Finite Difference Methods For Nonlinear Evolution Equations

Author : Zhi-Zhong Sun
Publisher : Walter de Gruyter GmbH & Co KG
ISBN 13 : 3110796015
Total Pages : 432 pages
Book Rating : 4.18/5 ( download)

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Download or read book Finite Difference Methods for Nonlinear Evolution Equations written by Zhi-Zhong Sun and published by Walter de Gruyter GmbH & Co KG. This book was released on 2023-05-08 with total page 432 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introduces recent research results of finite difference methods including important nonlinear evolution equations in applied science. The presented difference schemes include nonlinear difference schemes and linearized difference schemes. Features widely used nonlinear evolution equations such as Burgers equation, regular long wave equation, Schrodinger equation and more. Each PDE model includes details on efficiency, stability, and convergence.

Author : Zhi-Zhong Sun
Publisher : Walter de Gruyter GmbH & Co KG
ISBN 13 : 3110796112
Total Pages : 499 pages
Book Rating : 4.17/5 ( download)

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Download or read book Finite Difference Methods for Nonlinear Evolution Equations written by Zhi-Zhong Sun and published by Walter de Gruyter GmbH & Co KG. This book was released on 2023-05-08 with total page 499 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nonlinear evolution equations are widely used to describe nonlinear phenomena in natural and social sciences. However, they are usually quite difficult to solve in most instances. This book introduces the finite difference methods for solving nonlinear evolution equations. The main numerical analysis tool is the energy method. This book covers the difference methods for the initial-boundary value problems of twelve nonlinear partial differential equations. They are Fisher equation, Burgers' equation, regularized long-wave equation, Korteweg-de Vries equation, Camassa-Holm equation, Schrödinger equation, Kuramoto-Tsuzuki equation, Zakharov equation, Ginzburg-Landau equation, Cahn-Hilliard equation, epitaxial growth model and phase field crystal model. This book is a monograph for the graduate students and science researchers majoring in computational mathematics and applied mathematics. It will be also useful to all researchers in related disciplines.

Author : Tristan Kremp
Publisher : Cuvillier Verlag
ISBN 13 : 3736936850
Total Pages : 240 pages
Book Rating : 4.50/5 ( download)

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Download or read book Quasi-spectral Finite Difference Methods written by Tristan Kremp and published by Cuvillier Verlag. This book was released on 2011-03-25 with total page 240 pages. Available in PDF, EPUB and Kindle. Book excerpt: The doctoral thesis „Quasi-spectral finite difference methods: Convergence analysis and application to nonlinear optical pulse propagation“ by Tristan Kremp addresses the theory and application of so-called quasi-spectral finite differences. Contrary to the common Taylor approach, these are by construction exact for trigonometric instead of algebraic polynomials. With any fixed discretization spacing, this allows for a higher accuracy, e.g., when differencing functions that have a band-pass like Fourier spectrum. In this dissertation, the convergence of such quasi-spectral finite differences is proven for the first time. It is shown that the highest possible order of convergence is the same as for the Taylor approach, i.e., it is basically identical to the total number of summands in the finite difference. This order is achieved if all frequencies, for which the quasi-spectral finite difference is exact, vanish sufficiently fast in comparison to the discretization spacing. This condition can be easily incorporated in the finite difference weights construction, which can be achieved by spectral interpolation or least-squares optimization, respectively. Employing previously unknown Haar (or Chebyshev) systems that consist of combinations of algebraic and trigonometric monomials, the equivalence of both methods of construction is proven. In a semidiscretization framework, these finite differences are, for the first time, combined with exponential split-step integrators for an efficient solution of linear or nonlinear evolution equations. It is shown that a simple modification of the common symmetric split-step integrator guarantees its second-order convergence even in the presence of general nonlinearities. An important example of such a partial differential equation is the nonlinear Schrödinger equation (NLSE). In contrast to the standard literature, the NLSE is derived here directly from Maxwell’s equations, without the common assumption that the second spatial derivative in the direction of the propagation can be neglected, and without the assumption that the multiplicative nonlinear term behaves as a constant with respect to the Fourier transformation. A practically relevant application is the propagation of wavelength division multiplexing (WDM) signals in optical fibers. Compared to other semidiscretization techniques such as finite elements, wavelet collocation and the pseudo-spectral methods (split-step Fourier method) that are mostly employed by the industry, the quasi-spectral finite differences allow, at the same accuracy, for a substantial reduction of the computation time.

Author : Reinhard Racke
Publisher : Birkhäuser
ISBN 13 : 3319218735
Total Pages : 315 pages
Book Rating : 4.31/5 ( download)

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Download or read book Lectures on Nonlinear Evolution Equations written by Reinhard Racke and published by Birkhäuser. This book was released on 2015-08-31 with total page 315 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book mainly serves as an elementary, self-contained introduction to several important aspects of the theory of global solutions to initial value problems for nonlinear evolution equations. The book employs the classical method of continuation of local solutions with the help of a priori estimates obtained for small data. The existence and uniqueness of small, smooth solutions that are defined for all values of the time parameter are investigated. Moreover, the asymptotic behaviour of the solutions is described as time tends to infinity. The methods for nonlinear wave equations are discussed in detail. Other examples include the equations of elasticity, heat equations, the equations of thermoelasticity, Schrödinger equations, Klein-Gordon equations, Maxwell equations and plate equations. To emphasize the importance of studying the conditions under which small data problems offer global solutions, some blow-up results are briefly described. Moreover, the prospects for corresponding initial boundary value problems and for open questions are provided. In this second edition, initial-boundary value problems in waveguides are additionally considered.

