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1988 | Buch

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Second Course in Ordinary Differential Equations for Scientists and Engineers

verfasst von: Mayer Humi, William Miller

Verlag: Springer US

Buchreihe : Universitext

Enthalten in: Professional Book Archive

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Über dieses Buch

The world abounds with introductory texts on ordinary differential equations and rightly so in view of the large number of students taking a course in this subject. However, for some time now there is a growing need for a junior-senior level book on the more advanced topics of differential equations. In fact the number of engineering and science students requiring a second course in these topics has been increasing. This book is an outgrowth of such courses taught by us in the last ten years at Worcester Polytechnic Institute. The book attempts to blend mathematical theory with nontrivial applications from varipus disciplines. It does not contain lengthy proofs of mathemati~al theorems as this would be inappropriate for its intended audience. Nevertheless, in each case we motivated these theorems and their practical use through examples and in some cases an "intuitive proof" is included. In view of this approach the book could be used also by aspiring mathematicians who wish to obtain an overview of the more advanced aspects of differential equations and an insight into some of its applications. We have included a wide range of topics in order to afford the instructor the flexibility in designing such a course according to the needs of the students. Therefore, this book contains more than enough material for a one semester course.

Inhaltsverzeichnis

Frontmatter
Chapter 0. Review
Abstract
Probably one of the least understood topics in a beginning ordinary differential equations course is finding a series solution of a given equation. Unfortunately this lack of knowledge hinders a student’s understanding of such important functions as Bessel’s function, Legendre polynomials and other such functions which arise in modern engineering problems.
Mayer Humi, William Miller
Chapter 1. Boundary Value Problems
Abstract
When a person begins the study of ordinary differential equations, he is usually confronted first by initial value problems, i.e. a differential equation plus conditions which the solution must satisfy at a given point x = x0.
Mayer Humi, William Miller
Chapter 2. Special Functions
Abstract
In the year 1812 Johann Frederick Carl Gauss published a comprehensive work studying the hypergeometric series which has the form
$$1+\frac{\alpha \beta}{\gamma}\ {x}+\frac{\alpha(1+\alpha)\beta(1+\beta)}{2!\gamma(1+\gamma)}{x}^{2}\ +\frac{\alpha(1+\alpha)(2+\alpha)\beta (1+\beta)(2+\beta)}{3!\gamma(1+\gamma)(2+\gamma)}{x}^{3}\ +\ ...$$
where α, β, γ, x are real numbers. The name hypergeometric comes from the fact that this series is a generalization of the geometric series. In fact, if we set α = 1 and β = γ in (1.1) above we have
$$1\ +{x}\ +\ {x}^{2}\ +\ {x}^{3}\ +\ ...$$
which is the well-known geometric series.
Mayer Humi, William Miller
Chapter 3. Systems of Ordinary Differential Equations
Abstract
In this chapter we discuss systems of first order ordinary differential equations and methods for their solutions. The importance of such systems stems from the following observations:
1.
A coupled system of differential equations of any order can always be rewritten as a system of first order equations.
 
Mayer Humi, William Miller
Chapter 4. Applications of Symmetry Principles to Differential Equations
Abstract
The application of symmetry principles to differential equations might lead to their simplification or solution in many important cases. Furthermore, it yields insights into the properties of these solutions as well as inter-relations in between them. To see how this is done we present, in this chapter, a short introduction to Lie groups and Lie algebras and then explore the various applications of these algebraic tools to several classes of first and second order equations.
Mayer Humi, William Miller
Chapter 5. Equations with Periodic Coefficients
Abstract
In this chapter we discuss differential equations whose coefficients are periodic and the properties of their solutions. Such equations appear in various fields of science, e.g., solid state physics, celestial mechanics and others. Our objective is then to investiage the implications of periodicity on these systems properties and behavior. From another point of view, many physical systems are invariant with respect to certain transformations of the independent and dependent variables. Accordingly, the corresponding differential equations which model these systems are invariant under the same transformations. However, surprisingly not all solutions to these equations are invariant with respect to these same transformations. We shall illustrate this phenomena (and its partial resolution) within the context of periodic equations.
Mayer Humi, William Miller
Chapter 6. Greens’s Functions
Abstract
In many important applications it is desirable to find an integral representation for the solution to a boundary value problem. To achieve this goal we first discuss in this chapter “nonsmooth” solutions to such problems and then show how the existence of such solutions enable us to solve our original problem.
Mayer Humi, William Miller
Chapter 7. Perturbation Theory
Abstract
Many mathematical models of real life systems are approximations. These approximations are generally made to simplify the model equations and make them mathematically tractable. However, after the initial successes of such approximations in predicting the general behavior of the system under consideration attention must turn to an examination of the effects of these approximations on the general behavior of the system and its equilibrium states. In most cases, however, the new additional terms in the model equations lead to nonlinearities which preclude any attempt to solve these equations in a closed form. As a result a practical technique had evolved to estimate the effect of these additional terms through some perturbation expansion. In this chapter we shall give a brief introduction to this branch of applied mathematics mostly through examples.
Mayer Humi, William Miller
Chapter 8. Phase Diagrams and Stability
Abstract
In most realistic models of scientific and engineering systems one is confronted with the necessity to analyze and solve systems of nonlinear differential equations. However, since the search for exact analytic solutions of such systems is, in most instances hopeless, it is natural to inquire in retrospect what is the most crucial information that has to be extracted from these equations. One discovers then that many such systems have transient states which are time dependent and equilibrium states which are time independent states. The equilibrium states are usually the most significant from a practical point of view and their stability against small perturbations and/or small changes in the system parameters is a central problem in the design and analysis of these systems.
Mayer Humi, William Miller
Chapter 9. Catastrophes and Bifurcations
Abstract
In the previous chapter we discussed various methods to analyze the stability of the equilibrium states of a dynamical system when the values of the system parameters are known and fixed. The objective of catastrophe and bifurcation theory is to investigate what happens to the type, number, and stability of the equilibrium states as a result of a continuous change in the system parameters. In other words catastrophe theory is concerned with the “dynamical analysis” of the equilibrium states as a function of the system parameters as compared to the “static analysis” of these states which were performed in the last chapter.
Mayer Humi, William Miller
Chapter 10. Sturmian Theory
Abstract
In many applications we deal with differential equations or systems which cannot be solved in closed form. Previously we showed how to cope with this situation when one considers only the equilibrium solutions of such equations. However, in many other instances when the time dependent solution is needed it is important to be able to derive upper and lower bounds as well as the properties (e.g. number of zeros) of the solution by solving proper approximations to the original equations. In this chapter we present an introduction to the theory developed by Sturm and others in an attempt to answer such questions.
Mayer Humi, William Miller
Backmatter
Metadaten
Titel
Second Course in Ordinary Differential Equations for Scientists and Engineers
verfasst von
Mayer Humi
William Miller
Copyright-Jahr
1988
Verlag
Springer US
Electronic ISBN
978-1-4612-3832-4
Print ISBN
978-0-387-96676-2
DOI
https://doi.org/10.1007/978-1-4612-3832-4