Differential Equations with MATLAB: Exploration, Applications, and Theory

495 Pages·2014·10.877 MB·English

by  McKibben, Mark A., Webster, Micah D

Mathematics
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Differential Equations with MATLAB®: Exploration, Applications, and Theory
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provides you with an understanding of the practical and theoretical aspects
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circuits. Focusing on linear PDEs, the second part covers PDEs that arise in
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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
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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
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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.
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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