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E-raamat: Differential Equations with MATLAB: Exploration, Applications, and Theory

(Goucher College, Baltimore, Maryland, USA), (West Chester University, Pennsylvania, USA)
  • Formaat: 497 pages
  • Sari: Textbooks in Mathematics
  • Ilmumisaeg: 08-Sep-2014
  • Kirjastus: CRC Press Inc
  • Keel: eng
  • ISBN-13: 9781040162071
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Differential Equations with MATLAB: Exploration, Applications, and TheorySuurem pilt
  • Formaat: 497 pages
  • Sari: Textbooks in Mathematics
  • Ilmumisaeg: 08-Sep-2014
  • Kirjastus: CRC Press Inc
  • Keel: eng
  • ISBN-13: 9781040162071
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In a textbook for students who have completed multivariate calculus, McKibben and Webster seek to instill a sense of intuition and practical and theoretical knowledge pertaining to mathematical models involving ordinary and partial differential equations. They establish a broad context within which they analyze more than 20 distinct models spanning disciplines, describe the thought process of building the surrounding theory from the ground up, and actively involve students in the mathematical enterprise throughout the text. Annotation ©2015 Ringgold, Inc., Portland, OR (protoview.com)

A unique textbook for an undergraduate course on mathematical modeling, Differential Equations with MATLAB: Exploration, Applications, and Theory provides students with an understanding of the practical and theoretical aspects of mathematical models involving ordinary and partial differential equations (ODEs and PDEs). The text presents a unifying picture inherent to the study and analysis of more than 20 distinct models spanning disciplines such as physics, engineering, and finance.

The first part of the book presents systems of linear ODEs. The text develops mathematical models from ten disparate fields, including pharmacokinetics, chemistry, classical mechanics, neural networks, physiology, and electrical circuits. Focusing on linear PDEs, the second part covers PDEs that arise in the mathematical modeling of phenomena in ten other areas, including heat conduction, wave propagation, fluid flow through fissured rocks, pattern formation, and financial mathematics.

The authors engage students by posing questions of all types throughout, including verifying details, proving conjectures of actual results, analyzing broad strokes that occur within the development of the theory, and applying the theory to specific models. The authors’ accessible style encourages students to actively work through the material and answer these questions. In addition, the extensive use of MATLAB® GUIs allows students to discover patterns and make conjectures.

Arvustused

"The purpose of this book is to illustrate the use of MATLAB in the study of several models involving ordinary and partial differential equations. It includes different disciplines such as physics, engineering and finance. It may be useful for engineers, physicists and applied mathematicians and also for advanced undergraduate (or beginning graduate) students who are interested in the utilization of MATLAB in differential equations. ... The volume incorporates many figures and MATLAB exercises and many questions are raised throughout the text so that readers can do their own computer experiments." Antonio Canada Villar (Granada), writing in Zentralblatt MATH 1320 1

