Fundamentals of differential equations and boundary value problems
著者
書誌事項
Fundamentals of differential equations and boundary value problems
Pearson Addison-Wesley, c2008
5th ed
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注記
Includes index
XISBN from back cover
内容説明・目次
内容説明
Fundamentals of Differential Equations presents the basic theory of differential equations and offers a variety of modern applications in science and engineering. Available in two versions, these flexible texts offer the instructor many choices in syllabus design, course emphasis (theory, methodology, applications, and numerical methods), and in using commercially available computer software.
Fundamentals of Differential Equations, Seventh Edition is suitable for a one-semester sophomore- or junior-level course. Fundamentals of Differential Equations with Boundary Value Problems, Fifth Edition, contains enough material for a two-semester course that covers and builds on boundary value problems. The Boundary Value Problems version consists of the main text plus three additional chapters (Eigenvalue Problems and Sturm-Liouville Equations; Stability of Autonomous Systems; and Existence and Uniqueness Theory).
目次
- 1. Introduction Background Solutions and Initial Value Problems Direction Fields The Approximation Method of Euler Chapter Summary Technical Writing Exercises Group Projects for Chapter 1 A. Taylor Series Method B. Picard's Method C. Magnetic Dipole D. The Phase Line 2. First-Order Differential Equations Introduction: Motion of a Falling Body Separable Equations Linear Equations Exact Equations Special Integrating Factors Substitutions and Transformations Chapter Summary Technical Writing Exercises Group Projects for Chapter 2 A. Differential Equations in Clinical Medicine B. Torricelli's Law of Fluid Flow C. The Snowplow Problem D. Two Snowplows E. Clairaut Equations and Singular Solutions F. Multiple Solutions of a First-Order Initial Value Problem G. Designing a Solar Collector H. Asymptotic Behavior of Solutions to Linear Equations I. Utility Functions and Risk Aversion 3. Mathematical Models and Numerical Methods Involving First Order Equations Mathematical Modeling Compartmental Analysis Heating and Cooling of Buildings Newtonian Mechanics Electrical Circuits Improved Euler's Method Higher-Order Numerical Methods: Taylor and Runge-Kutta Chapter Summary Technical Writing Exercises Group Projects for Chapter 3 A. Dynamics of HIV Infection B. Aquaculture C. Curve of Pursuit D. Aircraft Guidance in a Crosswind E. Feedback and the Op Amp F. Bang-Bang Controls G. Market Equilibrium: Stability and Time Paths H. Stability of Numerical Methods I. Period Doubling and Chaos 4. Linear Second-Order Equations Introduction: The Mass-Spring Oscillator Homogeneous Linear Equations
- The General Solution Auxiliary Equations with Complex Roots Nonhomogeneous Equations: The Method of Undetermined Coefficients The Superposition Principle and Undetermined Coefficients Revisited Variation of Parameters Variable-Coefficient Equations Qualitative Considerations for Variable-Coefficient and Nonlinear Equations A Closer Look at Free Mechanical Vibrations A Closer Look at Forced Mechanical Vibrations Chapter Summary Technical Writing Exercises Group Projects for Chapter 4 A. Nonlinear Equations Solvable by First-Order Techniques B. Apollo Reentry C. Simple Pendulum D. Linearization of Nonlinear Problems E. Convolution Method F. Undetermined Coefficients Using Complex Arithmetic G. An Alternative to the Method of Undetermined Coefficients H. Asymptotic Behavior of Solutions 5. Introduction to Systems and Phase Plane Analysis Interconnected Fluid Tanks Elimination Method for Systems with Constant Coefficients Solving Systems and Higher-Order Equations Numerically Introduction to the Phase Plane Applications to Biomathematics: Epidemic and Tumor Growth Models Coupled Mass-Spring Systems Electrical Systems Dynamical Systems, Poincare Maps, and Chaos Chapter Summary Technical Writing Exercises Group Projects for Chapter 5 A. Designing a Landing System for Interplanetary Travel B. Things That Bob C. Hamiltonian