Modelling with Ordinary Differential Equations
A Comprehensive Approach
2nd Edition
Will be published approx. on 21. October 2026
528 pages
E-Book
978-1-040-67991-3 (ISBN)
Description
"Alfio Borzi's Modelling with Ordinary Differential Equations is a remarkably comprehensive text that succeeds in something rare: it covers the classical foundations of ODE theory with precision and rigour, while consistently opening windows to advanced and genuinely original applications that are hard to find in comparable textbooks"
-Professor Volker Schulz, Universitaet Trier
"Expertly written, organized and presented, Modelling with Ordinary Differential Equations: A Comprehensive Approach is an ideal textbook for college and university Numerical Analysis & Scientific Computing curriculums. [. . . ] unreservedly recommended as a critically important addition to academic library collections"
-Midwest Book Review
Modelling with Ordinary Differential Equations: A Comprehensive Approach aims to provide a broad and self-contained introduction to the mathematical tools necessary to investigate and apply ODE models. The book starts by establishing the existence of solutions in various settings and analysing their stability properties. The next step is to illustrate modelling issues arising in the calculus of variation and optimal control theory that are of interest in many applications. This discussion is continued with an introduction to inverse problems governed by ODE models and to differential games.
The book is completed with an illustration of stochastic differential equations and the development of neural networks to solve ODE systems. Many numerical methods are presented to solve the classes of problems discussed in this book.
New to the Second Edition:
This second edition has been thoroughly revised, reorganised, and expanded with new material across most chapters. The theoretical foundations are strengthened by additional results on existence and uniqueness, Lipschitz conditions, blow-up phenomena, and Green functions, as well as new sections on compartment models and Hamiltonian systems. A major addition is a new chapter on mechanics, ranging from classical to relativistic and quantum frameworks. The numerical analysis part now includes the Verlet method, while stability theory has been extended to cover chaos and synchronization.
Optimal control is treated in greater depth, with new material on the HJB equation, controllability, and free-horizon problems. Further additions include evolutionary differential games and enhanced inverse problem examples.The chapter on neural networks has been also expanded, introducing residual networks, and reservoir computing for dynamical systems.
-Professor Volker Schulz, Universitaet Trier
"Expertly written, organized and presented, Modelling with Ordinary Differential Equations: A Comprehensive Approach is an ideal textbook for college and university Numerical Analysis & Scientific Computing curriculums. [. . . ] unreservedly recommended as a critically important addition to academic library collections"
-Midwest Book Review
Modelling with Ordinary Differential Equations: A Comprehensive Approach aims to provide a broad and self-contained introduction to the mathematical tools necessary to investigate and apply ODE models. The book starts by establishing the existence of solutions in various settings and analysing their stability properties. The next step is to illustrate modelling issues arising in the calculus of variation and optimal control theory that are of interest in many applications. This discussion is continued with an introduction to inverse problems governed by ODE models and to differential games.
The book is completed with an illustration of stochastic differential equations and the development of neural networks to solve ODE systems. Many numerical methods are presented to solve the classes of problems discussed in this book.
New to the Second Edition:
This second edition has been thoroughly revised, reorganised, and expanded with new material across most chapters. The theoretical foundations are strengthened by additional results on existence and uniqueness, Lipschitz conditions, blow-up phenomena, and Green functions, as well as new sections on compartment models and Hamiltonian systems. A major addition is a new chapter on mechanics, ranging from classical to relativistic and quantum frameworks. The numerical analysis part now includes the Verlet method, while stability theory has been extended to cover chaos and synchronization.
Optimal control is treated in greater depth, with new material on the HJB equation, controllability, and free-horizon problems. Further additions include evolutionary differential games and enhanced inverse problem examples.The chapter on neural networks has been also expanded, introducing residual networks, and reservoir computing for dynamical systems.
