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Differential Equations with Boundary Value Problems
Selwyn Hollis, Armstrong Atlantic State University

ISBN-10: 0130159271
ISBN-13: 9780130159274

Publisher: Prentice Hall
Copyright: 2002
Format: Cloth; 635 pp
Published: 11/27/2001
Status: Out of Print

Suggested retail price: $131.80
We're sorry, this product is no longer available. Please contact your Pearson rep if you are using this product and need instructor resources.

For undergraduate (sophomore/junior) courses in Differential Equations. For students majoring in Mathematics, Engineering, Physical Sciences, Biological Science, or Computer Science. Assumes knowledge of Calculus.

A solid introduction to Differential Equations and their applications emphasizing analytical, qualitative, and numerical methods.

Applications—Numerous applications such as population models, circuits, and mechanical systems are emphasized throughout. A full chapter on applications of nonlinear systems contains elementary treatments of several models from mathematical biology (Lotka-Volterra population models, the SIR, SIS, and SIRS models of the spread of infectious disease, the chemostat, and the Fitzhugh-Nagumo model of neural impulses), chemical reactions (the Brussekator and the Field-Noyes model of the Belousov-Zhabotinski reaction), and complex mechanical systems (the double pendulum and the three-body problem).

Modern Coverage—Numerical methods are introduced early in the text, including a discussion of error estimates for the Euler, Heun, and Runge-Kutta methods. Systems and the phase plane are also introduced early, first in the context of pairs first-order equations, and then in the context of second-order linear equations.

Flexibility—The organization of the text allows professors a great deal of flexibility in its use. With a judicious choice of sections, the text can be used in a traditional applications-oriented course, a modern technology-based course, or a more theoretical course for mathematics majors.

Over 1000 problems of varying levels of difficulty—Many are routine computational exercises; many others reinforce concepts, explore related concepts, or preview coming concepts.

Illustrations—Visualization is an important theme. Includes more than 200 figures, most created with Mathematica by the author.

1. Introduction.

Prologue: What Are Differential Equations? Four Introductory Models. Fundamental Concepts and Terminology.

2. Linear First-Order Equations.

Methods of Solution. Some Elementary Applications. Generalized Solutions.

3. Nonlinear First-Order Equations I.

Direction Fields and Numerical Approximation. Separable Equations. Bernoulli and Riccati Equations. Reduction of Order. Nonlinear First-order Equations in Applications.

4. Nonlinear First-Order Equations II.

Construction of Local Solutions. Existence and Uniqueness. Qualitative and Asymptotic Behavior. The Logistic Population Model. Numerical Methods. A First Look at Systems.

5. Linear Second-Order Equations I.

Introduction: Modeling Vibrations. State Variables and Numerical Approximation. Operators and Linearity. Solutions and Linear Independence. Variation of Constants and Green's Functions. Power-Series Solutions. Polynomial Solutions.

6. Linear Second-Order Equations II.

Homogeneous Equations with Constant Coefficients. Exponential Shift. Complex Roots. Real Solutions from Complex Solutions. Unforced Vibrations. Periodic Force and Response.

7. The Laplace Transform.

Definition and Basic Properties. More Transforms and Further Properties. Heaviside Functions and Piecewise-Defined Inputs. Periodic Inputs. Impulses and the Dirac Distribution. Convolution.

8. Linear First-Order Systems.

Introduction. Two Ad Hoc Methods. Vector-Valued Functions and Linear Independence. Evolution Matrices and Variation of Constants. Autonomous Systems: Eigenvalues and Eigenvectors. eAT</D> and the Cayley-Hamilton Theorem. Asymptotic Stability.

9. Geometry of Autonomous Systems in the Plane.

The Phase Plane. Phase Portraits of Homogeneous Linear Systems. Phase Portraits of Nonlinear Systems. Limit Cycles. Beyond the Plane.

10. Nonlinear Systems in Applications.

Lotka-Volterra Systems in Ecology. Infectious Disease and Epidemics. Other Biological Models. Chemical Systems. Mechanics.

11. Diffusion Problems and Fourier Series.

The Basic Diffusion Problem. Solutions by Separation of Variables. Fourier Series. Fourier Sine and Cosine Series. Sturm-Liouville Eigenvalue Problems. Singular Sturm-Liouville Problems. Eigenfunction Expansions.

12. Further Topics in PDEs.

The Wave Equation. The 2-D Laplace Equation. The 2-D Diffusion Equation.

Appendices.

Differential Equations (Mathematics)

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