×

Differential equations: A dynamical systems approach. Part 1: Ordinary differential equations. (English) Zbl 0724.34001

Texts in Applied Mathematics, 5. New York etc.: Springer-Verlag. xviii, 348 p. DM 78.00 (1990).
It is a first course in differential equations, intendent mainly to students of applied mathematics. In this part one-dimensional ODEs are considered. The main stress is put into geometrical aspects of the theory as well as to practical calculations using numerical methods. In spite of the subtitle, it does not present standard results from the theory of flows. Most part of the book is devoted to examples, many of them can be followed by personal computers. Chapter 1 presents the geometry of the extended phase plane and introduces new concepts of fence, funnel and antifunnel. Chapter 2 presents few analytical methods - separating of variables, exact and first order linear equations, solutions by power series. The Euler method, the midpoint Euler method and the Runge-Kutta method are introduced in Chapter 3. The main theorems on existence and uniqueness of solutions are presented in Chapter 4. Chapter 5 is devoted to iterations of functions of one variable. Relations to numerical methods are considered and simple facts concerning iterations in the complex domain are presented.

MSC:

34-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations
37E99 Low-dimensional dynamical systems
65J99 Numerical analysis in abstract spaces
65L05 Numerical methods for initial value problems involving ordinary differential equations
34A26 Geometric methods in ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
PDFBibTeX XMLCite