János
Karsai
Models of Impulsive Phenomena
Mathematica experiments
Typotex Publishing
Company, Budapest, 2002
Table of contents
Preface
How
to use the book and the electronic attachment
I. Modeling
preliminaries
1. Visualization exercises
1.1. Computer
visualization: advantages and dangers
1.2. Animation
1.3. Visualization
of the path of a motion
2. Elements of computer-aided investigation of
differential equations
2.1. Concepts,
definitions
2.2. The
built-in Mathematica functions
2.3. The
ODESolve package
II. Properties
of impulsive systems
3. Introduction
3.1. Some
examples
3.2.
The Dirac-δ function
3.3. The
Dirac-δ in differential equations
4. Impulses at fixed instants
4.1. Definitions,
basic properties
4.2. The
IDESolve package
4.3. Modeling
scheme for 1D systems
4.4. Modeling
scheme for 2D systems
5. State-dependent impulses
5.1. Impulses
at variable instants
5.2. Autonomous
impulsive systems
5.3. General
impulses
5.4. Visualization
of general impulses
5.5. General
solving program: IDERKSolve
5.6. The
beating phenomenon
6. Computer-aided study of phase mapping
6.1. Theoretical
overview
6.2. Phase
maps of ordinary differential equations
6.3. Phase
maps of impulsive systems
7. Linear impulsive systems
7.1. General
properties
7.2. Linear
periodic systems
8. Stability
8.1. Concepts,
definitions
8.2. Stability
of linear systems
8.3. Stability
by linear approximation
8.4. Linear approximation in the case of variable impulse instants
9. Liouville-formula, phase-volume method
9.1. Theoretical
overview
9.2. Phase-volume
experiments for differential equations
9.3. Phase-volume
experiments for impulsive systems
10. Stability by the Liapunov’s direct method
10.1. Overview,
stability theorems
10.2. Experiments
for differential equations
10.3. Experiments
for impulses at fixed instants
10.4. Experiments
for impulses at variable instants
III. Some applications
11. Drug dosage
11.1. Compartmental
systems
11.2. Infusion
or injection: systems of one compartment
11.3. Infusion
or tablets: systems of two compartments
12. Impulses in population dynamics
12.1. Control
of the Malthusian model
12.2. Population
growth in restricted life space
13. Oscillators with impulsive perturbation
13.1. Oscillator
equations of one degree of freedom
13.2. Harmonic
oscillators with impulsive damping
13.3. Beating
in oscillators
13.4. Impulsive
external forces
13.5. Impulsive
relaxation oscillations
14. Beating, reflection, control
14.1. Introductory
examples
14.2. Playing
with a ball
14.3. Systems
of impulsive control
IV. Appendix
15. References
16. Mathematica
functions developed for the book
17. Notations
18. Index
V. Electronic
attachment. Add-on chapters
19. Mathematica fundamentals
19.1. Concepts,
data, variables, functions, lists
19.2. 2D
graphics
19.3. 3D
graphics
19.4. Solving
equations
19.5 Calculus:
limit, derivative, integral, series
20. Some useful method in modeling
20.1. Data:
plotting and transformations
20.2. Linear
approximation
20.3. Fixed
points, zero points, and iterations
20.4. Maxima
and minima
20.4. Curve
fitting
21. Modeling schemes for ordinary differential
equations
21.1. Modeling
scheme for 1D equations
21.2.
Modeling scheme for 2D equations