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Applied Differential Equations with Boundary Value Problems [Kõva köide]

(Brown University, RI, USA)
  • Formaat: Hardback, 698 pages, kõrgus x laius: 280x210 mm, kaal: 1990 g, 20 Tables, black and white; 260 Illustrations, black and white
  • Sari: Textbooks in Mathematics
  • Ilmumisaeg: 16-Oct-2017
  • Kirjastus: Chapman & Hall/CRC
  • ISBN-10: 1498733654
  • ISBN-13: 9781498733656
Teised raamatud teemal:
Applied Differential Equations with Boundary Value ProblemsSuurem pilt
  • Formaat: Hardback, 698 pages, kõrgus x laius: 280x210 mm, kaal: 1990 g, 20 Tables, black and white; 260 Illustrations, black and white
  • Sari: Textbooks in Mathematics
  • Ilmumisaeg: 16-Oct-2017
  • Kirjastus: Chapman & Hall/CRC
  • ISBN-10: 1498733654
  • ISBN-13: 9781498733656
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781498733656.html
Märksõnad:
Applied Differential Equations with Boundary Value Problems presents a contemporary treatment of ordinary differential equations (ODEs) and an introduction to partial differential equations (PDEs), including their applications in engineering and the sciences. This new edition of the authors popular textbook adds coverage of boundary value problems.

The text covers traditional material, along with novel approaches to mathematical modeling that harness the capabilities of numerical algorithms and popular computer software packages. It contains practical techniques for solving the equations as well as corresponding codes for numerical solvers. Many examples and exercises help students master effective solution techniques, including reliable numerical approximations.