Systems D. Strange Behavior of Competing Species - Part 1 E. Cleaning Up the Great Lakes 6. Theory of Higher-Order Linear Differential Equations Basic Theory of Linear Differential Equations Homogeneous Linear Equations with Constant Coefficients Undetermined Coefficients and the Annihilator Method Method of Variation of Parameters Chapter Summary Technical Writing Exercises Group Projects for Chapter 6 A. Computer Algebra Systems and Exponential Shift B. Justifying the Method of Undetermined Coefficients C. Transverse Vibrations of a Beam 7. Laplace Transforms Introduction: A Mixing Problem Definition of the Laplace Transform Properties of the Laplace Transform Inverse Laplace Transform Solving Initial Value Problems Transforms of Discontinuous and Periodic Functions Convolution Impulses and the Dirac Delta Function Solving Linear Systems with Laplace Transforms Chapter Summary Technical Writing Exercises Group Projects for Chapter 7 A. Duhamel's Formulas B. Frequency Response Modeling C. Determining System Parameters 8. Series Solutions of Differential Equations Introduction: The Taylor Polynomial Approximation Power Series and Analytic Functions Power Series Solutions to Linear Differential Equations Equations with Analytic Coefficients Cauchy-Euler (Equidimensional) Equations Method of Frobenius Finding a Second Linearly Independent Solution Special Functions Chapter Summary Technical Writing Exercises Group Projects for Chapter 8 A. Spherically Symmetric Solutions to Shrodinger's Equation for the Hydrogen Atom B. Airy's Equation C. Buckling of a Tower D. Aging Spring and Bessel Functions 9. Matrix Methods for Linear Systems Introduction Review 1: Linear Algebraic Equations Review 2: Matrices and Vectors Linear Systems in Normal Form Homogeneous Linear Systems with Constant Coefficients Complex Eigenvalues Nonhomogeneous Linear Systems The Matrix Exponential Function Chapter Summary Technical Writing Exercises Group Projects for Chapter 9 A. Uncoupling Normal Systems B. Matrix Laplace Transform Method C. Undamped Second-Order Systems D. Strange Behavior of Competing Species - Part II 10. Partial Differential Equations Introduction: A Model for Heat Flow Method of Separation of Variables Fourier Series Fourier Cosine and Sine Series The Heat Equation The Wave Equation Laplace's Equation Chapter Summary Technical Writing Exercises Group Projects for Chapter 10 A. Steady-State Temperature Distribution in a Circular Cylinder B. A Laplace Transform Solution of the Wave Equation C. Green's Function D. Numerical Method for ?u=f on a Rectangle 11. Eigenvalue Problems and Sturm-Liouville Equations Introduction: Heat Flow in a Nonuniform Wire Eigenvalues and Eigenfunctions Regular Sturm-Liouville Boundary Value Problems Nonhomogeneous Boundary Value Problems and the Fredholm Alternative Solution by Eigenfunction Expansion Green's Functions. Singular Sturm-Liouville Boundary Value Problems. Oscillation and Comparison Theory. Chapter Summary Technical Writing Exercises Group Projects for Chapter 11 A. Hermite Polynomials and the Harmonic Oscillator B. Continuous and Mixed Spectra C. Picone Comparison Theorem D. Shooting Method E. Finite-Difference Method for Boundary Value Problems 12. Stability of Autonomous Systems Introduction: Competing Species Linear Systems in the Plane Almost Linear Systems Energy Methods Lyapunov's Direct Method Limit Cycles and Periodic Solutions Stability of Higher-Dimensional Systems Chapter Summary Technical Writing Exercises Group Projects for Chapter 12 A. Solutions and Korteweg-de Vries Equation B. Burger's Equation C. Computing Phase Plane Diagrams D. Ecosystem on Planet GLIA-2 13. Existence and Uniqueness Theory Introduction: Successive Approximations Picard's Existence and Uniqueness Theorem Existence of Solutions of Linear Equations Continuous Dependence of Solutions Chapter Summary Technical Writing Exercises Appendices A. Newton's Method B. Simpson's Rule C. Cramer's Rule D. Method of Least Squares E. Runge-Kutta Precedure for n Equations Answers to Odd-Numbered Problems Index
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