Mathematical Methods for Scientists and Engineers: Linear and Nonlinear Systems

Mathematical Methods for Scientists and Engineers: Linear and Nonlinear Systems

by Peter B. Kahn


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Appropriate for advanced undergraduate and graduate students in a variety of scientific and engineering fields, this text introduces linear and nonlinear problems and their associated models. The first part covers linear systems, emphasizing perturbation or approximation techniques and asymptotic methods. The second part comprises nonlinear problems, including weakly nonlinear oscillatory systems and nonlinear difference equations. The two parts, both of which include exercises, merge smoothly, and many of the nonlinear techniques arise from the study of the linear systems. 1990 edition. 70 figures. 4 tables. Appendix. Index.

Product Details

ISBN-13: 9780486435169
Publisher: Dover Publications
Publication date: 06/14/2004
Series: Dover Books on Mathematics Series
Pages: 496
Product dimensions: 6.20(w) x 9.24(h) x 0.97(d)

Table of Contents

Part ILinear Systems1
0Miscellaneous Resources3
0.2Numerical Results3
0.3Trigonometric Identities and Differential Equations4
0.4Taylor-Series Expansions6
0.6Graphs and Analytical Expressions8
1Matrix Theory14
1.1Notation and Preliminary Remarks15
1.1.1Matrices and Vectors
1.1.2Matrix Operations
1.1.3The Trace and the Determinant
1.1.4The Inverse
1.1.5Unitary Matrices
1.1.6Ill-Conditioned Matrices
1.1.7Eigenvalues and Eigenvectors
1.1.9Similarity Transformations
1.2Eigenvalues and Eigenvectors27
1.2.1Eigenvalues and Eigenvectors
1.2.2Notation for the Scalar Product
1.2.3The Concept of Linear Dependence and Linear Independence
1.3The Gram-Schmidt Procedure33
1.4Diagonalization of Matrices34
1.5The Cayley-Hamilton (CH) Theorem39
1.5.1Powers of a Matrix
1.5.2Difference Equations
1.5.3Functions of a Matrix
1.6Perturbation Theory54
1.7Concluding Remarks59
1.7.1Rayleigh's Inequality and Related Material
1.7.2Location of the Eigenvalues of a Matrix
2The Gamma and Related Functions71
2.1The Gamma Function71
2.2Dirichlet Integrals75
2.3Beta Functions79
2.4Applications and Exercises84
2.5The Riemann Zeta Function88
2.6The Dirac Delta Function91
3Elements of Asymptotics100
3.2The Asymptotic Symbols102
3.2.2The Big "O" Symbol and the Little "o" Symbol
3.3The Error Function115
3.4.1The Geometric Sum
3.4.2Connection Between a Sum and an Integral: "The Trapezoidal Rule"
3.5Asymptotic Series133
4Evaluation of Sums: The Euler - MacLaurin Sum Expansion138
4.1Derivation of the Euler-MacLaurin Summation Expansion139
4.2Sums That Depend Upon a Parameter150
4.3Divergent Sums156
5Evaluation of Integrals: The Laplace Method168
5.1General Ideas169
5.1.1Some Observations
5.1.2Introduction of the Breakpoint, [diamond]
5.1.3Primary Conclusions
5.2The Laplace Method186
5.2.1Finding the Dominant Behavior
5.2.2An Interior Maximum
5.2.3Maximum at an End Point
5.2.4The Next Complications
5.3Problems That Need Some Preparation194
5.4Higher-Order Terms197
6Differential Equations207
6.1Elimination of the Middle Term208
6.1.1Transformation of Variables
6.1.2An Oscillation Theorem
6.2Inhomogeneous Equations214
6.2.1The Second Solution
6.2.2Method of Variation of Parameters
6.3The Liouville-Green Transformation217
6.4Bellman's Inequality--Gronwall's Lemma223
