Differential Equations : A Modeling Perspective （2 HAR/CDR）

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Differential Equations : A Modeling Perspective （2 HAR/CDR）

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  • 製本 Hardcover:ハードカバー版／ページ数 704 p.
  • 言語 ENG
  • 商品コード 9780471433323
  • DDC分類 515.35

Full Description

This effective and practical new edition continues to focus on differential equations as a powerful tool in constructing mathematical models for the physical world. It emphasizes modeling and visualization of solutions throughout. Each chapter introduces a model and then goes on to look at solutions of the differential equations involved using an integrated analytical, numerical, and qualitative approach. The authors present the material in a way that's clear and understandable to students at all levels. Throughout the text the authors convey their enthusiasm and excitement for the study of ODEs.

Contents

1: Modeling and Differential Equations. 1.1 The Modeling Approach. 1.2 A Modeling Adventure. 1.3 Models and Initial Value Problems. 1.4 The Modeling Process: Differential Systems. SPOTLIGHT ON MODELING: RADIOCARBON DATING. SPOTLIGHT ON MODELING: COLD MEDICATION I. 2: First-Order Differential Equations. 2.1 Linear Differential Equations. 2.2 Linear Differential Equations: Qualitative Analysis. 2.3 Existence and Uniqueness of Solutions. 2.4 Visualizing Solution Curves: Slope Fields. 2.5 Separable Differential Equations: Planar Systems. 2.6 A Predator-Prey Model: the Lotka-Volterra System. 2.7 Extension of Solutions: Long-Term Behavior. 2.8 Qualitative Analysis: State Lines, Sign Analysis. 2.9 Bifucations: A Harvested Logistic Model. Snapshot on Solution Formula Techniques. SPOTLIGHT ON APPROXIMATE NUMERICAL SOLUTIONS. SPOTLIGHT ON COMPUTER IMPLEMENTATION. SPOTLIGHT ON STEADY STATES: LINEAR ODES. SPOTLIGHT ON MODELING: COLD MEDICATION II. SPOTLIGHT ON CHANGE OF VARIABLES: PURSUIT MODELS. SPOTLIGHT ON CONTINUITY IN THE DATA. 3: Second-Order Differential Equations. 3.1 Models of Springs. 3.2 Undriven Constant-Coefficient Linear Differential Equations. 3.3 Visualizing Graphs of Solutions: Direction Fields. 3.4 Periodic Solutions: Simple Harmonic Motion. 3.5 Driven Linear ODEs: Undetermined Coefficients I. 3.6 Driven Linear ODEs: Undetermined Coefficients II. 3.7 Theory of Second-Order Linear Differential Equations. 3.8 Nonlinear Second-Order Differential Equations. A Snapshot Look at Constant-Coefficient Polynomial Operators. SPOTLIGHT ON MODELING: VERTICAL MOTION. SPOTLIGHT ON MODELING: SHOCK ABSORBERS. SPOTLIGHT ON EINSTEIN'S FIELD EQUATIONS. 4: Applications of Second-Order Differential Equations. 4.1 Newton's Laws: The Pendulum. 4.2 Beats and Resonance. 4.3 Frequency Response Modeling. 4.4 Electrical Circuits. Snapshot on Mechanical and Electrical Models. SPOTLIGHT ON MODELING: TUNING A CIRCUIT. 5: The Laplace Transform. 5.1 The Laplace Transform: Solving IVPs. 5.2 Working with the Transform. 5.3 Transforms of Periodic Functions. 5.4 Convolution. SPOTLIGHT ON THE DELTA FUNCTION. SPOTLIGHT ON MODELING: TIME DELAYS AND COLLISIONS. 6: Linear Systems of Differential Equations. 6.1 Compartment Models: Tracking Lead. 6.2 Eigenvalues, Eigenvectors and Eigenspaces of Matrices. 6.3 Undriven Linear Differential Systems: Real Eigenvalues. 6.4 Undriven Linear Systems: Complex Eigenvalues. 6.5 Orbital Portraits for Planar Systems. 6.6 Driven Systems: The Matrix Exponential. 6.7 Steady States. 6.8 The Theory of General Linear Systems. SPOTLIGHT ON VECTORS, MATRICES, INDEPENDENCE. SPOTLIGHT ON LINEAR ALGEBRAIC EQUATIONS. SPOTLIGHT ON BIFURCATIONS: SENSITIVITY. 7: Nonlinear Differential Systems. 7.1 Chemical Kinetics: The Fundamental Theorem. 7.2 Properties of Autonomous Systems, Direction Fields. 7.3 Interacting Species: Cooperation, Competition. SPOTLIGHT ON MODELING: DESTRUCTIVE COMPETITION. SPOTLIGHT ON MODELING: BIFURCATION AND SENSITIVITY. 8: Stability. 8.1 Stability and Linear Autonomous Systems. 8.2 Stability and Nonlinear Autonomous Systems. Stability of PlanarNonlinear Systems. 8.3 Conservative Systems. SPOTLIGHT ON LYAPUNOV FUNCTIONS. SPOTLIGHT ON ROTATING BODIES. 9: Nonlinear Systems: Cycles and Chaos. 9.1 Cycles. 9.2 Solution Behavior in Planar Autonomous Systems. 9.3 Bifucations. 9.4 Chaos. SPOTLIGHT ON CHAOTIC SYSTEMS. 10: Fourier Series and Partial Differential Equations. 10.1 Vibrations of a Guitar String. 10.2 Fourier Trigonometric Series. 10.3 Half-Range and Exponential Fourier Series. 10.4 Temperature in a Thin Rod. 10.5 Sturm-Liouville Problems. 10.6 The Method of Eigenfunction Expansions. SPOTLIGHT ON DECAY ESTIMATES. SPOTLIGHT ON THE OPTIMAL DEPTH FOR A WINE CELLAR. SPOTLIGHT ON APPROXIMATION OF FUNCTIONS. 11: Series Solutions. 11.1 The Method of Power Series. 11.2 Series Solutions Near an Ordinary Point. 11.3 Regular Singular Points: Euler's ODE. 11.4 Series Solutions Near Regular Singular Points. SPOTLIGHT ON LEGENDRE POLYNOMIALS. SPOTLIGHT ON BESSEL FUNCTIONS. Appendix A: Basic Theory of Initial Value Problems. A.1 Uniqueness. A.2 The Picard Process for Solving an Initial Value Problem. A.3 Extention of Solutions. Appendix B: Background Information. B.1 Power Series. B.2 Results from Calculus. Answers to Selected Problems. Index. Web Spotlights. 27 Additional SPOTLIGHTS appear on the text's Web site at www.wiley.com/college/borrelli.