Mathematical Modelling

Accessible text features over 100 reality-based examples pulled from the science, engineering, and operations research fields. Prerequisites: ordinary differential equations, continuous probability. Numerous references. Includes 27 black-and-white figures. 1978 edition.

Almost every year, a new book on mathematical modeling is published, so, why another? The answer springs directly from the fact that it is very rare to find a book that covers modeling with all types of differential equations in one volume. Until now. Mathematical Modeling: Models, Analysis and Applications covers modeling with all kinds of differential equations, namely ordinary, partial, delay, and stochastic. The book also contains a chapter on discrete modeling, consisting of differential equations, making it a complete textbook on this important skill needed for the study of science, engineering, and social sciences. More than just a textbook, this how-to guide presents tools for mathematical modeling and analysis. It offers a wide-ranging overview of mathematical ideas and techniques that provide a number of effective approaches to problem solving. Topics covered include spatial, delayed, and stochastic modeling. The text provides real-life examples of discrete and continuous mathematical modeling scenarios. MATLAB® and Mathematica® are incorporated throughout the text. The examples and exercises in each chapter can be used as problems in a project. Since mathematical modeling involves a diverse range of skills and tools, the author focuses on techniques that will be of particular interest to engineers, scientists, and others who use models of discrete and continuous systems. He gives students a foundation for understanding and using the mathematics that is the basis of computers, and therefore a foundation for success in engineering and science streams.

This book presents mathematical modelling and the integrated process of formulating sets of equations to describe real-world problems. It describes methods for obtaining solutions of challenging differential equations stemming from problems in areas such as chemical reactions, population dynamics, mechanical systems, and fluid mechanics. Chapters 1 to 4 cover essential topics in ordinary differential equations, transport equations and the calculus of variations that are important for formulating models. Chapters 5 to 11 then develop more advanced techniques including similarity solutions, matched asymptotic expansions, multiple scale analysis, long-wave models, and fast/slow dynamical systems. Methods of Mathematical Modelling will be useful for advanced undergraduate or beginning graduate students in applied mathematics, engineering and other applied sciences.

Mathematical modeling is becoming increasingly versatile and multi-disciplinary. This text demonstrates the broadness of this field as the authors consider the principles of model construction and use common approaches to build models from a range of subject areas. The book reflects the interests and experiences of the authors, but it explores mathematical modeling across a wide range of applications, from mechanics to social science. A general approach is adopted, where ideas and examples are favored over rigorous mathematical procedures. This insightful book will be of interest to specialists, teachers, and students across a wide range of disciplines..

Each Chapter Of The Book Deals With Mathematical Modelling Through One Or More Specified Techniques. Thus There Are Chapters On Mathematical Modelling Through Algebra, Geometry, Trigonometry And Calculus, Through Ordinary Differential Equations Of First And Second Order, Through Systems Of Differential Equations, Through Difference Equations, Through Partial Differential Equations, Through Functional Equations And Integral Equations, Through Delay-Differential, Differential-Difference And Integro-Differential Equations, Through Calculus Of Variations And Dynamic Programming, Through Graphs, Through Mathematical Programming, Maximum Principle And Maximum Entropy Principle.Each Chapter Contains Mathematical Models From Physical, Biological, Social, Management Sciences And Engineering And Technology And Illustrates Unity In Diversity Of Mathematical Sciences.The Book Contains Plenty Of Exercises In Mathematical Modelling And Is Aimed To Give A Panoramic View Of Applications Of Modelling In All Fields Of Knowledge. It Contains Both Probabilistic And Deterministic Models.The Book Presumes Only The Knowledge Of Undergraduate Mathematics And Can Be Used As A Textbook At Senior Undergraduate Or Post-Graduate Level For A One Or Two- Semester Course For Students Of Mathematics, Statistics, Physical, Social And Biological Sciences And Engineering. It Can Also Be Useful For All Users Of Mathematics And For All Mathematical Modellers.

A large amount of relevant mathematical problems arise from the polymer industry with respect to the quality of manufactured polymer parts. This book provides the first unified presentation of the mathematical modeling of polymerization, crystallization and extrusion of polymer melts, by means of advanced methods, presented in an accessible way for applied scientists and engineers.

Approach your problems from the right It isn't that they can't see the solution. It end and begin with the answers. Then is that they can't see the problem. one day, perhaps you will find the final question. G.K. Chesterton. The Scandal of Father Brown 'The point of a Pin'. 'The Hermit Clad in Crane Feathers' in R. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, cod ing theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical pro gramming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces.

