Traditional Functional-Discrete Methods for the Problems of Mathematical Physics E-Book
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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
This book is devoted to the construction and study of approximate methods for solving mathematical physics problems in canonical domains. It focuses on obtaining weighted a priori estimates of the accuracy of these methods while also considering the influence of boundary and initial conditions. This influence is quantified by means of suitable weight functions that characterize the distance of an inner point to the boundary of the domain. New results are presented on boundary and initial effects for the finite difference method for elliptic and parabolic equations, mesh schemes for equations with fractional derivatives, and the Cayley transform method for abstract differential equations in Hilbert and Banach spaces. Due to their universality and convenient implementation, the algorithms discussed throughout can be used to solve a wide range of actual problems in science and technology. The book is intended for scientists, university teachers, and graduate and postgraduate students who specialize in the field of numerical analysis.
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Seitenzahl: 171
Veröffentlichungsjahr: 2024
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Table of Contents
Cover
Table of Contents
Title Page
Copyright Page
Preface: New Aspects of the Traditional Functional-Discrete Methods for the Problems of Mathematical Physics
Introduction
1 Elliptic Equations in Canonical Domains with the Dirichlet Condition on the Boundary or its Part
1.1. A standard finite-difference scheme for Poisson’s equation with mixed boundary conditions
1.2. A nine-point finite-difference scheme for Poisson’s equation with the Dirichlet boundary condition
1.3. A finite-difference scheme of the higher order of approximation for Poisson’s equation with the Dirichlet boundary condition
1.4. A finite-difference scheme for the equation with mixed derivatives
2 Parabolic Equations in Canonical Domains with the Dirichlet Condition on the Boundary or its Part
2.1. A standard finite-difference scheme for the one-dimensional heat equation with mixed boundary conditions
2.2. A standard finite-difference scheme for the two-dimensional heat equation with mixed boundary conditions
2.3. A standard finite-difference scheme for the two-dimensional heat equation with the Dirichlet boundary condition
3 Differential Equations with Fractional Derivatives
3.1. BVP for a differential equation with constant coefficients and a fractional derivative of order ½
3.2. BVP for a differential equation with constant coefficients and a fractional derivative of order α ∈ (0,1)
3.3. BVP for a differential equation with variable coefficients and a fractional derivative of order α ∈ (0,1)
3.4. Two-dimensional differential equation with a fractional derivative
3.5. The Goursat problem with fractional derivatives
4 The Abstract Cauchy Problem
4.1. The approximation of the operator exponential function in a Hilbert space
4.2. Inverse theorems for the operator sine and cosine functions
4.3. The approximation of the operator exponential function in a Banach space
4.4. Conclusion
5 The Cayley Transform Method for Abstract Differential Equations
5.1. Exact and approximate solutions of the BVP in a Hilbert space
5.2. Exact and approximate solutions of the BVP in a Banach space
References
Index
Other titles from ISTE in Mathematics and Statistics
End User License Agreement
List of Tables
Chapter 1
Table 1.1. Numerical calculations for Lemma 1.9
Chapter 3
Table 3.1. Numerical example for problem [3.55]
List of Illustrations
Chapter 5
Figure 5.1. To the definition of a strongly positive operator
Guide
Cover Page
Table of Contents
Title Page
Copyright Page
Preface
Introduction
Begin Reading
Index
Other titles from ISTE in Mathematics and Statistics
WILEY END USER LICENSE AGREEMENT
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Series Editor
Nikolaos Limnios
Traditional Functional-Discrete Methods for the Problems of Mathematical Physics
New Aspects
Volodymyr Makarov
Nataliya Mayko
First published 2023 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUKwww.iste.co.uk
John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USAwww.wiley.com
© ISTE Ltd 2023The rights of Volodymyr Makarov and Nataliya Mayko to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s), contributor(s) or editor(s) and do not necessarily reflect the views of ISTE Group.
Library of Congress Control Number: 2023943904
British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-78630-933-4
PrefaceNew Aspects of the Traditional Functional-Discrete Methods for the Problems of Mathematical Physics
This book is based on the authors’ latest research focusing on obtaining weighted accuracy estimates of numerical methods for solving boundary value and initial value problems. The idea of such estimates is based on Volodymyr Makarov’s observation that due to the Dirichlet boundary condition for a differential equation in a canonical domain (e.g. on an interval or in a rectangle), the accuracy of the approximate solution in the mesh nodes near the boundary of the domain is higher compared to the accuracy in the mesh nodes away from the boundary. The study commenced about 30 years ago with the finite-difference scheme for the two-dimensional elliptic equation with the generalized solution from Sobolev spaces and later continued for other types of problems: quasilinear stationary and non-stationary equations with boundary conditions, boundary value problems for equations with fractional derivatives, the Cauchy problem and boundary value problems for abstract differential equations in Hilbert and Banach spaces, etc. For brevity, to name the influence that boundary and initial conditions have on the accuracy of the approximate solution, we choose to use the wording boundary effect or initial effect. Thus, we obtain a priori accuracy weighted estimates, taking into account the boundary and initial effects. These effects are quantitatively described by means of a suitable weight function, which characterizes the distance of a point to the boundary of the domain.
To our best knowledge, there are very few publications addressing these issues. It is our hope that the present book will meet this need and thus help to inspire new generations of students, researchers and practitioners. We also sincerely hope that our approach, methods and techniques developed in the book will contribute not only to the theory of the numerical analysis but also to its applications, since awareness of the boundary and initial effects makes it possible to use a greater mesh step near the boundary of the domain. Since the finite-difference approximations and the mesh schemes proposed and studied in this book are traditional and not exotic, they can be used for solving a wide range of problems in physics, engineering, chemistry, biology, finance, etc.
The target audience of our book is graduate and postgraduate students, specialists in numerical analysis, computational and applied mathematics, and engineers. As in books like ours, the analytical and numerical components are closely intertwined, we expect the potential reader to have fluency in both univariate and multivariate analysis, familiarity with ordinary and partial differential equations, basic knowledge of functional analysis, advanced knowledge of numerical analysis, and be at ease with modern scientific computing. These mathematical prerequisites will make the text much easier to understand.
We are deeply grateful to Professor Nikolaos Limnios and Professor Dmytro Koroliuk for their suggestion to submit the manuscript, to Professor Ivan Gavrilyuk for many fruitful discussions, and to Professor Vyacheslav Ryabichev for his valuable software advice and constant professional assistance. We also express our gratitude to the team at ISTE Group for useful recommendations and careful preparation of our book for publication. We are immensely thankful to our families for everyday understanding, support and encouragement.
Volodymyr MAKAROV,
Institute of Mathematics of the National Academy of Sciences of Ukraine,
Nataliya MAYKO,
Taras Shevchenko National University of Kyiv,
Kyiv, Ukraine
July 2023
