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E-raamat: A Course in Differential Equations with Boundary Value Problems

(Arizona State University, Phoenix, USA), (California State Polytechnic University, Pomona, USA), (California State Polytechnic University, Pomona, USA)
  • Formaat: 788 pages
  • Sari: Textbooks in Mathematics
  • Ilmumisaeg: 24-Jan-2017
  • Kirjastus: Chapman & Hall/CRC
  • ISBN-13: 9781498736060
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A Course in Differential Equations with Boundary Value ProblemsSuurem pilt
  • Formaat: 788 pages
  • Sari: Textbooks in Mathematics
  • Ilmumisaeg: 24-Jan-2017
  • Kirjastus: Chapman & Hall/CRC
  • ISBN-13: 9781498736060
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A Course in Differential Equations with Boundary Value Problems, 2nd Edition adds additional content to the authors successful A Course on Ordinary Differential Equations, 2nd Edition. This text addresses the need when the course is expanded.

The focus of the text is on applications and methods of solution, both analytical and numerical, with emphasis on methods used in the typical engineering, physics, or mathematics students field of study. The text provides sufficient problems so that even the pure math major will be sufficiently challenged.

The authors offer a very flexible text to meet a variety of approaches, including a traditional course on the topic. The text can be used in courses when partial differential equations replaces Laplace transforms. There is sufficient linear algebra in the text so that it can be used for a course that combines differential equations and linear algebra.

Most significantly, computer labs are given in MATLAB®, Mathematica®, and Maple. The book may be used for a course to introduce and equip the student with a knowledge of the given software. Sample course outlines are included.

