Each section ends with a collection of problems, partly intended to help the reader to gain understanding and experience with the material, partly drafting demonstrations of the mor The exposition of the book is mostly geometric, though the algebraic side of the constructions is also prominently featured. on several occasions the reader is introduced to adjacent areas, such as intersection theory for divisors on the projective plane or geometric theory of holomorphic vector bundles with meromorphic connections. The book provides the reader with the principal tools of the modern theory of analytic differential equations and intends to serve as a standard source for references in this area.
Download or read book Elementary Differential Equations and Boundary Value Problems written by William E. Boyce and published by John Wiley & Sons. This book was released on 2017-08-21 with total page 624 pages. Available in PDF, EPUB and Kindle. Book excerpt: Elementary Differential Equations and Boundary Value Problems 11e, like its predecessors, is written from the viewpoint of the applied mathematician, whose interest in differential equations may sometimes be quite theoretical, sometimes intensely practical, and often somewhere in between. The authors have sought to combine a sound and accurate (but not abstract) exposition of the elementary theory of differential equations with considerable material on methods of solution, analysis, and approximation that have proved useful in a wide variety of applications. While the general structure of the book remains unchanged, some notable changes have been made to improve the clarity and readability of basic material about differential equations and their applications. In addition to expanded explanations, the 11th edition includes new problems, updated figures and examples to help motivate students. The program is primarily intended for undergraduate students of mathematics, science, or engineering, who typically take a course on differential equations during their first or second year of study. The main prerequisite for engaging with the program is a working knowledge of calculus, gained from a normal two or three semester course sequence or its equivalent. Some familiarity with matrices will also be helpful in the chapters on systems of differential equations.
Download or read book Modeling Life written by Alan Garfinkel and published by Springer. This book was released on 2017-09-06 with total page 456 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book develops the mathematical tools essential for students in the life sciences to describe interacting systems and predict their behavior. From predator-prey populations in an ecosystem, to hormone regulation within the body, the natural world abounds in dynamical systems that affect us profoundly. Complex feedback relations and counter-intuitive responses are common in nature; this book develops the quantitative skills needed to explore these interactions. Differential equations are the natural mathematical tool for quantifying change, and are the driving force throughout this book. The use of Euler’s method makes nonlinear examples tractable and accessible to a broad spectrum of early-stage undergraduates, thus providing a practical alternative to the procedural approach of a traditional Calculus curriculum. Tools are developed within numerous, relevant examples, with an emphasis on the construction, evaluation, and interpretation of mathematical models throughout. Encountering these concepts in context, students learn not only quantitative techniques, but how to bridge between biological and mathematical ways of thinking. Examples range broadly, exploring the dynamics of neurons and the immune system, through to population dynamics and the Google PageRank algorithm. Each scenario relies only on an interest in the natural world; no biological expertise is assumed of student or instructor. Building on a single prerequisite of Precalculus, the book suits a two-quarter sequence for first or second year undergraduates, and meets the mathematical requirements of medical school entry. The later material provides opportunities for more advanced students in both mathematics and life sciences to revisit theoretical knowledge in a rich, real-world framework. In all cases, the focus is clear: how does the math help us understand the science?
Download or read book Elementary Differential Equations and Boundary Value Problems Binder Ready Version written by William E. Boyce and published by Wiley. This book was released on 2012-10-02 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The 10th edition of Elementary Differential Equations and Boundary Value Problems, like its predecessors, is written from the viewpoint of the applied mathematician, whose interest in differential equations may sometimes be quite theoretical, sometimes intensely practical, and often somewhere in between. The authors have sought to combine a sound and accurate exposition of the elementary theory of differential equations with considerable material on methods of solution, analysis, and approximation that have proved useful in a wide variety of applications. While the general structure of the book remains unchanged, some notable changes have been made to improve the clarity and readability of basic material about differential equations and their applications. In addition to expanded explanations, the 10th edition includes new problems, updated figures and examples to help motivate students. The book is written primarily for undergraduate students of mathematics, science, or engineering, who typically take a course on differential equations during their first or second year of study. WileyPLUS sold separately from text.