Reviews / Votes
"Alfio Borzi's Modelling with Ordinary Differential Equations is a remarkably comprehensive text that succeeds in something rare: it covers the classical foundations of ODE theory with precision and rigour, while consistently opening windows to advanced and genuinely original applications that are hard to find in comparable textbooks. The opening chapter on the modelling process already sets the tone - it is superb and immediately conveys the spirit that carries the entire book. The treatment of compartment models and the master equation, Hamiltonian systems, and - further along - the excursions into special relativity and quantum mechanics are captivating and showcase the author's ability to connect apparently distant topics through the unifying language of ODEs. Indeed, a recurring strength of this book is that the final two or three subsections of most chapters present fascinating perspectives that one does not easily encounter elsewhere. Chapter 9 on the calculus of variations is particularly original and engaging; the chapter on optimal control provides the Pontryagin Maximum Principle with a proof in admirably compact form - something I have long been looking for. The final chapter on neural networks and ODEs brings the subject into a very timely and modern context. Overall, this book is an extraordinary resource and a true Fundgrube (treasure trove) for anyone interested in the many facets of ODE modelling. If I were to use it as a course textbook, I would need at least two semesters with four hours of lectures and two hours of tutorials each - which speaks for itself. Warmly recommended."-Professor Volker Schulz, Universitaet Trier
"Expertly written, organized and presented, Modelling with Ordinary Differential Equations: A Comprehensive Approach is an ideal textbook for college and university Numerical Analysis & Scientific Computing curriculums. [. . . ] unreservedly recommended as a critically important addition to academic library collections"
-Midwest Book Review
More details
Series
Edition
2nd edition
Language
English
Place of publication
London
United Kingdom
Publishing group
Taylor & Francis Ltd
Target group
College/higher education
Product notice
Reflowable
Illustrations
2 Tables, black and white; 57 Line drawings, color; 16 Line drawings, black and white; 2 Halftones, color; 59 Illustrations, color; 16 Illustrations, black and white
ISBN-13
978-1-040-67991-3 (9781040679913)
Copyright in bibliographic data is held by Nielsen Book Services Limited or its licensors: all rights reserved.
Schweitzer Classification
Other editions
Additional editions
Book
approx. 10/2026
2nd Edition
Chapman & Hall/CRC
€74.50
Not yet published
Book
approx. 10/2026
2nd Edition
Chapman & Hall/CRC
€197.50
Not yet published
Person
Alfio Borzi, born 1965 in Catania (Italy), is the professor and chair of Scientific Computing at the Institute for Mathematics of the University of Wuerzburg, Germany. He studied Mathematics and Physics in Catania and Trieste where he received his PhD in Mathematics from Scuola Internazionale Superiore di Studi Avanzati (SISSA).
He served as Research Officer at the University of Oxford (UK) and as Assistant Professor at the University of Graz (Austria) where he completed his habilitation and was appointed as Associate Professor. Since 2011 he has been Professor of Scientific Computing at the University of Wuerzburg.
Alfio Borzi is the author of 4 mathematics books and numerous articles in journals. The main topics of his research and teaching activities are modelling and numerical analysis, numerical optimisation and machine learning, optimal control theory and scientific computing. He is member of the editorial board for SIAM Review and of the book series Advances in Mechanics and Mathematics (AMMA).
He served as Research Officer at the University of Oxford (UK) and as Assistant Professor at the University of Graz (Austria) where he completed his habilitation and was appointed as Associate Professor. Since 2011 he has been Professor of Scientific Computing at the University of Wuerzburg.
Alfio Borzi is the author of 4 mathematics books and numerous articles in journals. The main topics of his research and teaching activities are modelling and numerical analysis, numerical optimisation and machine learning, optimal control theory and scientific computing. He is member of the editorial board for SIAM Review and of the book series Advances in Mechanics and Mathematics (AMMA).
Content
1. Introduction 2. Elementary solution methods for simple ODEs 3. Theory of ordinary differential equations 4. Systems of ordinary differential equations 5. Higher-order ordinary differential equations 6. Mechanics and second-order ODEs 7. Numerical solution of ODE problems 8. Stability of ODE systems 9. ODEs and the calculus of variations 10. Optimal control of ODE models 11. Inverse problems with ODE models 12. Differential games 13. Stochastic differential equations 14. Neural networks and ODE problems
