Differential equations with Mathematica:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Amsterdam [u.a.]
Elsevier Acad. Press
2004
|
| Edition: | 3. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 876 S. |
| ISBN: | 0120415623 |
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Record in the Search Index
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Titel: Differential equations with Mathematica
Autor: Abell, Martha L
Jahr: 2004
Contents
Preface.xiii
1 Introduction to Differential Equations.I
1.1 Definitions and Concepts .2
1.2 Solutions of Differential Equations.6
1.3 Initial and Boundary-Value Problems .IB
1.4 Direction Fields.26
2 First-Order Ordinary Differential Equations.41
2.1 Theory of First-Order Equations: A Brief Discussion.41
2.2 Separation of Variables.46
Application: Kidney Dialysis .55
2.3 Homogeneous Equations.59
Application: Models of Pursuit .64
2.4 Exact Equations.69
2.5 Linear Equations .74
'2.5.1 Integrating Factor Approach.75
2.5.2 Variation of Parameters and the Method of Undetermined Coefficients . 86
Application: Antibiotic Production.89
2.6 Numerical Approximations of Solutions to First-Order Equations . 92
2.6.1 Built-in Methods.92
Contents
Application: Modeling the Spread of a Disease.
2.6.2 Other Numerical Methods .
3 Applications of First-Order Ordinary Differential Equations.119
119
3.1 Orthogonal Trajectories.
Application: Oblique Trajectories .
3.2 Population Growth and Decay .132
3.2.1 The Malthas Model.132
3.2.2 The Logistic Equation.^
Application: Harvesting.