Author : Ronald E. Mickens
Publisher : World Scientific
ISBN 13 : 9789810241339
Total Pages : 268 pages
Book Rating : 4.3X/5 ( download)

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Download or read book Applications of Nonstandard Finite Difference Schemes written by Ronald E. Mickens and published by World Scientific. This book was released on 2000 with total page 268 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main purpose of this book is to provide a concise introduction to the methods and philosophy of constructing nonstandard finite difference schemes and illustrate how such techniques can be applied to several important problems. Chapter I gives an overview of the subject and summarizes previous work. Chapters 2 and 3 consider in detail the construction and numerical implementation of schemes for physical problems involving convection-diffusion-reaction equations, that arise in groundwater pollution and scattering of electromagnetic waves using Maxwell's equations. Chapter 4 examines certain mathematical issues related to the nonstandard discretization of competitive and cooperative models for ecology. The application chapters illustrate well the power of nonstandard methods. In particular, for the same accuracy as obtained by standard techniques, larger step sizes can be used. This volume will satisfy the needs of scientists, engineers, and mathematicians who wish to know how to construct nonstandard schemes and see how these are applied to obtain numerical solutions of the differential equations which arise in the study of nonlinear dynamical systems modeling important physical phenomena.

Author : Kazufumi Ito
Publisher : World Scientific
ISBN 13 : 9789812380265
Total Pages : 524 pages
Book Rating : 4.64/5 ( download)

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Download or read book Evolution Equations and Approximations written by Kazufumi Ito and published by World Scientific. This book was released on 2002 with total page 524 pages. Available in PDF, EPUB and Kindle. Book excerpt: Annotation Ito (North Carolina State U.) and Kappel (U. of Graz, Austria) offer a unified presentation of the general approach for well-posedness results using abstract evolution equations, drawing from and modifying the work of K. and Y. Kobayashi and S. Oharu. They also explore abstract approximation results for evolution equations. Their work is not a textbook, but they explain how instructors can use various sections, or combinations of them, as a foundation for a range of courses. Annotation copyrighted by Book News, Inc., Portland, OR

Author : Kazufumi Ito
Publisher : World Scientific
ISBN 13 : 9814488380
Total Pages : 518 pages
Book Rating : 4.89/5 ( download)

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Download or read book Evolution Equations And Approximations written by Kazufumi Ito and published by World Scientific. This book was released on 2002-05-24 with total page 518 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents an approximation theory for a general class of nonlinear evolution equations in Banach spaces and the semigroup theory, including the linear (Hille-Yosida), nonlinear (Crandall-Liggett) and time-dependent (Crandall-Pazy) theorems.The implicit finite difference method of Euler is shown to generate a sequence convergent to the unique integral solution of evolution equations of the maximal monotone type. Moreover, the Chernoff theory provides a sufficient condition for consistent and stable time integration of time-dependent nonlinear equations. The Trotter-Kato theorem and the Lie-Trotter type product formula give a mathematical framework for the convergence analysis of numerical approximations of solutions to a general class of partial differential equations. This book contains examples demonstrating the applicability of the generation as well as the approximation theory.In addition, the Kobayashi-Oharu approach of locally quasi-dissipative operators is discussed for homogeneous as well as nonhomogeneous equations. Applications to the delay differential equations, Navier-Stokes equation and scalar conservation equation are given.

Author : Ronald E Mickens
Publisher : World Scientific
ISBN 13 : 981122255X
Total Pages : 332 pages
Book Rating : 4.59/5 ( download)

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Download or read book Nonstandard Finite Difference Schemes: Methodology And Applications written by Ronald E Mickens and published by World Scientific. This book was released on 2020-11-11 with total page 332 pages. Available in PDF, EPUB and Kindle. Book excerpt: This second edition of Nonstandard Finite Difference Models of Differential Equations provides an update on the progress made in both the theory and application of the NSFD methodology during the past two and a half decades. In addition to discussing details related to the determination of the denominator functions and the nonlocal discrete representations of functions of dependent variables, we include many examples illustrating just how this should be done.Of real value to the reader is the inclusion of a chapter listing many exact difference schemes, and a chapter giving NSFD schemes from the research literature. The book emphasizes the critical roles played by the 'principle of dynamic consistency' and the use of sub-equations for the construction of valid NSFD discretizations of differential equations.

Author : Songmu Zheng
Publisher : CRC Press
ISBN 13 : 0203492226
Total Pages : 304 pages
Book Rating : 4.22/5 ( download)

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Download or read book Nonlinear Evolution Equations written by Songmu Zheng and published by CRC Press. This book was released on 2004-07-08 with total page 304 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nonlinear evolution equations arise in many fields of sciences including physics, mechanics, and material science. This book introduces some important methods for dealing with these equations and explains clearly and concisely a wide range of relevant theories and techniques. These include the semigroup method, the compactness and monotone operator

Author : J. Kral
Publisher : Springer Science & Business Media
ISBN 13 : 1461344255
Total Pages : 138 pages
Book Rating : 4.54/5 ( download)

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Download or read book Nonlinear Evolution Equations and Potential Theory written by J. Kral and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 138 pages. Available in PDF, EPUB and Kindle. Book excerpt: Preface.- Gottfried Anger: Direct and inverse problems in potential theory.- Viorel Barbu: Regularity results for sane differential equations associated with maximal monotone operators in Hilbert spaces.- Haim Brezis: Classes d'interpolation associées à un opérateur monotone et applications.- Siegfried Dnümmel: On inverse problems for k-dimensional potentials.- Jozef Ka?ur: Application of Rothe's method to nonlinear parabolic boundary value problems.- Josef Král: Potentials and removability of singularities.- Vladimir Lovicar: Theorem of Fréchet and asymptotically almost periodid solutions of.