Features





MATLAB®, Mathematica®, and Maple are incorporated at the end of each chapter All three software packages have parallel code and exercises There are numerous problems of varying difficulty for both the applied and pure math major, as well as problems for engineering, physical science and other students. An appendix that gives the reader a "crash course" in the three software packages Chapter reviews at the end of each chapter to help the students review Projects at the end of each chapter that go into detail about certain topics and introduce new topics that the students are now ready to see Answers to most of the odd problems in the back of the book
About the Authors xiii
Preface xv
1 Traditional First-Order Differential Equations
1(68)
1.1 Introduction to First-Order Equations
1(6)
1.2 Separable Differential Equations
7(6)
1.3 Linear Equations
13(11)
1.4 Some Physical Models Arising as Separable Equations
24(9)
1.5 Exact Equations
33(10)
1.6 Special Integrating Factors and Substitution Methods
43(26)
1.6.1 Bernoulli Equation
45(2)
1.6.2 Homogeneous Equations of the Form g(y/x)
47(7)
Chapter 1 Review
54(1)
Computer Labs: MATLAB®, Maple™, Mathematica
55(11)
Chapter 1 Projects
66(1)
Project 1A Particles in the Atmosphere
66(1)
Project 1B Insights into Graphing
67(2)
2 Geometrical and Numerical Methods for First-Order Equations
69(60)
2.1 Direction Fields---Geometry of Differential Equations
69(4)
2.2 Existence and Uniqueness for First-Order Equations
73(5)
2.3 First-Order Autonomous Equations---Geometrical Insight
78(13)
2.3.1 Graphing Factored Polynomials
85(3)
2.3.2 Bifurcations of Equilibria
88(3)
2.4 Modeling in Population Biology
91(8)
2.4.1 Nondimensionalization
94(5)
2.5 Numerical Approximation: Euler and Runge--Kutta Methods
99(8)
2.6 Introduction to Autonomous Second-Order Equations
107(22)
Chapter 2 Review
112(2)
Computer Labs: MATLAB, Maple, Mathematica
114(13)
Chapter 2 Projects
127(1)
Project 2A Spruce Budworm
127(1)
Project 2B Multistep Methods of Numerical Approximation
127(2)
3 Elements of Higher-Order Linear Equations
129(72)
3.1 Introduction to Higher-Order Equations
129(8)
3.1.1 Operator Notation
133(4)
3.2 Linear Independence and the Wronskian
137(8)
3.3 Reduction of Order---the Case n = 2
145(5)
3.4 Numerical Considerations for nth-Order Equations
150(3)
3.5 Essential Topics from Complex Variables
153(7)
3.6 Homogeneous Equations with Constant Coefficients
160(9)
3.7 Mechanical and Electrical Vibrations
169(32)
Chapter 3 Review
183(2)
Computer Labs: MATLAB, Maple, Mathematica
185(13)
Chapter 3 Projects
198(1)
Project 3A Runge--Kutta Order 2
198(1)
Project 3B Stiff Differential Equations
199(2)
4 Techniques of Nonhomogeneous Higher-Order Linear Equations
201(68)
4.1 Nonhomogeneous Equations
201(9)
4.2 Method of Undetermined Coefficients via Superposition
210(11)
4.3 Method of Undetermined Coefficients via Annihilation
221(9)
4.4 Exponential Response and Complex Replacement
230(9)
4.5 Variation of Parameters
239(9)
4.6 Cauchy--Euler (Equidimensional) Equation
248(4)
4.7 Forced Vibrations
252(17)
Chapter 4 Review
257(2)
Computer Labs: MATLAB, Maple, Mathematica
259(8)
Chapter 4 Projects
267(1)
Project 4A Forced Duffing Equation
267(1)
Project 4B Forced van der Pol Oscillator
268(1)
5 Fundamentals of Systems of Differential Equations
269(84)
5.1 Useful Terminology
269(8)
5.2 Gaussian Elimination
277(6)
5.3 Vector Spaces and Subspaces
283(9)
5.3.1 The Nullspace and Column Space
287(5)
5.4 Eigenvalues and Eigenvectors
292(10)
5.5 A General Method, Part I: Solving Systems with Real and Distinct or Complex Eigenvalues
302(6)
5.6 A General Method, Part II: Solving Systems with Repeated Real Eigenvalues
308(11)
5.7 Matrix Exponentials
319(9)
5.8 Solving Linear Nonhomogeneous Systems of Equations
328(25)
Chapter 5 Review
334(4)
Computer Labs: MATLAB, Maple, Mathematica
338(12)
Chapter 5 Projects
350(1)
Project 5A Transition Matrix and Stochastic Processes
350(2)
Project 5B Signal Processing
352(1)
6 Geometric Approaches and Applications of Systems of Differential Equations
353(66)
6.1 An Introduction to the Phase Plane
353(8)
6.2 Nonlinear Equations and Phase Plane Analysis
361(9)
6.2.1 Systems of More Than Two Equations
366(4)
6.3 Bifurcations
370(9)
6.4 Epidemiological Models
379(13)
6.5 Models in Ecology
392(27)
Chapter 6 Review
400(2)
Computer Labs: MATLAB, Maple, Mathematica
402(13)
Chapter 6 Projects
415(1)
Project 6A An MSEIR Model
415(1)
Project 6B Routh--Hurwitz Criteria
416(3)
7 Laplace Transforms
419(68)
7.1 Introduction
419(8)
7.2 Fundamentals of the Laplace Transform
427(11)
7.3 The Inverse Laplace Transform
438(10)
7.3.1 Laplace Transform Solution of Linear Differential Equations
442(6)
7.4 Translated Functions, Delta Function, and Periodic Functions
448(8)
7.5 The s-Domain and Poles
456(5)
7.6 Solving Linear Systems Using Laplace Transforms
461(4)
7.7 The Convolution
465(22)
Chapter 7 Review
470(2)
Computer Labs: MATLAB, Maple, Mathematica
472(10)
Chapter 7 Projects
482(1)
Project 7A Carrier-Borne Epidemics
482(1)
Project 7B Integral Equations
483(4)
8 Series Methods
487(66)
8.1 Power Series Representations of Functions
487(10)
8.2 The Power Series Method
497(8)
8.3 Ordinary and Singular Points
505(8)
8.4 The Method of Frobenius
513(16)
8.5 Bessel Functions
529(24)
Chapter 8 Review
541(1)
Computer Labs: MATLAB, Maple, Mathematica
541(8)
Chapter 8 Projects
549(1)
Project 8A Asymptotic Series
549(1)
Project 8B Hypergeometric Functions
550(3)
9 Boundary-Value Problems and Fourier Series
553(40)
9.1 Two-Point Boundary-Value Problems
553(5)
9.2 Orthogonal Functions and Fourier Series
558(7)
9.3 Even, Odd, and Discontinuous Functions
565(6)
9.4 Simple Eigenvalue-Eigenfunction Problems
571(4)
9.5 Sturm--Liouville Theory
575(6)
9.6 Generalized Fourier Series
581(12)
Computer Labs: MATLAB, Maple, Mathematica
586(6)
Chapter 9 Projects
592(1)
Project 9 Chebyshev Polynomials
592(1)
10 Partial Differential Equations
593(60)
10.1 Separable Linear Partial Differential Equations
593(7)
10.2 Heat Equation
600(9)
10.3 Wave Equation
609(11)
10.4 Laplace Equation
620(10)
10.5 Nonhomogeneous Boundary Conditions
630(6)
10.6 Non-Cartesian Coordinate Systems
636(17)
Computer Labs: MATLAB, Maple, Mathematica
645(7)
Chapter 10 Projects
652(1)
Project 10 Finite Difference Approximation
652(1)
A An Introduction to MATLAB®, Maple™, and Mathematica
653(20)
A.1 MATLAB
653(8)
A.1.1 Some Helpful MATLAB Commands
655(3)
A.1.2 Programming with a script and a function in MATLAB
658(3)
A.2 Maple
661(5)
A.2.1 Some Helpful Maple Commands
662(2)
A.2.2 Programming in Maple
664(2)
A.3 Mathematica
666(7)
A.3.1 Some Helpful Mathematica Commands
668(1)
A.3.2 Programming in Mathematica
669(4)
B Selected Topics from Linear Algebra
673(34)
B.1 A Primer on Matrix Algebra
673(12)
B.2 Matrix Inverses, Cramer's Rule
685(7)
B.2.1 Calculating the Inverse of a Matrix
685(4)
B.2.2 Cramer's Rule
689(3)
B.3 Linear Transformations
692(9)
B.4 Coordinates and Change of Basis
701(6)
B.4.1 Similarity Transformations
703(4)
Computer Labs: MATLAB, Maple, Mathematica 707(10)
Answers to Odd Problems 717(38)
References 755(4)
Index 759
Stephen A. Wirkus is an associate professor of mathematics at Arizona State University, where he has been a recipient of the Professor of the Year Award and NSF AGEP Mentor of the Year Award. He has published over 30 papers and technical reports. He completed his Ph.D. at Cornell University under the direction of Richard Rand.

Randall J. Swift is a professor of mathematics and statistics at California State Polytechnic University, Pomona, where he has been a recipient of the Ralph W. Ames Distinguished Research Award. He has authored more than 80 journal articles, three research monographs, and three textbooks. He completed his Ph.D. at the University of California, Riverside under the direction of M.M. Rao.
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