Nonstandard Finite Difference Schemes Methodology and Applications 1111889911 99778899881111222222553355 ttpp..iinndddd 11 2233//99//2200 1111::1122 AAMM b2530 International Strategic Relations and China’s National Security: World at the Crossroads TTTThhhhiiiissss ppppaaaaggggeeee iiiinnnntttteeeennnnttttiiiioooonnnnaaaallllllllyyyy lllleeeefffftttt bbbbllllaaaannnnkkkk b2530_FM.indd 6 01-Sep-16 11:03:06 AM Nonstandard Finite Difference Schemes Methodology and Applications Ronald E Mickens Clark Atlanta University, USA World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO 1111889911 99778899881111222222553355 ttpp..iinndddd 22 2233//99//2200 1111::1122 AAMM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Names: Mickens, Ronald E., 1943– author. Title: Nonstandard finite difference schemes : methodology and applications / Ronald E. Mickens, Clark Atlanta University, USA. Description: New Jersey : World Scientific, [2021] | Includes bibliographical references and index. Identifiers: LCCN 2020044299 | ISBN 9789811222535 (hardcover) | ISBN 9789811222542 (ebook) Subjects: LCSH: Finite differences. | Nonstandard mathematical analysis. Classification: LCC QA431 .M4285 2021 | DDC 515/.625--dc23 LC record available at https://lccn.loc.gov/2020044299 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2021 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. 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For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/11891#t=suppl Printed in Singapore LLaaiiFFuunn -- 1111889911 -- NNoonnssttaannddaarrdd FFiinniittee DDiiffffeerreennccee SScchheemmeess..iinndddd 11 1122//1100//22002200 99::4477::4400 aamm September17,2020 15:29 ws-book9x6 NonstandardFiniteDifferenceSchemes: MethodologyandApplications 11891-main pagev This book is dedicated to my wife Maria, my son James Williamson, my daughter Leah Maria. v b2530 International Strategic Relations and China’s National Security: World at the Crossroads TTTThhhhiiiissss ppppaaaaggggeeee iiiinnnntttteeeennnnttttiiiioooonnnnaaaallllllllyyyy lllleeeefffftttt bbbbllllaaaannnnkkkk b2530_FM.indd 6 01-Sep-16 11:03:06 AM September17,2020 15:29 ws-book9x6 NonstandardFiniteDifferenceSchemes: MethodologyandApplications 11891-main pagevii Preface This book is an expanded second edition of my previous volume: R. E. Mickens, Nonstandard Finite Difference Models of Differential Equa- tions (World Scientific, Singapore, 1994). The purpose of the original book was to introduce a new methodology for constructing finite differ- ence schemes for numerically integrating differential equations and this set of procedures became known as nonstandard finite difference (NSFD) schemes. However, since 1994 great progress has been made in both the understanding and the types of differential equations for which the NSFD methodology has been applied. Applications have been extended to de- layed, fractional derivative, and stochastic differential equations. Further, two major summaries and reviews have been written by Professor Kailash C. Patidar. This current book is a record of the progress made on the NSFD methodology since 1994. We estimate that more than one thousand publications have appeared which make use of the general NSFD methodology. Of interest is the fact that concepts such as dynamic consistency, denominator functions, and nonlocal discretizations of functions of the dependent variables are quickly making their way into the vocabulary of the numerical integration of dif- ferential equations often without explicit acknowledgement of their genesis within the original context of the NSFD methodology. This is a clear indi- cation of the increasing acceptance of this set of techniques. It should be clearly and strongly understood that the NSFD method- ology is, in particular, a general philosophy as to how finite difference discretizations should be formulated and constructed for differential equa- tions. It, in general, is not a fixed set of rules or procedures to carry out such constructions. In particular, it must be understood that each specific vii September28,2020 8:37 ws-book9x6 NonstandardFiniteDifferenceSchemes: MethodologyandApplications 11891-main pageviii viii Nonstandard Finite Difference Schemes: Methodology and Applications differentialequationmustbetreateddifferentlyandjusthowthisshouldbe accomplished depends critically on the background knowledge base of the researcherandtheirinsightsintothederivationofthedifferentialequations and its proposed applications. This second edition reformulates just what is the NSFD methodology. Thus, the new interpretation places the principle of dynamic consistency (DC)astheessentialfoundationonwhichalltheothercriticalfeaturesare to be based and/or derived, i.e., the calculation of denominator functions for the discrete derivatives and the nonlocal discretizations for functions of the dependent variables. Chapters1–8,intheoriginalvolumeremainthesameexceptforcorrec- tions and very minor additions. These Chapters still provide an excellent introduction to the genesis, construction, and history of NSFD schemes. A new Chapter 0 has been added. It gives a brief overview of what this volume is about and gives a summary of the new materials. Chapter 9 gives a fuller and deeper explanation of the NSFD methodology. The main purposeofChapter10istosummarizealargenumberofbothordinaryand partial differential equations for which “exact finite difference schemes” are known. These results are of value, since one of the techniques used in the construction of NSFD schemes is to incorporate exact schemes of subequations into the full discretization. Finally, Chapter 11 gives a representative sampling of the use of the NSFD methodology to construct schemes for a number of important appli- cations in the natural and engineering sciences. For those who wish a quick historical based introduction to the NSFD methodology and the important advances made in its techniques and ap- plications, we suggest reading the following four articles: 1) R. E. Mickens, “Exact solutions to a finite-difference model of a nonlinear reaction-advection equation: Implications for numerical analysis,” Numerical Methods for Partial Differential Equations 5 (1989), 313–325. 2) R.E.Mickens,“Dynamicconsistency: Afundamentalprinciplefor constructing NSFD schemes for differential equations,” Journal of Difference Equations and Applications 11 (2005), 645–653. 3) R. E. Mickens, “Numerical integration of population models sat- isfying conservation laws: NSFD methods,” Journal of Biological Dynamics 1 (2007), 427–436. September28,2020 8:37 ws-book9x6 NonstandardFiniteDifferenceSchemes: MethodologyandApplications 11891-main pageix Preface ix 4) R. E. Mickens, “Calculation of denominator functions for NSFD schemesfordifferentialequationssatisfyingapositivitycondition,” Numerical Methods for Partial Differential Equations 23 (2007), 672–691. Iwishtothankthehundredsofindividualswhohaveused,applied,and extended the NSFD methodology. In particular, I am thankful for the re- search productions of the following colleagues and their collaborators: Ron Buckmire,BentoM.Chen-Charpentier,JamesB.Cole,MatthiasEhrhardt, SaberN.Elaydi,AbbaB.Gumel,PedroM.Jordan,HristoV.Kojouharov, Jean M.-S. Lubuma, Jorge E. Macias-Diaz, Kale Oyedeji, Kailash C. Pati- dar, Lih-Ing W. Roeger, and Talitha M. Washington. Also, special thanks are extended to Imani Beverly and Bryan Brione for their help in obtaining manuscripts, books, and other technical items which greatly aided in my writing the original manuscript of this book. Asalways,IamparticularlygratefultoAnnetteRohrsforhandlingthe typingandpreliminaryeditorialprocessingofthefullmanuscript. Without her assistance, this volume would not exist. Ronald E. Mickens Atlanta, Georgia September 2020