Mathematical Modeling
Heinz, S. 2011 Mathematical Modeling. Springer-Verlag, Heidelberg, Dordrecht, London, New York (ISBN: 978-3-642-20310-7)
The whole picture of Mathematical Modeling is systematically and thoroughly explained in this text for undergraduate and graduate students of mathematics, engineering, economics, finance, biology, chemistry, and physics. This textbook gives an overview of the spectrum of modeling techniques, deterministic and stochastic methods, and first-principle and empirical solutions. Complete range: The text continuously covers the complete range of basic modeling techniques: it provides a consistent transition from simple algebraic analysis methods to simulation methods used for research. Such an overview of the spectrum of modeling techniques is very helpful for the understanding of how a research problem considered can be appropriately addressed. Complete methods: Real-world processes always involve uncertainty, and the consideration of randomness is often relevant. Many students know deterministic methods, but they do hardly have access to stochastic methods, which are described in advanced textbooks on probability theory. The book develops consistently both deterministic and stochastic methods. In particular, it shows how deterministic methods are generalized by stochastic methods. Complete solutions: A variety of empirical approximations is often available for the modeling of processes. The question of which assumption is valid under certain conditions is clearly relevant. The book provides a bridge between empirical modeling and first-principle methods: it explains how the principles of modeling can be used to explain the validity of empirical assumptions. The basic features of micro-scale and macro-scale modeling are discussed - which is an important problem of current research. |
Reviews:
"Reading this book "Mathematical Modeling" by Stefan Heinz, reminded me about the best teachers I have had. They not only explained how to solve equations, but also provided the story why a certain model is possible or useful and what are the famous historical or contemporary examples." … "Having pointed out why the book is of particular interest for teachers and researchers in fluid dynamics, it is worth coming back to main merits of the book for a wider audience in science and engineering. As mentioned it treats both deterministic and stochastic models. But it also presents a complete range of modeling techniques, from simple algebraic analysis to statistical equations systems requiring numerical simulation. And finally, it is really great that the presentation is complete. Not only are the assumptions and steps in the derivations, but also the reasoning leading to the use of a specific type of mode made fully clear. 570 exercises questions are provided organized in 220 problems, equally divided over the chapters. They are of the right level of difficulty to test the understanding of the reader." … "In conclusion I strongly recommend the book to anyone wanting to acquire good understanding what mathematical modeling is about."
"Mathematical modeling, combined with computational simulation, has become an area of specialty in almost all disciplines: engineering, physics, chemistry, business, finance, economics, biology, psychology, and medicine, to name a few. At several universities, including mine, computational modeling and simulation (CMS) is now an independent major in which students can obtain a crossdisciplinary Ph.D. degree. The basic foundation for this degree is covered by mathematics and computer science course work; the remaining components are covered in other courses in the student's specialty area. I find this text very useful in our CMS program as it is accessible to almost all of the students, regardless of background."… "The book has several other pleasing features: conciseness of writing, clarity of figures, presentation of relevant examples, and inclusion of a large number of exercises at the end of each chapter. An instruction manual is also available that includes solutions to the exercise problems."… "Overall, I like this book and recommend it as a text in mathematical modeling. I also recommend it for those studying turbulent fluid mechanics, stochastic vibration & structures, noise, meteorology, random data management, and stock market analysis."
"The book under review is aimed at students in mathematics, physics, engineering, biology, chemistry, economics, finance. … well-written, generously illustrated, and contains many carefully explained examples that are often based on real data. Each chapter starts with a brief ‘Motivation’ section and concludes with a ‘Summary’ that emphasizes the most important ideas and techniques. … This textbook nicely complements existing literature on mathematical modeling and can be used both as a main source or as a supplementary text … in applied mathematics and mathematical modeling."
"This book is aimed at advanced undergraduate and graduate students in mathematics who have a background in single and multivariable calculus, linear algebra, and ordinary differential equations. … focuses on the construction of mathematical models, the derivation of analytical solutions to tractable models, and the qualitative analysis of the behavior of solutions to the less tractable models. … The entire book could be covered in a two-semester sequence of courses at the graduate level. Summing Up: Recommended. Upper-division undergraduates and graduate students."
"This book may serve as a reference for those seeking examples of modeling in different specializations, or as a resource for exercises; several of the latter are at the end of each chapter."
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