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Mathematics TEXTBOOKS in MATHEMATICS TEXTBOOKS in MATHEMATICS DDiiffffeerreennttiiaall EEqquuaattiioonnss Differential Equations with MATLAB®: Exploration, Applications, and Theory DD provides you with an understanding of the practical and theoretical aspects of mathematical models involving ordinary and partial differential equations ii wwiitthh MMAATTLLAABB®® ff (ODEs and PDEs). The text presents a unifying picture inherent to the study and ww ff analysis of more than 20 distinct models spanning disciplines such as physics, ee ii tt rr engineering, and finance. hh ee EExxpplloorraattiioonn,, AApppplliiccaattiioonnss,, aanndd TThheeoorryy nn The first part of the book presents systems of linear ODEs. The text develops MM tt mathematical models from ten disparate fields, including pharmacokinetics, ii aa chemistry, classical mechanics, neural networks, physiology, and electrical AA ll circuits. Focusing on linear PDEs, the second part covers PDEs that arise in TT EE the mathematical modeling of phenomena in ten other areas, including heat LL qq conduction, wave propagation, fluid flow through fissured rocks, pattern AA uu formation, and financial mathematics. aa BB The authors pose questions of all types throughout, including verifying details, tt ®® ii proving conjectures of actual results, analyzing broad strokes that occur within oo the development of the theory, and applying the theory to specific models. Their nn accessible style encourages you to actively work through the material and answer ss these questions. In addition, the extensive use of MATLAB GUIs allows you to discover patterns and make conjectures. Features • Develops your intuition by building the mathematical theory from the ground up and illustrating the theorems with examples • Demonstrates the analysis of mathematical models through concrete applications MM • Uses MATLAB GUIs that enable you to discover patterns and make WW cc conjectures ee KK bb ii • Provides heuristic commentary and motivation in each chapter ss bb tete bb • Includes numerous hands-on activities that facilitate exploration of the tools rr ee nn and models MMaarrkk AA.. MMccKKiibbbbeenn MMiiccaahh DD.. WWeebbsstteerr K15453 K15453_cover.indd 1 7/25/14 11:22 AM Differential Equations with MATLAB® Exploration, Applications, and Theory TEXTBOOKS in MATHEMATICS Series Editors: Ken Rosen and Al Boggess PUBLISHED TITLES RISK ANALYSIS IN ENGINEERING AND ECONOMICS, SECOND EDITION Bilal M. Ayyub INTRODUCTION TO THE CALCULUS OF VARIATIONS AND CONTROL WITH MODERN APPLICATIONS John T. Burns MIMETIC DISCRETIZATION METHODS Jose E. Castillo AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS WITH MATLAB®, SECOND EDITION Mathew Coleman RISK MANAGEMENT AND SIMULATION Aparna Gupta ABSTRACT ALGEBRA: AN INQUIRY-BASED APPROACH Jonathan K. Hodge, Steven Schlicker, and Ted Sundstrom QUADRACTIC IRRATIONALS: AN INTRODUCTION TO CLASSICAL NUMBER THEORY Franz Holter-Koch GROUP INVERSES OF M-MATRICES AND THEIR APPLICATIONS Stephen J. Kirkland AN INTRODUCTION TO NUMBER THEORY WITH CRYPTOGRAPHY James Kraft and Larry Washington REAL ANALYSIS AND FOUNDATIONS, THIRD EDITION Steven G. Krantz ELEMENTS OF ADVANCED MATHEMATICS, THIRD EDITION Steven G. Krantz APPLYING ANALYTICS: A PRACTICAL APPROACH Evan S. Levine ADVANCED LINEAR ALGEBRA Nicholas Loehr DIFFERENTIAL EQUATIONS WITH MATLAB®: EXPLORATION, APPLICATIONS, AND THEORY Mark A. McKibben and Micah D. Webster PUBLISHED TITLES CONTINUED APPLICATIONS OF COMBINATORIAL MATRIX THEORY TO LAPLACIAN MATRICES OF GRAPHS Jason J. Molitierno ABSTRACT ALGEBRA: AN INTERACTIVE APPROACH William Paulsen ADVANCED CALCULUS: THEORY AND PRACTICE John Srdjan Petrovic COMPUTATIONS OF IMPROPER REIMANN INTEGRALS Ioannis Roussos TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk TEXTBOOKS in MATHEMATICS Differential Equations with MATLAB® Exploration, Applications, and Theory Mark A. McKibben West Chester University West Chester, Pennsylvania, USA Micah D. Webster Goucher College Baltimore, Maryland, USA MATLAB®, Simulink®, and Stateflow® are trademarks of The MathWorks, Inc. and are used with per- mission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB®, Simulink®, and Stateflow® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB®, Simulink®, and Stateflow® software. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2015 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20140724 International Standard Book Number-13: 978-1-4665-5708-6 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information stor- age or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copy- right.