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Advanced Differential Equations
- 1st Edition - April 13, 2022
- Author: Youssef N. Raffoul
- Language: English
- Paperback ISBN:9 7 8 - 0 - 3 2 3 - 9 9 2 8 0 - 0
- eBook ISBN:9 7 8 - 0 - 3 2 3 - 9 9 2 8 1 - 7
Advanced Differential Equations provides coverage of high-level topics in ordinary differential equations and dynamical systems. The book delivers difficult material in an access… Read more
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Request a sales quoteAdvanced Differential Equations provides coverage of high-level topics in ordinary differential equations and dynamical systems. The book delivers difficult material in an accessible manner, utilizing easier, friendlier notations and multiple examples. Sections focus on standard topics such as existence and uniqueness for scalar and systems of differential equations, the dynamics of systems, including stability, with examples and an examination of the eigenvalues of an accompanying linear matrix, as well as coverage of existing literature. From the eigenvalues' approach, to coverage of the Lyapunov direct method, this book readily supports the study of stable and unstable manifolds and bifurcations.
Additional sections cover the study of delay differential equations, extending from ordinary differential equations through the extension of Lyapunov functions to Lyapunov functionals. In this final section, the text explores fixed point theory, neutral differential equations, and neutral Volterra integro-differential equations.
- Includes content from a class-tested over multiple years with advanced undergraduate and graduate courses
- Presents difficult material in an accessible manner by utilizing easier, friendlier notations, multiple examples and thoughtful exercises of increasing difficulty
- Provides content that is appropriate for advanced classes up to, and including, a two-semester graduate course in exploring the theory and applications of ordinary differential equations
- Requires minimal background in real analysis and differential equations
- Offers a partial solutions manual for student study
Advanced students in undergraduate and graduate programs taking courses on Advanced Ordinary Differential Equations in Math, Science, and Engineering
- Cover image
- Title page
- Table of Contents
- Copyright
- Dedication
- Preface
- Chapter 1: Preliminaries and Banach spaces
- Abstract
- 1.1. Preliminaries
- 1.2. Escape velocity
- 1.3. Applications to epidemics
- 1.4. Metrics and Banach spaces
- 1.5. Variation of parameters
- 1.6. Special differential equations
- 1.7. Exercises
- Bibliography
- Chapter 2: Existence and uniqueness
- Abstract
- 2.1. Existence and uniqueness of solutions
- 2.2. Existence on Banach spaces
- 2.3. Existence theorem for linear equations
- 2.4. Continuation of solutions
- 2.5. Dependence on initial conditions
- 2.6. Exercises
- Bibliography
- Chapter 3: Systems of ordinary differential equations
- Abstract
- 3.1. Existence and uniqueness
- 3.2. x′=A(t)x
- 3.3. x′=A(t)x+g(t)
- 3.4. Discussion
- 3.5. Exercises
- Bibliography
- Chapter 4: Stability of linear systems
- Abstract
- 4.1. Definitions and examples
- 4.2. x′=A(t)x
- 4.3. Floquet theory
- 4.4. Exercises
- Chapter 5: Qualitative analysis of linear systems
- Abstract
- 5.1. Preliminary theorems
- 5.2. Near-constant systems
- 5.3. Perturbed linear systems
- 5.4. Autonomous systems in the plane
- 5.5. Hamiltonian and gradient systems
- 5.6. Exercises
- Chapter 6: Nonlinear systems
- Abstract
- 6.1. Bifurcations in scalar systems
- 6.2. Stability of systems by linearization
- 6.3. An SIR epidemic model
- 6.4. Limit cycle
- 6.5. Lotka–Volterra competition model
- 6.6. Bifurcation in planar systems
- 6.7. Manifolds and Hartman–Grobman theorem
- 6.8. Exercises
- Bibliography
- Chapter 7: Lyapunov functions
- Abstract
- 7.1. Lyapunov method
- 7.2. Global asymptotic stability
- 7.3. Instability
- 7.4. ω-limit set
- 7.5. Connection between eigenvalues and Lyapunov functions
- 7.6. Exponential stability
- 7.7. Exercises
- Chapter 8: Delay differential equations
- Abstract
- 8.1. Introduction
- 8.2. Method of steps
- 8.3. Existence and uniqueness
- 8.4. Stability using Lyapunov functions
- 8.5. Stability using fixed point theory
- 8.6. Exponential stability
- 8.7. Existence of positive periodic solutions
- 8.8. Exercises
- Bibliography
- Chapter 9: New variation of parameters
- Abstract
- 9.1. Applications to ordinary differential equations
- 9.2. Applications to delay differential equations
- 9.3. Exercises
- Bibliography
- Bibliography
- Index
- No. of pages: 364
- Language: English
- Edition: 1
- Published: April 13, 2022
- Imprint: Academic Press
- Paperback ISBN: 9780323992800
- eBook ISBN: 9780323992817
YR
Youssef N. Raffoul
Youssef N. Raffoul holds the rank of Professor and is Graduate Program Director in the Department of Mathematics at the University of Dayton, US.
Prof. Raffoul joined the faculty in 1999. He obtained a B.S. and M.S. from the University of Dayton in mathematics in 1987 and 1989. After receiving his Ph.D. in mathematics from Southern Illinois University at Carbondale in 1996, he joined the faculty of the Mathematics Department at Tougaloo College in Mississippi, where he became the department chair for two years until he came to Dayton. Prof. Raffoul has published extensively in the area of functional differential and difference equations and has won several awards for his research, most recently the Career in Science Award from the Lebanese Government. Prof. Raffoul published four books: Qualitative Theory of Volterra Difference Equations; with Professor Murat Adivar, Stability, Periodicity, and Boundedness in Functional Dynamical Systems on Time Scales; Advanced Differential Equations (Academic Press, 2022); and his fourth book, Applied Mathematics for Scientists and Engineers.
Affiliations and expertise
Professor and Graduate Program Director, Department of Mathematics, University of Dayton, OH, USARead Advanced Differential Equations on ScienceDirect