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Differential Equations and Boundary Value Problems : Computing and Modeling,9780130652454
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Differential Equations and Boundary Value Problems : Computing and Modeling

by ;
Edition:
4th
ISBN13:

9780130652454

ISBN10:
0130652458
Format:
Hardcover
Pub. Date:
1/1/2008
Publisher(s):
Pearson College Div
List Price: $144.20
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Summary

This practical book reflects the new technological emphasis that permeates differential equations, including the wide availability of scientific computing environments likeMaple, Mathematica, and MATLAB; it does not concentrate on traditional manual methods but rather on new computer-based methods that lead to a wider range of more realistic applications. The book starts and ends with discussions of mathematical modeling of real-world phenomena, evident in figures, examples, problems, and applications throughout the book. For mathematicians and those in the field of computer science.

Table of Contents

Application Modules viii
Preface xi
First-Order Differential Equations
1(76)
Differential Equations and Mathematical Models
1(9)
Integrals as General and Particular Solutions
10(8)
Slope Fields and Solution Curves
18(13)
Separable Equations and Applications
31(15)
Linear First-Order Equations
46(12)
Substitution Methods and Exact Equations
58(19)
Mathematical Models and Numerical Methods
77(67)
Population Models
77(13)
Equilibrium Solutions and Stability
90(8)
Acceleration-Velocity Models
98(12)
Numerical Approximation: Euler's Method
110(12)
A Closer Look at the Euler Method
122(10)
The Runge-Kutta Method
132(12)
Linear Equations of Higher Order
144(98)
Introduction: Second-Order Linear Equations
144(14)
General Solutions of Linear Equations
158(12)
Homogeneous Equations with Constant Coefficients
170(12)
Mechanical Vibrations
182(13)
Nonhomogeneous Equations and Undetermined Coefficients
195(14)
Forced Oscillations and Resonance
209(13)
Electrical Circuits
222(7)
Endpoint Problems and Eigenvalues
229(13)
Introduction to Systems of Differential Equations
242(39)
First-Order Systems and Applications
242(12)
The Method of Elimination
254(11)
Numerical Methods for Systems
265(16)
Linear Systems of Differential Equations
281(85)
Matrices and Linear Systems
281(19)
The Eigenvalue Method for Homogeneous Systems
300(15)
Second-Order Systems and Mechanical Applications
315(13)
Multiple Eigenvalue Solutions
328(16)
Matrix Exponentials and Linear Systems
344(14)
Nonhomogeneous Linear Systems
358(8)
Nonlinear Systems and Phenomena
366(69)
Stability and the Phase Plane
366(12)
Linear and Almost Linear Systems
378(15)
Ecological Models: Predators and Competitors
393(13)
Nonlinear Mechanical Systems
406(17)
Chaos in Dynamical Systems
423(12)
Laplace Transform Methods
435(62)
Laplace Transforms and Inverse Transforms
435(11)
Transformation of Initial Value Problems
446(11)
Translation and Partial Fractions
457(10)
Derivatives, Integrals, and Products of Transforms
467(8)
Periodic and Piecewise Continuous Input Functions
475(11)
Impulses and Delta Functions
486(11)
Power Series Methods
497(75)
Introduction and Review of Power Series
497(13)
Series Solutions Near Ordinary Points
510(13)
Regular Singular Points
523(16)
Method of Frobenius: The Exceptional Cases
539(15)
Bessel's Equation
554(9)
Applications of Bessel Functions
563(9)
Fourier Series Methods
572(73)
Periodic Functions and Trigonometric Series
572(9)
General Fourier Series and Convergence
581(8)
Fourier Sine and Cosine Series
589(12)
Applications of Fourier Series
601(5)
Heat Conduction and Separation of Variables
606(15)
Vibrating Strings and the One-Dimensional Wave Equation
621(14)
Steady-State Temperature and Laplace's Equation
635(10)
Eigenvalues and Boundary Value Problems
645(66)
Sturm-Liouville Problems and Eigenfunction Expansions
645(13)
Applications of Eigenfunction Series
658(10)
Steady Periodic Solutions and Natural Frequencies
668(10)
Cylindrical Coordinate Problems
678(15)
Higher-Dimensional Phenomena
693(18)
References for Further Study 711(3)
Appendix: Existence and Uniqueness of Solutions 714(15)
Answers to Selected Problems 729
Index 1

Excerpts

Many introductory differential equations courses in the recent past have emphasized the formal solution of standard types of differential equations using a (seeming) grab-bag of systematic solution techniques. Many students have concentrated on learning to match memorized methods with memorized equations. The evolution of the present text is based on experience teaching a course with a greater emphasis on conceptual ideas and the use of applications and computing projects to involve students in more intense and sustained problem-solving experiences. The availability of technical computing environments likeMaple, Mathematica,and MATLAB is reshaping the role and applications of differential equations in science and engineering and has shaped our approach in this text. New technology motivates a shift in emphasis from traditional manual methods to both qualitative and computer-based methods that render accessible a wider range of more realistic applications; permit the use of both numerical computation and graphical visualization to develop greater conceptual understanding; and encourage empirical investigations that involve deeper thought and analysis than standard textbook problems. Major Features The following features of this text are intended to support a contemporary differential equations course that augments traditional core skills with conceptual perspectives that students will need for the effective use of differential equations in their subsequent work and study: Coverage of seldom-used topics has been trimmed and new topics added to place a greater emphasis on core techniques as well as qualitative aspects of the subject associated with direction fields, solution curves, phase plane portraits, and dynamical systems. We combine symbolic, graphic, and numeric solution methods wherever it seems advantageous. A fresh computational flavor should be evident in figures, examples, problems, and applications throughout the text. About 15% of the examples in the text are new or newly revised for this edition. The organization of the book places an increased emphasis on linear systems of differential equations, which are covered in Chapters 4 and 5 (together with the necessary linear algebra), followed by a substantial treatment in Chapter 6 of nonlinear systems and phenomena (including chaos in dynamical systems). This book begins and ends with discussions and examples of the mathematical modeling of real-world phenomena. Students learn through mathematical modeling and empirical investigation to balance the questions of what equation to formulate, how to solve it, and whether a solution will yield useful information.,/LI> The first course in differential equations should also be a window on the world of mathematics. While it is neither feasible nor desirable to include proofs of the fundamental existence and uniqueness theorems along the way in an elementary course, students need to see precise and clear-cut statements of these theorems and to understand their role in the subject. We include appropriate existence and uniqueness proofs in the Appendix and occasionally refer to them in the main body of the text. While our approach reflects the widespread use of new computer methods for the solution of differential equations, certain elementary analytical methods of solution (as in Chapters 1 and 3) are important for students to learn. Effective and reliable use of numerical methods often requires preliminary analysis using standard elementary techniques; the construction of a realistic numerical model often is based on the study of a simpler analytical model. We therefore continue to stress the mastery of traditional solution techniques (especially through the inclusion of extensive problem sets). Computing Features The following features highlight the flavor of computing technology that distinguishes much of


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