Introduction to Mathematical Modeling and Computer Simulations is written as a textbook for readers who want to understand the main principles of Modeling and Simulations in settings that are important for the applications, without using the profound mathematical tools required by most advanced texts. It can be particularly useful for applied mathematicians and engineers who are just beginning their careers. The goal of this book is to outline Mathematical Modeling using simple mathematical descriptions, making it accessible for first- and second-year students.
Table of Contents
I General Principles and Methods
1. Principles of Mathematical Modeling
2. Numerical and symbolic computations
II Basic Applications
3. Application of calculus to classic mechanics
4. Ordinary differential equations and their applications
5. Stochastic models
6. One-dimensional stationary problems
III Advanced Applications
7. Vector analysis
8. Heat Equation
9. Asymptotic methods in composites
Dr. Natalia Rylko and Dr. Wojciech Nawalaniec are members of the scientific group headed by Vladimir Mityushev, Professor of the Pedagogical University of Krakow, Poland. Their academic and scientific activity is based on interdisciplinary international research devoted to applied mathematics and computer simulations. The main scientific achievements of the group concern random composites and porous media, analytical formulae for the effective properties of regular and random composites, pattern formation, and the collective behaviour of bacteria.
"The textbook is addressed to students of Applied Mathematics, of Science and Engineering at the undergraduate level. The reader is familiarized with basic principles of modeling. Moreover, basic methods of analysis, numerics and statistics are presented for the treatment of applied problems of general interest. Some of these interesting problems are heat conduction, biological processes and composite materials. It is a nice idea to use in addition to numerical methods program packages of symbolic computations as Mathematica and Matlab for simulation. A lot of interesting exercises motivate the reader to further studies. Through this book the reader learns skills of mathematicians and engineers as well. It is very suitable for students interested in interdisciplinary research."
—Michael Reissig, Professor of Partial Differential Equations, Technical University Bergakademie Freiberg
"There are many specialised books on mathematical modelling in specific narrow areas, this book takes a broader point of view to touch upon many of the key ideas in language suitable for an undergraduate or non-specialist audience. This book therefore nicely fills a gap, moreover it is rich in examples and physical motivations. Importantly it contains both Mathematica and Matlab support to enable simulation, and these complement and reinforce the modelling. Courses in mathematical modelling could benefit from adopting this book as it provides a valuable, and accessible, teaching resource."
— Richard Craster, Professor of Applied Mathematics, FIMA, FCGI, Leverhulme Trust Research Fellow, Imperial College London
"The book introduces to a Reader the problems of mathematical modeling and computer simulations in a friendly and easy manner, but still mathematically precise. I haven't seen before so systematic and self-contained presentation of the mathematical modeling foundations.
Most of the available books, which I know, concern mathematical modeling or applied mathematics without computer simulations or, from the other hand, they present numerical methods and computer simulations without mathematical background. This special book joins mathematics and computer simulations and proposes the powerful methods and advantages of both subjects. The authors explain exhaustively and clearly how to develop and apply mathematical models, how to create the corresponding codes for symbolic and numerical computations and finally how to analyze the results. A number of examples and exercises help the reader to understand everything without too much effort.
Summarizing, this book will occupy for sure an important place in my collection, and I recommend it to the students and teachers of applied mathematics, technology, informatics and any other field of science where the computer simulations of processes may be useful."
— Edyta Hetmaniok, Silesian University
"This book hits many of the major topics that should be in an introductory undergraduate course on Mathematical Modeling at the sophomore or junior level. It is especially useful in the present time where computational literacy is essential in modeling. A nice one semester course could be developed around Chapters 1-6 with a choice of one advanced topic in Chapters 7-9 to end the course with an interesting application. The use of Mathematica and Matlab to enhance the course material is appropriate and does not feel overwhelming. This text will allow beginning applied mathematics students access to modeling early in their careers and set them up for success with advanced material as they mature in their studies. Other texts in modeling currently available have not placed the added emphasis on developing the computational aspects as the model is developed, but rather treat simulation as only a way to solve the model. I also like that each chapter sets the stage for a fundamental application and then develops the requisite theory and computation."
—Shawn D. Ryan, Assistant Professor of Mathematics, Cleveland State University
"At present, there is an increasing interest in interdisciplinary research, which is caused by the need to study the properties of objects that are complex in structure and properties, to understand the processes occurring inside them and to be able to describe these processes in a qualified manner. The aim of this book is an attempt to eliminate the problems of interdisciplinary research.
The first problem is to formulate the subject of research in such a way that it can be studied by means of both mechanics and mathematics. The second problem is to provide the knowledge of mechanics and mathematicians to create fruitful communications so that they can professionally participate in the discussion and receipt of information. The third problem is to ensure the transfer of applied results of interdisciplinary research into technological implementation, to carry out a qualified evaluation of the obtained results.
This book is useful not only for experienced mathematicians and mechanics, dealing with applied problems, but also for students, studying abstract objects to their inexperienced look, to get an idea of how these objects are used in mechanics, to generate interest and acquire skills for future research. The book is written in an accessible language, is provided with various entertaining both simple and complex examples of physical models and methods of mathematical modeling of their properties, and also contains exercises.
The practice of theoretical research of various applied objects and participation in interdisciplinary projects allowed the authors of this book to accumulate quite a rich experience in this field and share this experience with other researchers. I strongly recommend this textbook for everybody who wants apply mathematical models."
—Ekaterina Pesetskaya, Department of Mathematical Physics, A. Razmadze Mathematical Institute, Tbilisi State University