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that pro- vides licenses and registration for a variety of users. For organizations that have been granted a photo- copy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com To our mentors... Sergiu Aizicovici, Sam “Cal” Calavitta, Patrick Guidotti, and David Keck. TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk Contents List of Figures xv List of Tables xxiii Preface xxv Author Bios xxix I Ordinary Differential Equations 1 1 Welcome! 3 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 This Book Is a Field Guide. What Does That Mean For YOU? . . . . . . . 4 1.3 Mired in Jargon—A Quick Language Lesson! . . . . . . . . . . . . . . . . . 5 1.4 Introducing MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 A First Look at Some Elementary Mathematical Models . . . . . . . . . . 7 2 A Basic Analysis Toolbox 13 2.1 Some Basic Mathematical Shorthand . . . . . . . . . . . . . . . . . . . . . 14 2.2 Set Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 The Space (R, |·|) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4.1 Order Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4.2 Absolute Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4.3 Completeness Property of (R, |·|) . . . . . . . . . . . . . . . . . . . 20 2.5 A Closer Look at Sequences in (R, |·|) . . . . . . . . . . . . . . . . . . . . . 21 2.5.1 Sequences and Subsequences . . . . . . . . . . . . . . . . . . . . . . 21 2.5.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5.3 Properties of Convergent Sequences . . . . . . . . . . . . . . . . . . 25 2.5.4 Cauchy Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.5.5 A Brief Look at Infinite Series . . . . . . . . . . . . . . . . . . . . . 27 2.5.6 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.6 The Spaces (cid:0)RN, (cid:107)·(cid:107) (cid:1) and (cid:16)MN(R), (cid:107)·(cid:107) (cid:17) . . . . . . . . . . . . . . 30 RN MN(R) 2.6.1 The Space (cid:0)RN, (cid:107)·(cid:107) (cid:1) . . . . . . . . . . . . . . . . . . . . . . . . . 30 RN (cid:16) (cid:17) 2.6.2 The Space MN(R), (cid:107)·(cid:107) . . . . . . . . . . . . . . . . . . . . . 33 MN(R) 2.7 Calculus of RN-valued and MN(R)-valued Functions . . . . . . . . . . . . . 35 2.7.1 Notation and Interpretation . . . . . . . . . . . . . . . . . . . . . . 36 2.7.2 Limits and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.7.3 The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ix

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Mathematics TEXTBOOKS in MATHEMATICS TEXTBOOKS in MATHEMATICS DDiiffffeerreennttiiaall EEqquuaattiioonnss Differential Equations with MATLAB®: Exploration, Applications, and Theory DD provides you with an understanding of the practical and theoretical aspects of mathematical models involving ordinary and partial differential equations ii wwiitthh MMAATTLLAABB®® ff (ODEs and PDEs). The text presents a unifying picture inherent to the study and ww ff analysis of more than 20 distinct models spanning disciplines such as physics, ee ii tt rr engineering, and finance. hh ee EExxpplloorraattiioonn,, AApppplliiccaattiioonnss,, aanndd TThheeoorryy nn The first part of the book presents systems of linear ODEs. The text develops MM tt mathematical models from ten disparate fields, including pharmacokinetics, ii aa chemistry, classical mechanics, neural networks, physiology, and electrical AA ll circuits. Focusing on linear PDEs, the second part covers PDEs that arise in TT EE the mathematical modeling of phenomena in ten other areas, including heat LL qq conduction, wave propagation, fluid flow through fissured rocks, pattern AA uu formation, and financial mathematics. aa BB The authors pose questions of all types throughout, including verifying details, tt ®® ii proving conjectures of actual results, analyzing broad strokes that occur within oo the development of the theory, and applying the theory to specific models. Their nn accessible style encourages you to actively work through the material and answer ss these questions. In addition, the extensive use of MATLAB GUIs allows you to discover patterns and make conjectures. Features • Develops your intuition by building the mathematical theory from the ground up and illustrating the theorems with examples • Demonstrates the analysis of mathematical models through concrete applications MM • Uses MATLAB GUIs that enable you to discover patterns and make WW cc conjectures ee KK bb ii • Provides heuristic commentary and motivation in each chapter ss bb tete bb • Includes numerous hands-on activities that facilitate exploration of the tools rr ee nn and models MMaarrkk AA.. MMccKKiibbbbeenn MMiiccaahh DD.. WWeebbsstteerr K15453 K15453_cover.indd 1 7/25/14 11:22 AM Differential Equations with MATLAB® Exploration, Applications, and Theory TEXTBOOKS in MATHEMATICS Series Editors: Ken Rosen