This Book >>

  • 244pp. Published December 2017
  • Scientific Research Publishing, Inc.,USA.
  • Category: Physics & Mathematics
  • ISBN: 978-1-61896-439-7
  • (Paperback) USD 89.00
  • ISBN: 978-1-61896-440-3
  • (E-Book) USD 29.00

Connect with SCIRP >>

Home > Books > Mathematical Modeling of Discontinuous P...
Mathematical Modeling of Discontinuous Processes
  • Description
  • E-Book
  • Author(s) Information

In this monograph as a mathematical apparatus are used and investigated several classes of differential equations. The most significant feature of these differential equations is the presence of impulsive effects.


The main goals and the results achieved in the monograph are related to the use of this class of equation for an adequate description of the dynamics of several types of processes that are subject to discrete external interventions and change the speed of development. In all proposed models the following requirements have met:

1) Presented and studied mathematical models in the book are extensions of existing known in the literature models of real objects and related processes.

2) Generalizations of the studied models are related to the admission of external impulsive effects, which lead to “jump-like” change the quantity characteristics of the described object as well as the rate of its modification.

3) Sufficient conditions which guarantee certain qualities of the dynamics of the quantities of the modeled objects are found.

4) Studies of the qualities of the modification of the modeled objects are possible to be successful by differential equations with variable structure and impulsive effects.

5) The considerations relating to the existence of the studied properties of dynamic objects cannot be realized without introducing new concepts and proving of appropriate theorems.


The main objectives can be conditionally divided into several parts: 1) New classes of differential equations with variable structure and impulses are introduced and studied; 2) Specific properties of the above-mentioned class of differential equations are introduced and studied.


The present monograph consists of an introduction and seven chapters. Each chapter contains several sections.

Components of the Book:
  • FRONT MATTER
    • Notations
    • Introduction
  • Chapter 1. Vibrations of the solutions of autonomous differential equations with variable structure and impulses. Application
    • 1. Vibrations of the solutions of autonomous differential equations with variable structure and impulses
    • 2. Gompertz model with variable structure and impulsive effects
  • Chapter 2. Continuous dependence of the solutions of nonautonomous differential equations with variable structure and impulses on switching functions
    • 1. The existence of switching moments for nonautonomous differential equations with variable structure and impulses
    • 2. Continuous dependence of the solutions of nonautonomous differential equations with variable structure and impulses
    • 3. Continuous dependence on permanent acting perturbations of bounded solutions of differential equations with variable structure.
  • Chapter 3. Stability and boundedness of solutions of differential equations with variable structure and impulses at unfixed switching moments via sequences of Lyapunov functions
    • 1. Stability of the zero solution of differential equations with variable structure and impulses at unfixed switching moments
    • 2. Boundedness of solutions of differential equations with variable structure and impulses at unfixed switching moments
  • Chapter 4. Method of limiting equations for stability of impulsive differential equations with variable impulsive moments of impulsive effects. Application
    • 1. Continuability of solutions of impulsive differential equations with variable moments of impulsive effects
    • 2. Stability of impulsive differential equations with variable moments of impulsive effects via limiting equations
    • 3. Stability of limiting equations, respective to impulsive differential equations with variable moments of impulsive effects
    • 4. Stable development of impulsive model of a predator-prey model
  • Chapter 5. Stability of nonzero solutions of impulsive differential equations with arbitrary impulsive moments. Application
    • 1. Stability of nonzero solutions of impulsive differential equations with arbitrary impulsive moments
    • 2. Logistic model of Verhulst with impulsive effects
  • Chapter 6. Asymptotic stability of nonzero solutions of differential equations with variable structure and unfixed moments of impulsive effects. Application
    • 1. Asymptotic stability of nonzero solutions of differential equations with variable structure and unfixed moments of impulsive effects
    • 2. Therapeutic model with variable structure and impulses
  • Chapter 7. Optimal impulsive effects and maximum intervals of existence of the solutions of impulsive differential equations. Application
    • 1. Optimal impulsive effects and maximal intervals of existence of solutions of impulsive differential equations1. Optimal impulsive effects and maximal intervals of existence of solutions of impulsive differential equations
    • 2. Optimal therapeutic model with impulsive effects
  • BACK MATTER
    • Bibliographical notes and comments
    • REFERENCES
Readership: Researchers who are interested in researches of impulsive effects and their differential equations.
comments powered by Disqus