Mathematical modeling of physical systems. An introduction. (English) Zbl 1044.00001

Basmadjian, Diran

Oxford: Oxford University Press (ISBN 0-19-515314-6/hbk). xv, 350 p. (2003).

There are many books published on this topic in the 1950’s and 1960’s and their titles were Mathematical methods in Engineering, in Heat Transfer, etc. Those books were written by authorities in their field of research, so they were popular.
But this textbook is different from those published earlier, though the aim is the same. The author explains in the preface, why such methods are important? In physical systems, one encounters many difficulties while manufacturing a model. To do experiments and then finalise a suitable model, it takes a long time. So to avoid delay and save funds, one has to deal with this situation by mathematical modelling based on known mathematical equations and many times one has to add some more mathematical terms as per physical situation. This equation is then to be solved under proper physical conditions known as initial and boundary conditions. Then there are two ways of solving this system of equations viz., analytical and numerical methods. However, analytical methods are preferable. Many times numerical methods are rather difficult.
In Chapter 1, the author gives us the idea of when to start modelling and when not. He cites some examples from different fields which are interesting and of practical importance which are not found in other books, hence the importance of this book. The reviewer feels that it can be used as a reference book and not as a textbook in a College/University, in different disciplines like atomic energy, electrical and mechanical engineering, mass transfer of a substance in a fluid, in motion, financial matters, in one’s own life, heating of fluid in a flow, etc.
In Chapter 2, the author introduces mathematical tools useful in solving different types of equations e.g.: vectors, matrices, Laplace transform and their applications in solving different types of differential equations, algebraic equations.
Chapter 3, introduces geometrical methods to deal with different types of problems of practical interest.
Chapter 4 deals with different types of forces that modify the physical system e.g.: (i) Gravitational force; (ii) Electric and magnetic forces; (iii) Stress and strain relation, only one interesting example can be cited fiz.: in satellite system, minimum escape velocity etc.. Many examples are solved based on these types of forces.
In Chapter 5, a new concept is introduced known as compartmental modelling. These methods are useful in chemical engineering, environmental engineering and biomedical engineering. The author discusses many examples, which are not found in books so far. One interesting example is discussed viz.: Controlled release drug delivery, blood clots in flowing blood, HIV etc. Examples are discussed on this topic and students in these fields will be benefitted by this knowledge.
In Chapter 6, examples are discussed where properties of fluids vary in direction due to temperature and pressure. Many examples are discussed.
In Chapter 7, some good examples based on simple networks are discussed. Interesting examples are “Dialysis”, “Radioactive Decay”, “Some Electrical Circuits”.
Chapter 8 is devoted to most important topics of dimensional analysis and numerical methods. What is dimensional analysis, how is it used to simplify the process? It is discussed in this chapter, by considering some examples. At the end, different examples are solved using different numerical procedures and the corresponding computer programs.

Reviewer: V. M. Soundalgekar (Thane)

MSC:

00A06	Mathematics for nonmathematicians (engineering, social sciences, etc.)
00A69	General applied mathematics
00A71	General theory of mathematical modeling

Keywords:

modelling; geometrical methods; compartmental modelling; dimensional and numerical method