Application: The logistic Difference Equation.l3^
3.3 Newton's Law of Cooling.332
3.4 Free-Falling Bodies.^3
4 Highei^Order Differential Equations.
4.1 Preliminary Definitions and Notation .I?3
4.1.1 Introduction .1^5
4.1.2 The nth-Order Ordinary Linear Differential Equation.ISO
4.1.3 Fundamental Set of Solutions .185
4.1.4 Existence of a Fundamental Set of Solutions.191
4.1.5 Reduction of Order.193
4.2 Solving Homogeneous Equations with Constant Coefficients . 196
4.2.2 Second-Order Equations.196
4.2.2 Higher-Order Equations.200
Application: Testing for Diabetes.211
4.3 Introduction to Solving Nonhomogeneous Equations
with Constant Coefficients.216
4.4 Nonhomogeneous Equations with Constant Coefficients:
The Method of Undetermined Coefficients .222
4.4.1 Second-Order Equations.223
4.4.2 Higher-Order Equations.239
4.5 Nonhomogeneous Equations with Constant Coefficients:
Variation of Parameters.248
4.5.1 Second-Order Equations.248
4.5.2 Higher-Order Nonhomogeneous Equations.252
Contents vii
4.6 Cauchy-Euler Equations.255
4.6.1 Second-Order Cauchy-Euler Equations.255
4.6.2 Higher-Order Cauchy-Euler Equations.261
4.6.3 Variation of Parameters.265
4.7 Series Solutions.268
4.7.1 Power Series Solutions about Ordinary Points.268
4.7.2 Series Solutions about Regular Singular Points.281
4.7.3 Method of Frobenius.283
Application: Zeros of the Bessel Functions of the First Kind.295
Application: The Wave Equation on a Circular Plate. 298
4.8 Nonlinear Equations. 304
5 Applications of Higher-Order Differential Equations .321
5.1 Harmonic Motion.321
5.1.1 Simple Harmonic Motion.321
5.1.2 Damped Motion.332
5.1.3 Forced Motion.346
5.1.4 Soft Springs.365
5.1.5 Hard Springs.368
5.1.6 Aging Springs.370
Application: Hearing Beats and Resonance.372
5.2 The Pendulum Problem.373
5.3 Other Applications.387
5.3.2 L-R-C Circuits.387
5.3.2 Deflection of a Beam.390
5.3.3 Bode Plots.393
5.3.4 The Catenary.398
6 Systems of Ordinary Differential Equations
6.1 Review of Matrix Algebra and Calculus.411
6.1.1 Defining Nested Lists, Matrices, and Vectors .411
6.1.2 Extracting Elements of Matrices .416
6.1.3 Basic Computations with Matrices .419
6.1.4 Eigenvalues and Eigenvectors.422
6.2.5 Matrix Calculus.426
6.2 Systems of Equations: Preliminary Definitions and Theory.427
6.2.2 Preliminary Theory.429
Contents
446
6.2.2 Linear Systems.-.
6.3 Homogeneous Linear Systems with Constant Coefficients.454
6.3.2 Distinct Real Eigenvalues.454
6.3.2 Complex Conjugate Eigenvalues .461
6.3.3 Alternate Method for Solving Initial-Value Problems.474
6.3.4 Repeated Eigenvalues.477
6.4 Nonhomogeneous First-Order Systems: Undetermined
Coefficients, Variation of Parameters, and the Matrix Exponential . 485
6.4.1 Undetermined Coefficients.485
6.4.2 Variation of Parameters . .490
6.4.3 The Matrix Exponential .498
6.5 Numerical Methods.506
6.5.2 Built-in Methods.506
Application: Controlling the Spread of a Disease.513
6.5.2 Euler's Method.525
6.5.3 Runge-Kutta Method.531
6.6 Nonlinear Systems, Linearization, and Classification of
Equilibrium Points.535
6.6.2 Real Distinct Eigenvalues.535
6.6.2 Repeated Eigenvalues.543
6.6.3 Complex Conjugate Eigenvalues .548
6.6.4 Nonlinear Systems.552
Applications of Systems of Ordinary Differential Equations.567
7.1 Mechanical and Electrical Problems with First-Order
Linear Systems .567
7.2.2 L-R-C Circuits with Loops.567
7.2.2 L-R-C Circuit with One Loop.568
7.2.3 L-R-C Circuit with Two Loops.571
7.1.4 Spring-Mass Systems.574
7.2 Diffusion and Population Problems with First-Order
Linear Systems.576
7.2.2 Diffusion through a Membrane.576
7.2.2 Diffusion through a Double-Walled Membrane .578
7.2.3 Population Problems.533
7.3 Applications that Lead to Nonlinear Systems.587
7.3.2 Biological Systems: Predator-Prey Interactions, The Lotka-Volterra System,
and Food Chains in the Chemostat.587
Contents '*
7.3.2 Physical Systems: Variable Damping.604
7.3.3 Differeittial Geometry: Curvature.611
8 Laplace Transform Methods.617
8.1 The Laplace Transform.618
8.1.1 Definition of the Laplace Transform.618
8.1.2 Exponential Order.621
8.1.3 Properties of the Laplace Transform.623
8.2 The Inverse Laplace Transform.629
8.2.1 Definition of the Inverse Laplace Transform.629
8.2.2 Laplace Transform of an Integral .635
8.3 Solving Initial-Value Problems with the Laplace Transform.637
8.4 Laplace Transforms of Step and Periodic Functions.645
8.4.1 Pieceivise-Defined Functions: The Unit Step Function.645
8.4.2 Solving Initial-Value Problems.649
8.4.3 Periodic Functions.652
8.4.4 Impulse Functions: The Delta Function.661
8.5 The Convolution Theorem.667
8.5.1 The Convolution Theorem.667
8.5.2 Integral and Integrodifferential Equations.669
8.6 Applications of Laplace Transforms, Part I.672
8.6.1 Spring-Mass Systems Revisited.672
8.6.2 L-R-C Circuits Revisited.679
8.6.3 Population Problems Revisited.687
Application: The Tautochrone.689
8.7 Laplace Transform Methods for Systems. 691
8.8 Applications of Laplace Transforms, Part II. 708
8.8.1 Coupled Spring-Mass Systems. 708
8.8.2 The Double Pendulum. 714
Application: Free Vibration of a Three-Story Building. 720
9 Eigenvalue Problems and Fourier Series
9.1 Boundary-Value Problems, Eigenvalue Problems,
Sturm-Liouville Problems.
