Advanced Differential Equations

Advanced Differential Equations provides unique coverage of high-level topics in ordinary differential equations and dynamical systems. The textbook deliberately presents difficult material in an accessible manner by utilizing easier, friendlier notations and multiple examples.

Divided into three parts, this valuable resource first focuses on standard topics such as existence and uniqueness for scalar and systems of differential equations, motivating the topics with examples and easing into the underlying theory.

In the second section, the textbook focuses on the dynamic of the systems, addressing stability thoroughly; again motivating the topics with interesting examples, examining the eigenvalues of an accompanying linear matrix, as well as covering the existing literature on the topic. From a discussion of the limitations of the eigenvalues’ approach, the text expands coverage the Lyapunov direct method, to support the study of stable and unstable manifolds and bifurcations.

The third part of the book is devoted to 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.

Requiring only minimal familiarity with real analysis and differential equations, Advanced Differential Equations provides students and professors with the background and practice through comprehensive exercises, such that, by the completion of Chapter 9, readers will be ready to conduct meaningful research in delay differential systems.

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.