List of Figures
xv
List of Tables
xxiii
Preface xxv
Author Bios xxix
I Ordinary Differential Equations
1(182)
1 Welcome!
3(10)
1.1 Introduction
3(1)
1.2 This Book Is a Field Guide. What Does That Mean For YOU?
4(1)
1.3 Mired in Jargon---A Quick Language Lesson!
5(1)
1.4 Introducing MATLAB
6(1)
1.5 A First Look at Some Elementary Mathematical Models
7(6)
2 A Basic Analysis Toolbox
13(38)
2.1 Some Basic Mathematical Shorthand
14(1)
2.2 Set Algebra
15(1)
2.3 Functions
16(2)
2.4 The Space (R, |·|)
18(3)
2.4.1 Order Properties
18(1)
2.4.2 Absolute Value
19(1)
2.4.3 Completeness Property of (R, I · I)
20(1)
2.5 A Closer Look at Sequences in (R, |·|)
21(9)
2.5.1 Sequences and Subsequences
21(1)
2.5.2 Convergence
22(3)
2.5.3 Properties of Convergent Sequences
25(1)
2.5.4 Cauchy Sequences
26(1)
2.5.5 A Brief Look at Infinite Series
27(2)
2.5.6 Power Series
29(1)
2.6 The Spaces (RN, ||·||RN) and (MN(R), ||·||MN(R))
30(5)
2.6.1 The Space (RN, ||·||RN)
30(3)
2.6.2 The Space (MN(R), ||·||MN(R))
33(2)
2.7 Calculus of (RN)-valued and MN(R)-valued Functions
35(11)
2.7.1 Notation and Interpretation
36(2)
2.7.2 Limits and Continuity
38(1)
2.7.3 The Derivative
39(1)
2.7.4 The Integral
40(2)
2.7.5 Sequences in RN and MN(R)
42(3)
2.7.6 Continuity---Revisited
45(1)
2.8 Some Elementary ODEs
46(3)
2.8.1 Separation of Variables
46(1)
2.8.2 First-Order Linear ODEs
47(1)
2.8.3 Higher-Order Linear ODEs
48(1)
2.9 Looking Ahead
49(2)
3 A First Wave of Mathematical Models
51(42)
3.1 Newton's Law of Heating and Cooling---Revisited
51(3)
3.2 Pharmocokinetics
54(3)
3.3 Uniform Mixing Models
57(3)
3.4 Combat! Nation in Balance
60(1)
3.5 Springs and Electrical Circuits---The Same, But Different
61(10)
3.5.1 Spring-Mass System
62(4)
3.5.2 Simple Electrical Circuits
66(2)
3.5.2.1 The Same, But Different!
68(3)
3.6 Boom!---Chemical Kinetics
71(4)
3.6.1 Reaction Models
72(3)
3.7 Going, Going, Gone! A Look at Projectile Motion
75(3)
3.7.1 Projectile Motion Models
75(3)
3.8 Shake, Rattle, Roll!
78(2)
3.8.1 Floor Displacement Model
78(2)
3.9 My Brain Hurts! A Look at Neural Networks
80(5)
3.10 Breathe In, Breathe Out---A Respiratory Regulation Model
85(3)
3.11 Looking Ahead!
88(5)
4 Finite-Dimensional Theory---Ground Zero: The Homogenous Case
93(36)
4.1 Introducing the Homogenous Cauchy Problem (HCP)
93(2)
4.2 Lessons Learned from a Special Case
95(1)
4.3 Defining the Matrix Exponential eAt
96(6)
4.3.1 One Approach---Taylor Series
96(3)
4.3.2 Another Approach
99(3)
4.4 Putzer's Algorithm
102(4)
4.5 Properties of eAt
106(6)
4.6 The Homogenous Cauchy Problem: Well-posedness
112(1)
4.7 Higher-Order Linear ODEs
113(4)
4.8 A Perturbed (HCP)
117(6)
4.9 What Happens to Solutions of (HCP) as Time Goes On and On and On...?
123(3)
4.10 Looking Ahead
126(3)
5 Finite-Dimensional Theory---Next Step: The Non-Homogenous Case
129(24)
5.1 Introducing... The Non-Homogenous Cauchy Problem (Non-CP)
129(2)
5.2 Carefully Examining the One-Dimensional Version of (Non-CP)
131(5)
5.2.1 Solving (Non-CP)---Calculus Based
131(1)
5.2.2 Solving (Non-CP)---Numerics Based
131(3)
5.2.3 Building an Existence Theory for One-Dimensional (Non-CP)
134(2)
5.2.4 Defining What is Meant By a Solution of (Non-CP)
136(1)
5.3 Existence Theory for General (Non-CP)
136(8)
5.3.1 Constructing a Solution of (Non-CP)
136(1)
5.3.2 Computing with the Variation of Parameters Formula
137(5)
5.3.3 An Existence-Uniqueness Theorem for (Non-CP)
142(2)
5.4 Dealing with a Perturbed (Non-CP)
144(6)
5.5 What Happens to Solutions of (Non-CP) as Time Goes On and On and On...?
150(3)
6 A Second Wave of Mathematical Models---Now, with Nonlinear Interactions
153(12)
6.1 Newton's Law of Heating and Cooling Subjected to Polynomial Effects
153(2)
6.2 Pharmocokinetics with Concentration-Dependent Dosing
155(1)
6.3 Springs with Nonlinear Restoring Forces
156(1)
6.4 Circuits with Quadratic Resistors
157(1)
6.5 Enyzme Catalysts
158(4)
6.6 Projectile Motion---Revisited
162(1)
6.7 Floor Displacement Model with Nonlinear Shock Absorbers
163(2)
7 Finite-Dimensional Theory---Last Step: The Semi-Linear Case
165(18)
7.1 Introducing the Even-More General Semi-Linear Cauchy Problem (Semi-CP)