9.1.1 Boundary-Value Problems .
727
727
Contents
9.1.2 Eigenvalue Problems.
9.1.3 Sturm-Liouville Problems .
9.2 Fourier Sine Series and Cosine Series.787
9.2.1 Fourier Sine Series.
9.2.2 Fourier Cosine Series.
749
9.3 Fourier Series.
749
9.3.2 Fourier Series .
9.3.2 Even, Odd, and Periodic Extensions.758
9.3.3 Differentiation and Integration of Fourier Series.764
9.3.4 Parseval's Equality.7^8
9.4 Generalized Fourier Series.77^
10 Partial Differential Equations .788
10.1 Introduction to Partial Differential Equations and
Separation of Variables.788
10.1.1 Introduction.788
10.1.2 Separation of Variables.788
10.2 The One-Dimensional Fleat Equation.787
10.2.1 The Heat Equation with Homogeneous Boundary Conditions.787
10.2.2 Nonhomogeneous Boundary Conditions.791
10.2.3 Insulated Boundary.795
10.3 The One-Dimensional Wave Equation.799
20.3.1 The Wave E^tmfion.799
10.3.2 D'Alembert's Solution .806
10.4 Problems in Two Dimensions: Laplace's Equation.810
10.4.2 Laplace's Equation.810
10.5 Two-Dimensional Problems in a Circular Region.817
10.5.1 Laplace's Equation in a Circular Region.817
10.5.2 The Wave Equation in a Circular Region .821
10.5.3 Other Partial Differential Equations.836
Appendix: Getting Started.841
Introduction to Mathematica.841
A Note Regarding Different Versions of Mathematica.843
Getting Started with Mathematica.843
Five Basic Rules of Mathematica Syntax.849
Contents xl
Loading Packages.850
A Word of Caution.853
Getting Help from Mathematica.854
Mathematica Help .858
The Mathematica Menu.863
Bibliography.865
Index.867 |
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| spelling | Abell, Martha L. 1962- Verfasser (DE-588)130087025 aut Differential equations with Mathematica Martha L. Abell ; James P. Braselton 3. ed. Amsterdam [u.a.] Elsevier Acad. Press 2004 XVI, 876 S. txt rdacontent n rdamedia nc rdacarrier Mathematica (Computer file) Datenverarbeitung Differential equations Data processing Datenverarbeitung (DE-588)4011152-0 gnd rswk-swf Mathematica Programm (DE-588)4268208-3 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 s Mathematica Programm (DE-588)4268208-3 s DE-604 Datenverarbeitung (DE-588)4011152-0 s 1\p DE-604 Braselton, James P. Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016581630&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Abell, Martha L. 1962- Braselton, James P. Differential equations with Mathematica Mathematica (Computer file) Datenverarbeitung Differential equations Data processing Datenverarbeitung (DE-588)4011152-0 gnd Mathematica Programm (DE-588)4268208-3 gnd Differentialgleichung (DE-588)4012249-9 gnd |
| subject_GND | (DE-588)4011152-0 (DE-588)4268208-3 (DE-588)4012249-9 |
| title | Differential equations with Mathematica |
| title_auth | Differential equations with Mathematica |
| title_exact_search | Differential equations with Mathematica |
| title_exact_search_txtP | Differential equations with Mathematica |
| title_full | Differential equations with Mathematica Martha L. Abell ; James P. Braselton |
| title_fullStr | Differential equations with Mathematica Martha L. Abell ; James P. Braselton |
| title_full_unstemmed | Differential equations with Mathematica Martha L. Abell ; James P. Braselton |
| title_short | Differential equations with Mathematica |
| title_sort | differential equations with mathematica |
| topic | Mathematica (Computer file) Datenverarbeitung Differential equations Data processing Datenverarbeitung (DE-588)4011152-0 gnd Mathematica Programm (DE-588)4268208-3 gnd Differentialgleichung (DE-588)4012249-9 gnd |
| topic_facet | Mathematica (Computer file) Datenverarbeitung Differential equations Data processing Mathematica Programm Differentialgleichung |
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