Bainov / Hristova Differential Equations with Maxima

1. Auflage 2011
ISBN: 978-1-4398-6758-7
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 312 Seiten

Reihe: Chapman & Hall/CRC Pure and Applied Mathematics

Differential equations with "maxima"—differential equations that contain the maximum of the unknown function over a previous interval—adequately model real-world processes whose present state significantly depends on the maximum value of the state on a past time interval. More and more, these equations model and regulate the behavior of various technical systems on which our ever-advancing, high-tech world depends. Understanding and manipulating the theoretical results and investigations of differential equations with maxima opens the door to enormous possibilities for applications to real-world processes and phenomena.

Presenting the qualitative theory and approximate methods, Differential Equations with Maxima begins with an introduction to the mathematical apparatus of integral inequalities involving maxima of unknown functions. The authors solve various types of linear and nonlinear integral inequalities, study both cases of single and double integral inequalities, and illustrate several direct applications of solved inequalities. They also present general properties of solutions as well as existence results for initial value and boundary value problems.

Later chapters offer stability results with definitions of different types of stability with sufficient conditions and include investigations based on appropriate modifications of the Razumikhin technique by applying Lyapunov functions. The text covers the main concepts of oscillation theory and methods applied to initial and boundary value problems, combining the method of lower and upper solutions with appropriate monotone methods and introducing algorithms for constructing sequences of successive approximations. The book concludes with a systematic development of the averaging method for differential equations with maxima as applied to first-order and neutral equations. It also explores different schemes for averaging, partial averaging, partially additive averaging, and partially multiplicative averaging.

A solid overview of the field, this book guides theoretical and applied researchers in mathematics toward further investigations and applications of these equations for a more accurate study of real-world problems.

Zielgruppe

Researchers and graduate students in mathematics; applied mathematicians and engineers in control theory; theoretical physicists.

Autoren/Hrsg.

Bainov, Drumi D.

Hristova, Snezhana G.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Technische Wissenschaften Technik Allgemein Mathematik für Ingenieure
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle

Inhaltsverzeichnis

Introduction

Integral Inequalities with Maxima
Linear Integral Inequalities with Maxima for Scalar Functions of One Variable
Nonlinear Integral Inequalities with Maxima for Scalar Functions of One Variable
Integral Inequalities with Maxima for Scalar Functions of Two Variables
Applications of the Integral Inequalities with Maxima

General Theory
Existence Theory for Initial Value Problems
Existence Theory for Boundary Value Problems
Differential Equations with "Maxima" via Weakly Picard Operator Theory

Stability Theory and Lyapunov Functions
Stability and Uniform Stability
Integral Stability in Terms of Two Measures
Stability and Cone Valued Lyapunov Functions
Practical Stability on a Cone

Oscillation Theory
Differential Equations with "Maxima" versus Differential Equations with Delay
Oscillations of Delay Differential Equations with "Maxima"
Oscillations of Forced n-th Order Differential Equations with "Maxima"
Oscillations and Almost Oscillations of n-th Order Differential Equations with "Maxima"
Oscillations of Differential Inequalities with "Maxima"

Asymptotic Methods
Monotone-Iterative Technique for Initial Value Problems
Monotone-Iterative Technique for a Periodic Boundary Value Problem
Monotone-Iterative Technique for Second Order Differential Equations with "Maxima"
Method of Quasilinearization for an Initial Value Problem
Method of Quasilinearization for a Periodic Boundary Value Problem

Averaging Method
Averaging Method for an Initial Value Problem
Averaging Method for Multipoint Boundary Value Problem
Partial Averaging Method
Partially Additive and Partially Multiplicative Averaging Method
Notes and Comments
Bibliography

Über Autor(innen)

Drumi D. Bainov, Medical University of Sofia, Bulgaria
Snezhana G. Hristova, Plovdiv University, Bulgaria