This book describes differential equations in the context of applications and presents the main techniques needed for modeling and systems analysis. It teaches students how to formulate a mathematical model, solve differential equations analytically and numerically, analyze them qualitatively, and interpret the results.
List of Symbols
xi
Preface xiii
1 Introduction
1(38)
1.1 Motivation
1(2)
1.2 Classification of Differential Equations
3(1)
1.3 Solutions to Differential Equations
4(4)
1.4 Particular and Singular Solutions
8(3)
1.5 Direction Fields
11(10)
1.6 Existence and Uniqueness
21(18)
Review Questions for
Chapter
36(3)
2 First Order Equations
39(98)
2.1 Separable Equations
39(14)
2.1.1 Autonomous Equations
47(6)
2.2 Equations Reducible to Separable Equations
53(20)
2.2.1 Equations with Homogeneous Coefficients
54(3)
2.2.2 Equations with Homogeneous Fractions
57(6)
2.2.3 Equations with Linear Coefficients
63(10)
2.3 Exact Differential Equations
73(7)
2.4 Simple Integrating Factors
80(6)
2.5 First-Order Linear Differential Equations
86(9)
2.6 Special Classes of Equations
95(16)
2.6.1 The Bernoulli Equation
95(4)
2.6.2 The Riccati Equation
99(6)
2.6.3 Equations with the Dependent or Independent Variable Missing
105(2)
2.6.4 Equations Homogeneous with Respect to Their Dependent Variable
107(2)
2.6.5 Equations Solvable for a Variable
109(2)
2.7 Qualitative Analysis
111(26)
2.7.1 Bifurcation Points
116(3)
2.7.2 Validity Intervals of Autonomous Equations
119(7)
Summary for
Chapter 2
126(2)
Review Questions for
Chapter 2
128(9)
3 Numerical Methods
137(50)
3.1 Difference Equations
138(8)
3.2 Euler's Methods
146(13)
3.3 The Polynomial Approximation
159(7)
3.4 Error Estimates
166(8)
3.5 The Runge-Kutta Methods
174(13)
Summary for
Chapter 3
182(1)
Review Questions for
Chapter 3
183(4)
4 Second and Higher Order Linear Differential Equations
187(82)
4.1 Second and Higher Order Differential Equations
188(10)
4.1.1 Linear Operators
190(1)
4.1.2 Exact Equations and Integrating Factors
191(2)
4.1.3 Change of Variables
193(5)
4.2 Linear Independence and Wronskians
198(6)
4.3 The Fundamental Set of Solutions
204(4)
4.4 Equations with Constant Coefficients
208(4)
4.5 Complex Roots
212(5)
4.6 Repeated Roots. Reduction of Order
217(7)
4.6.1 Reduction of Order
219(3)
4.6.2 Euler's Equations
222(2)
4.7 Nonhomogeneous Equations
224(15)
4.7.1 The Annihilator
225(3)
4.7.2 The Method of Undetermined Coefficients
228(11)
4.8 Variation of Parameters
239(7)
4.9 Bessel Equations
246(23)
4.9.1 Parametric Bessel Equation
249(1)
4.9.2 Bessel Functions of Half-Integer Order
250(1)
4.9.3 Related Differential Equations
250(5)
Summary for
Chapter 4
255(3)
Review Questions for
Chapter 4
258(11)
5 Laplace Transforms
269(72)
5.1 The Laplace Transform
270(12)
5.2 Properties of the Laplace Transform
282(9)
5.3 Discontinuous and Impulse Functions
291(11)
5.4 The Inverse Laplace Transform
302(12)
5.4.1 Partial Fraction Decomposition
302(4)
5.4.2 Convolution Theorem
306(2)
5.4.3 The Residue Method
308(6)
5.5 Homogeneous Differential Equations
314(6)
5.5.1 Equations with Variable Coefficients
318(2)
5.6 Nonhomogeneous Differential Equations
320(21)
5.6.1 Differential Equations with Intermittent Forcing Functions
324(7)
Summary for
Chapter 5
331(3)
Review Questions for
Chapter 5
334(7)
6 Introduction to Systems of ODEs
341(36)
6.1 Some ODE Models
341(15)
6.1.1 RLC-circuits
341(3)
6.1.2 Spring-Mass Systems
344(1)
6.1.3 The Euler-Lagrange Equation
345(1)
6.1.4 Pendulum
346(3)
6.1.5 Laminated Material
349(1)
6.1.6 Flow Problems
350(6)
6.2 Matrices
356(6)
6.3 Linear Systems of First Order ODEs
362(3)
6.4 Reduction to a Single ODE
365(6)
6.5 Existence and Uniqueness
371(6)
Summary for
Chapter 6
373(1)
Review Questions for
Chapter 6
374(3)
7 Topics from Linear Algebra
377(54)
7.1 The Calculus of Matrix Functions
377(3)
7.2 Inverses and Determinants
380(6)
7.2.1 Solving Linear Equations
383(3)
7.3 Eigenvalues and Eigenvectors
386(5)
7.4 Diagonalization
391(9)
7.5 Sylvester's Formula
400(5)
7.6 The Resolvent Method
405(7)
7.7 The Spectral Decomposition Method
412(19)
Summary for
Chapter 7
423(2)
Review Questions for
Chapter 7
425(6)
8 Systems of Linear Differential Equations
431(52)
8.1 Systems of Linear Equations
431(8)
8.1.1 The Euler Vector Equations
437(2)
8.2 Constant Coefficient Homogeneous Systems
439(16)
8.2.1 Simple Real Eigenvalues
444(3)
8.2.2 Complex Eigenvalues
447(2)
8.2.3 Repeated Eigenvalues
449(2)
8.2.4 Qualitative Analysis of Linear Systems
451(4)
8.3 Variation of Parameters
455(7)
8.3.1 Equations with Constant Coefficients
457(5)
8.4 Method of Undetermined Coefficients
462(3)
8.5 The Laplace Transformation
465(4)
8.6 Second Order Linear Systems
469(14)
Summary for
Chapter 8
475(2)
Review Questions for
Chapter 8
477(6)
9 Qualitative Theory of Differential Equations
483(62)
9.1 Autonomous Systems
483(9)
9.1.1 Two-Dimensional Autonomous Equations
485(7)
9.2 Linearization
492(8)
9.2.1 Two-Dimensional Autonomous Equations
493(4)
9.2.2 Scalar Equations
497(3)
9.3 Population Models
500(14)
9.3.1 Competing Species
500(4)
9.3.2 Predator-Prey Equations
504(6)
9.3.3 Other Population Models
510(4)
9.4 Conservative Systems
514(8)
9.4.1 Hamiltonian Systems
516(6)
9.5 Lyapunov's Second Method
522(7)
9.6 Periodic Solutions
529(16)
9.6.1 Equations with Periodic Coefficients
535(3)
Summary for
Chapter 9
538(1)
Review Questions for
Chapter 9
539(6)
10 Orthogonal Expansions
545(52)
10.1 Sturm-Liouville Problems
545(8)
10.2 Orthogonal Expansions
553(6)
10.3 Fourier Series
559(11)
10.3.1 Music as Motivation
559(2)
10.3.2 Sturm-Liouville Periodic Problem
561(1)
10.3.3 Fourier Series
562(8)
10.4 Convergence of Fourier Series
570(12)
10.4.1 Complex Fourier Series
574(3)
10.4.2 The Gibbs Phenomenon
577(5)
10.5 Even and Odd Functions
582(15)
Summary for
Chapter 10
590(1)
Review Questions for
Chapter 10
591(6)
11 Partial Differential Equations
597(32)
11.1 Separation of Variables for the Heat Equation
597(9)
11.1.1 Two-Dimensional Heat Equation
603(3)
11.2 Other Heat Conduction Problems
606(4)
11.3 Wave Equation
610(7)
11.3.1 Transverse Vibrations of Beams
615(2)
11.4 Laplace Equation
617(12)
11.4.1 Laplace Equation in Polar Coordinates
619(4)
Summary for
Chapter 11
623(1)
Review Questions for
Chapter 11
624(5)
12 Boundary Value Problems
629(40)
12.1 Green's Functions
629(6)
12.2 Green's Functions for Linear Systems
635(4)
12.3 Singular Sturm-Liouville Problems
639(8)
12.3.1 Green's Function
640(1)
12.3.2 Orthogonality of Bessel Functions
641(6)
12.4 Orthogonal Polynomials
647(12)
12.4.1 Chebyshev's Polynomials
647(2)
12.4.2 Legendre's Equation
649(4)
12.4.3 Hermite's Polynomials
653(2)
12.4.4 Laguerre's Polynomials
655(4)
12.5 Nonhomogeneous Boundary Value Problems
659(10)
Summary for
Chapter 12
666(1)
Review Questions for
Chapter 12
667(2)
Bibliography 669(2)
Index 671
Vladimir A. Dobrushkin is a Visiting Professor in the Center for Fluid MechanicsApplied Mathematics Department at Brown University.