and Al Boggess PUBLISHED TITLES RISK ANALYSIS IN ENGINEERING AND ECONOMICS, SECOND EDITION Bilal M. Ayyub INTRODUCTION TO THE CALCULUS OF VARIATIONS AND CONTROL WITH MODERN APPLICATIONS John T. Burns MIMETIC DISCRETIZATION METHODS Jose E. Castillo AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS WITH MATLAB®, SECOND EDITION Mathew Coleman RISK MANAGEMENT AND SIMULATION Aparna Gupta ABSTRACT ALGEBRA: AN INQUIRY-BASED APPROACH Jonathan K. Hodge, Steven Schlicker, and Ted Sundstrom QUADRACTIC IRRATIONALS: AN INTRODUCTION TO CLASSICAL NUMBER THEORY Franz Holter-Koch GROUP INVERSES OF M-MATRICES AND THEIR APPLICATIONS Stephen J. Kirkland AN INTRODUCTION TO NUMBER THEORY WITH CRYPTOGRAPHY James Kraft and Larry Washington REAL ANALYSIS AND FOUNDATIONS, THIRD EDITION Steven G. Krantz ELEMENTS OF ADVANCED MATHEMATICS, THIRD EDITION Steven G. Krantz APPLYING ANALYTICS: A PRACTICAL APPROACH Evan S. Levine ADVANCED LINEAR ALGEBRA Nicholas Loehr DIFFERENTIAL EQUATIONS WITH MATLAB®: EXPLORATION, APPLICATIONS, AND THEORY Mark A. McKibben and Micah D. Webster PUBLISHED TITLES CONTINUED APPLICATIONS OF COMBINATORIAL MATRIX THEORY TO LAPLACIAN MATRICES OF GRAPHS Jason J. Molitierno ABSTRACT ALGEBRA: AN INTERACTIVE APPROACH William Paulsen ADVANCED CALCULUS: THEORY AND PRACTICE John Srdjan Petrovic COMPUTATIONS OF IMPROPER REIMANN INTEGRALS Ioannis Roussos TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk TEXTBOOKS in MATHEMATICS Differential Equations with MATLAB® Exploration, Applications, and Theory Mark A. McKibben West Chester University West Chester, Pennsylvania, USA Micah D. Webster Goucher College Baltimore, Maryland, USA MATLAB®, Simulink®, and Stateflow® are trademarks of The MathWorks, Inc. and are used with per- mission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB®, Simulink®, and Stateflow® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB®, Simulink®, and Stateflow® software. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2015 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20140724 International Standard Book Number-13: 978-1-4665-5708-6 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information stor- age or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copy- right.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that pro- vides licenses and registration for a variety of users. For organizations that have been granted a photo- copy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com To our mentors... Sergiu Aizicovici, Sam “Cal” Calavitta, Patrick Guidotti, and David Keck. TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk Contents List of Figures xv List of Tables xxiii Preface xxv Author Bios xxix I Ordinary Differential Equations 1 1 Welcome! 3 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 This Book Is a Field Guide. What Does That Mean For YOU? . . . . . . . 4 1.3 Mired in Jargon—A Quick Language Lesson! . . . . . . . . . . . . . . . . . 5 1.4 Introducing MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 A First Look at Some Elementary Mathematical Models . . . . . . . . . . 7 2 A Basic Analysis Toolbox 13 2.1 Some Basic Mathematical Shorthand . . . . . . . . . . . . . . . . . . . . . 14 2.2 Set Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 The Space (R, |·|) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4.1 Order Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4.2 Absolute Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4.3 Completeness Property of (R, |·|) . . . . . . . . . . . . . . . . . . . 20 2.5 A Closer Look at Sequences in (R, |·|) . . . . . . . . . . . . . . . . . . . . . 21 2.5.1 Sequences and Subsequences . . . . . . . . . . . . . . . . . . . . . . 21 2.5.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5.3 Properties of Convergent Sequences . . . . . . . . . . . . . . . . . . 25 2.5.4 Cauchy Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.5.5 A Brief Look at Infinite Series . . . . . . . . . . . . . . . . . . . . . 27 2.5.6 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.6 The Spaces (cid:0)RN, (cid:107)·(cid:107) (cid:1) and (cid:16)MN(R), (cid:107)·(cid:107) (cid:17) . . . . . . . . . . . . . . 30 RN MN(R) 2.6.1 The Space (cid:0)RN, (cid:107)·(cid:107) (cid:1) . . . . . . . . . . . . . . . . . . . . . . . . . 30 RN (cid:16) (cid:17) 2.6.2 The Space MN(R), (cid:107)·(cid:107) . . . . . . . . . . . . . . . . . . . . . 33 MN(R) 2.7 Calculus of RN-valued and MN(R)-valued Functions . . . . . . . . . . . . . 35 2.7.1 Notation and Interpretation . . . . . . . . . . . . . . . . . . . . . . 36 2.7.2 Limits and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.7.3 The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ix

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