Ordinary differential equations: applications, models, and computing
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
Chapman & Hall/CRC
2010
|
| Series: | Textbooks in mathematics
A Chapman & Hall book |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XVII, 584 S. Ill., graph. Darst. CD-ROM (12 cm) |
| ISBN: | 9781439819081 |
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Record in the Search Index
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| adam_text | Mathematics
TEXTBOOKS in MATHEMATICS
Although the theory associated with nonlinear systems is advanced, generating
a numerical solution with a computer and interpreting that solution are fairly
elementary. Designed to be independent of any particular software package,
Ordinary Differential Equations: Applications, Models, and Computing
emphasizes the use of computer software in solving various types of differential
equations.
Providing an even balance between theory, computer solution, and application, the
text discusses the theorems and applications of the first-order initial value problem,
including learning theory models, population growth models, epidemic models,
and chemical reactions. It then examines the theory for n-th order linear differential
equations and the Laplace transform and its properties, before addressing several
linear differential equations with constant coefficients that arise in physical and
electrical systems. The author also presents systems of first-order differential
equations as well as linear systems with constant coefficients that arise in physical
systems, such as coupled spring-mass systems, pendulum systems, the path of
an electron, and mixture problems. The final chapter introduces techniques for
determining the behavior of solutions to systems of first-order differential equations
without first finding the solutions.
Features
•
Emphasizes the use of software to aid in problem solving
•
Includes numerical case studies that highlight possible pitfalls when
computing a numerical solution without first considering the appropriate
theory
•
Covers nonlinear differential equations and nonlinear systems
•
Shows how to solve various mathematical models, such as population
growth, epidemic, and predator-prey models
•
Discusses fundamental existence, uniqueness, and continuation theorems
»
Contains a CD-ROM with the software programs used in the text
»
Requires no prior knowledge of programming languages
Contents
Preface
xi
1
Introduction
1
1.1
Historical
Prologue
................... 1
1.2
Definitions and Terminology
............... 5
1.3
Solutions and Problems
................. 17
1.4
A Nobel Prize Winning Application
............ 29
2
The Initial Value Problem y
=
f(x,y); y(c)
=
d
33
2.1
Direction Fields
..................... 34
2.2
Fundamental Theorems
................. 42
2.3
Solution of Simple First-Order Differential Equations
.... 55
2.3.1
Solution of y =g(x)
................ 55
2.3.2
Solution of the Separable Equation y
=
g(x)/h(y)
... 58
2.3.3
Solution of the Linear Equation y
=
a(x)y
+
b(x)
... 64
2.4
Numerical Solution
................... 73
2.4.1
Single-Step Methods
................ 78
2.4.1.1
Taylor Series Method
............ 78
2.4.1.2
Runge-Kutta Methods
........... 86
2.4.2
Multistep Methods
................. 97
2.4.2.1
Adams-Bashforth Methods
.......... 97
2.4.2.2
Nyström
Methods
.............. 100
2.4.2.3
Adams-Moulton Methods
........... 101
2.4.3
Predictor-Corrector Methods
............. 102
2.4.4
Pitfalls of Numerical Methods
............ 107
VII
Vlil
3
Applications
of the Initial Value Problem
y = f(x,y); y(c) = d
117
3.1
Calculus Revisited
................... 117
3.2
Learning Theory Models
................. 129
3.3
Population Models
................... 131
3.4
Simple Epidemic Models
................. 138
3.5
Falling Bodies
..................... 142
3.6
Mixture Problems
................... 146
3.7
Curves of Pursuit
.................... 153
3.8
Chemical Reactions
................... 157
4
N-th Order Linear Differential Equations
163
4.1
Basic Theory
...................... 164
4.2
Roots of Polynomials
.................. 188
4.3
Homogeneous Linear Equations with Constant Coefficients
. . 200
4.4
Nonhomogeneous Linear Equations with Constant Coefficients
211
4.5
Initial Value Problems
.................. 219
5