165(2)
7.2 New Challenges
167(1)
7.3 Behind the Scenes: Issues and Resolutions Arising in the Study of (Semi-CP)
168(4)
7.4 Lipschitz to the Rescue!
172(3)
7.5 Gronwall's Lemma
175(1)
7.6 The Existence and Uniqueness of a Mild Solution for (Semi-CP)
176(1)
7.7 Dealing with a Perturbed (Semi-CP)
176(7)
II Abstract Ordinary Differential Equations
183(270)
8 Getting the Lay of a New Land
185(26)
8.1 A Hot Example
185(6)
8.2 The Hunt for a New Abstract Paradigm
191(4)
8.3 A Small Dose of Functional Analysis
195(15)
8.3.1 Moving Beyond Just RN and MN: Introducing the Notions of Banach Space and Hilbert Space
195(2)
8.3.1.1 The Notion of Convergence Revisited
197(1)
8.3.1.2 An Important Topological Notion---Closed Sets
198(1)
8.3.2 Hilbert Spaces
199(4)
8.3.3 Linear Operators
203(4)
8.3.4 Calculus in Abstract Spaces
207(1)
8.3.4.1 Limits
207(1)
8.3.4.2 Continuity
207(1)
8.3.4.3 The Derivative
208(1)
8.3.4.4 The Integral
209(1)
8.4 Looking Ahead
210(1)
9 Three New Mathematical Models
211(72)
9.1 Turning Up the Heat---Variants of the Heat Equation
211(19)
9.1.1 One-Dimensional Diffusion Equation
212(11)
9.1.2 Two-Dimensional Diffusion Equation
223(6)
9.1.3 Abstraction Formulation
229(1)
9.2 Clay Consolidation and Seepage of Fluid Through Fissured Rocks
230(28)
9.2.1 Seepage of Fluid Through Fissured Rocks
230(9)
9.2.2 Two-Dimensional Seepage of Fluid Through Fissured Rocks
239(6)
9.2.3 Hypoplasticity
245(7)
9.2.4 Two-Dimensional Hypoplasticity
252(6)
9.2.5 Abstract Formulation
258(1)
9.3 The Classical Wave Equation and its Variants
258(20)
9.3.1 One-Dimensional Wave Equation
258(13)
9.3.2 Two-Dimensional Wave Equations
271(5)
9.3.3 Abstract Formulation
276(2)
9.4 An Informal Recap: A First Step Toward Unification
278(5)
10 Formulating a Theory for (A-HCP)
283(20)
10.1 Introducing (A-HCP)
283(1)
10.2 Defining eAt
284(9)
10.3 Properties of eAt
293(1)
10.4 The Abstract Homogeneous Cauchy Problem: Well-posedness
294(4)
10.5 A Brief Glimpse of Long-Term Behavior
298(1)
10.6 Looking Ahead
299(4)
11 The Next Wave of Mathematical Models---With Forcing
303(82)
11.1 Turning Up the Heat---Variants of the Heat Equation
304(28)
11.1.1 One-Dimensional Diffusion with Forcing
304(16)
11.1.2 Two-Dimensional Diffusion with Forcing
320(11)
11.1.3 Abstract Formulation
331(1)
11.2 Seepage of Fluid Through Fissured Rocks
332(20)
11.2.1 One-Dimensional Fissured Rock Model with Forcing
332(10)
11.2.2 Higher-Dimensional Fissured Rock Model with Forcing
342(10)
11.2.3 Abstract Formulation
352(1)
11.3 The Classical Wave Equation and its Variants
352(33)
11.3.1 One-Dimensional Wave Equation with Source Effects
352(21)
11.3.2 Two-Dimensional Wave Equation with Source Effects
373(11)
11.3.3 Abstract Formulation
384(1)
12 Remaining Mathematical Models
385(12)
12.1 Population Growth---Fisher's Equation
385(2)
12.1.1 Two-Dimensional Fisher's Equation
387(1)
12.2 Zombie Apocalypse! Epidemiological Models
387(2)
12.3 How Did That Zebra Gets Its Stripes? A First Look at Spatial Pattern Formation
389(1)
12.4 Autocatalysis Combustion!
390(1)
12.5 Money, Money, Money---A Simple Financial Model
391(6)
12.5.1 Black and Scholes Equation and the Heat Equation
394(3)
13 Formulating a Theory for (A-NonCP)
397(12)
13.1 Introducing (A-NonCP)
397(3)
13.2 Existence and Uniqueness of Solutions of (A-NonCP)
400(1)
13.3 Dealing with a Perturbed (A-NonCP)
401(3)
13.4 Long-Term Behavior
404(1)
13.5 Looking Ahead
405(4)
14 A Final Wave of Models---Accounting for Semilinear Effects
409(44)
14.1 Turning Up the Heat---Semi-Linear Variants of the Heat Equation
409(3)
14.2 The Classical Wave Equation with Semilinear Forcing
412(4)
14.3 Population Growth---Fisher's Equation
416(12)
14.3.1 Two-Dimensional Fisher's Equation
422(6)
14.4 Zombie Apocalypse! Epidemiological Models
428(8)
14.5 How Did That Zebra Get Its Stripes? A First Look at Spatial Pattern Formation
436(10)
14.5.1 Two-Dimensional Gray-Scott Equation
441(5)
14.6 Autocatalysis---Combustion!
446(7)
Epilogue 453(2)
Appendix 455(4)
Bibliography 459(4)
Index 463
Mark McKibben, Micah D. Webster
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