6.5The WKB Approximation227
6.6Perturbation Theory231
Part IINonlinear Systems239
7The Simple Harmonic Oscillator and the Logistic Equation241
7.1The Simple Harmonic Oscillator (SHO)242
7.2Transformation of Our Equations243
7.4Linear Transformations245
7.5The Logistic Equation250
7.5.1The Exponential
7.5.2The Logistic Equation
8Aspects of Harmonic Motion and the Concept of Secular Terms255
8.1The Flashing Clock256
8.2Secular Terms258
8.3The Forced Harmonic Oscillator259
8.4The Altered Simple Harmonic Oscillator262
9Equilibrium Points and the Phase Plane271
9.1Equilibrium Points271
9.2The Phase Plane280
9.3Sign Conventions286
10Conservative Systems291
10.1Review of the Basic Ideas291
10.2Properties of Conservative Systems292
10.3Orientation to the Spirit of the Calculations298
10.3.2Organization of the Calculation
10.4The Poincare-Lindstedt Method301
10.4.1Computation of x[subscript 1]
10.5Another Viewpoint309
11Nonconservative Systems314
11.1Damped Harmonic Motion314
11.1.1Simplifying Assumptions
11.2Limit Cycles: A Nonlinear Phenomenon320
11.2.1Definition of a Limit Cycle
11.2.2An Example of a Limit Cycle
11.3Discussion of Figures 11.6a and 11.6b331
12The Method of Averaging (MOA)333
12.1Orientation and Introduction of Our Assumptions334
12.1.1The Elementary Method of Averaging (MOA)
12.1.2The Method of Krylov-Bogoliubov and Mitropolsky (KBM)
12.1.3The Method of Bogoliubov and Mitropolsky or The Method of Rapidly Rotating Phase (MRRP)
12.1.4The Basic Model
12.2The Method of Averaging (MOA)336
12.2.1Sign Convention I: dx / dt = y
12.2.2Sign Convention II: dx / dt = -y
12.2.3Basic Assumptions of the MOA
12.3Examples and Exercises341
12.4Conclusions and Cautions354
13The Method of Multiple Times Scales (MMTS)356
13.2The Concept of Time Scales: The MMTS357
13.3The Equivalence of the MOA and MMTS364
13.4Complex Notation367
13.5Exercises and Examples369
14Higher-Order Calculations372
14.1Second-Order Calculations for Conservative Systems372
14.1.1First Choice for the Homogeneous Solution: C = 0
14.1.2Second Choice for the Homogeneous Solution: C [not equal] 0
14.2The Role of the Initial Conditions in the Expansion Procedure379
14.3A Quadratic Oscillator383
14.4.1The Pendulum
14.4.2Nonlinear Damping
14.4.3An Oscillator with Quadratic and Cubic Terms
14.4.4Comment on the Method of Harmonic Balance
14.5Limit Cycles Arising from "Quadratic Terms"402
15Error Analysis407
15.1Outline of Our Approach408
15.2Basic Tools for the Analysis of the Error408
15.3Error Analysis for the Altered Simple Harmonic Oscillator411
15.4The Duffing Oscillator413
Bibliography for Chapters 7 to 15417
16One-Dimensional Iterative Maps and the Onset of Chaos421
16.3Review of the Logistic Differential Equation427
16.4Introduction of a Map431
16.5Fixed Points, Attractors, Repellers, and Multipliers432
16.6Period-2 Cycles and the Pathway to Chaos437
16.6.2Shift of the Origin
16.6.3Calculation of the Period-2 Cycle
16.7The Iteration Process443
16.8A Derivation of the Approximate Feigenbaum Numbers447
16.8.1The Scaled Map
16.8.2The Relationship Between the Numbers A[subscript i] and the Parameters a[subscript i]
16.8.3The Feigenbaum Number [delta]
16.8.4The Feigenbaum Number [alpha]
16.8.5The Convergence of the A Sequence
16.9Concluding Remarks456
AppendixA Discussion of Euler's Constant458

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