The Laplace Transform Method
223
5.1
The Laplace Transform and Its Properties
......... 223
5.2
Using the Laplace Transform and Its Inverse
to Solve Initial Value Problems
.............. 242
5.3
Convolution and the Laplace Transform
.......... 250
5.4
The Unit Function and Time-Delay Function
....... 257
5.5
Impulse Function
................... 267
6
Applications
of Linear Differential Equations
with Constant Coefficients
275
6.1
Second-Order Differential Equations
........... 275
6.1.1
Free Motion
................... 281
6.1.1.1
Free Undamped Motion
.......... 282
6.1.1.2
Free Damped Motion
........... 283
6.1.2
Forced Motion
................. 292
6.1.2.1
Undamped Forced Motion
......... 292
6.1.2.2
Damped Forced Motion
.......... 293
6.2
Higher Order Differential Equations
........... 297
7
Systems of First-Order Differential Equations
313
8
Linear Systems of First-Order Differential Equations
335
8.1
Matrices and Vectors
................. 335
8.2
Eigenvalues and Eigenvectors
.............. 349
8.3
Linear Systems with Constant Coefficients
........ 363
9
Applications of Linear Systems with Constant Coefficients
383
9.1
Coupled Spring-Mass Systems
.............. 383
9.2
Pendulum Systems
.................. 390
9.3
The Path of an Electron
................ 392
9.4
Mixture Problems
................... 397
10
Applications of Systems of Equations
407
10.1
Richardson s Arms Race Model
............. 407
10.2
Phase-Plane Portraits
................. 416
10.3
Modified Richardson s Arms Race Models
........ 433
10.4
Lanchester s Combat Models
.............. 445
10.5
Models for Interacting Species
............. 452
10.6
Epidemics
...................... 470
10.7
Pendulums
..................... 480
10.8
Dumng s Equation
.................. 489
10.9
Van
der Poľs
Equation
................ 490
10.10
Mixture Problems
.................. 491
10.11
The Restricted Three-Body Problem
.......... 494
Appendix A CSODE User s Guide
499
Appendix
В
PORTRAIT User s Guide
525
Appendix
С
Laplace Transforms
537
Answers to Selected Exercises
539
References
571
Index
575
|
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| isbn | 9781439819081 |
| language | English |
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| physical | XVII, 584 S. Ill., graph. Darst. CD-ROM (12 cm) |
| publishDate | 2010 |
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| series2 | Textbooks in mathematics A Chapman & Hall book |
| spelling | Roberts, Charles E. Verfasser aut Ordinary differential equations applications, models, and computing Charles E. Roberts Boca Raton [u.a.] Chapman & Hall/CRC 2010 XVII, 584 S. Ill., graph. Darst. CD-ROM (12 cm) txt rdacontent n rdamedia nc rdacarrier Textbooks in mathematics A Chapman & Hall book Includes bibliographical references and index Differential equations Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Gewöhnliche Differentialgleichung (DE-588)4020929-5 s DE-604 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018987946&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018987946&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Roberts, Charles E. Ordinary differential equations applications, models, and computing Differential equations Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| subject_GND | (DE-588)4020929-5 (DE-588)4123623-3 |
| title | Ordinary differential equations applications, models, and computing |
| title_auth | Ordinary differential equations applications, models, and computing |
| title_exact_search | Ordinary differential equations applications, models, and computing |
| title_full | Ordinary differential equations applications, models, and computing Charles E. Roberts |
| title_fullStr | Ordinary differential equations applications, models, and computing Charles E. Roberts |
| title_full_unstemmed | Ordinary differential equations applications, models, and computing Charles E. Roberts |
| title_short | Ordinary differential equations |
| title_sort | ordinary differential equations applications models and computing |
| title_sub | applications, models, and computing |
| topic | Differential equations Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| topic_facet | Differential equations Gewöhnliche Differentialgleichung Lehrbuch |
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