Beginning his work on the monograph to be published in English, this author tried to present more or less general notions of the possibilities of mathematics in the new and rapidly developing science of infectious immunology, describing the processes of an organism's defence against antigen invasions. The results presented in this monograph are based on the construc tion and application of closed models of immune response to infections which makes it possible to approach problems of optimizing the treat ment of chronic and hypertoxic forms of diseases. The author, being a mathematician, had creative long-Iasting con tacts with immunologists, geneticist, biologists, and clinicians. As far back as 1976 it resulted in the organization of a special seminar in the Computing Center of Siberian Branch of the USSR Academy of Sci ences on mathematical models in immunology. The seminar attracted the attention of a wide circle of leading specialists in various fields of science. All these made it possible to approach, from a more or less united stand point, the construction of models of immune response, the mathematical description of the models, and interpretation of results.

Title page -- Preface -- Contents -- Part I. Heart -- The Changing View of the HeartThrough the Centuries -- The Left Ventricular Ejection Effect -- Human Circulatory System Model Based On Frank's Mechanism -- Modelling Blood Flow in the Left Side of the Heart -- Part II: Arterial Tree -- Models of the Arterial Tree -- A One-Dimensional Fluid Dynamic Model of the Systemic Arteries -- Measurement of Arterial Compliance In Vivo -- Models of the Venous System -- Part III: Baroreceptor Control -- General Compartmental Models of the Cardiovascular System -- Modelling the Interaction Among Several Mechanisms in the Short-term Arterial Pressure Control -- Short term Autonomic Nervous Control of the Cardiovascular System: A System Theoretic Approach -- A Baroreflex Model of Short Term Blood Pressure and Heart Rate Variability -- Part IV: Applications for Simulators -- Mathematical Models Behind Advanced Simulators in Medicine -- Cognitive Studies of Ethical Reasoning Basedon the KARDIO-simulator -- Index -- Author Index

This volume is an interdisciplinary book which introduces, in a very readable way, state-of-the-art research in the fundamental topics of mathematical modelling of Biosystems. In short, the book offers an overview of mathematical and computational modelling of biosystems including biological phenomena in general. There is also a special introduction to Protein Physics which aims to explain the all-or-none first order phase transitions from native to denatured states.

The practice of modeling is best learned by those armed with fundamental methodologies and exposed to a wide variety of modeling experience. Ideally, this experience could be obtained by working on actual modeling problems. But time constraints often make this difficult. Applied Mathematical Modeling provides a collection of models illustrating the power and richness of the mathematical sciences in supplying insight into the operation of important real-world systems. It fills a gap within modeling texts, focusing on applications across a broad range of disciplines. The first part of the book discusses the general components of the modeling process and highlights the potential of modeling in practice. These chapters discuss the general components of the modeling process, and the evolutionary nature of successful model building. The second part provides a rich compendium of case studies, each one complete with examples, exercises, and projects. In keeping with the multidimensional nature of the models presented, the chapters in the second part are listed in alphabetical order by the contributor's last name. Unlike most mathematical books, in which you must master the concepts of early chapters to prepare for subsequent material, you may start with any chapter. Begin with cryptology, if that catches your fancy, or go directly to bursty traffic if that is your cup of tea. Applied Mathematical Modeling serves as a handbook of in-depth case studies that span the mathematical sciences, building upon a modest mathematical background. Readers in other applied disciplines will benefit from seeing how selected mathematical modeling philosophies and techniques can be brought to bear on problems in their disciplines. The models address actual situations studied in chemistry, physics, demography, economics, civil engineering, environmental engineering, industrial engineering, telecommunications, and other areas.

The book aims at showing the state-of-the-art in the field of modeling and applications in mathematics education. This is the first volume to do this. The book deals with the question of how key competencies of applications and modeling at the heart of mathematical literacy may be developed; with the roles that applications and modeling may play in mathematics teaching, making mathematics more relevant for students.

Certain basic modeling skills can be applied to a wide variety of problems. It focuses on those mathematical techniques which are applicable to models involving differential equations. Models in three different areas are considered: growth and decay process, interacting populations and heating/cooling problems. The main mathematical technique is solving differential equations, while the range of applications and mathematical techniques presented provides a broad appreciation of this type of modeling. This book contains three general sections: Compartmental Models, Population Models and Heat Transfer Models. Within each section, the process of constructing a model is presented in full detail. Applications and case studies are integral to this text, and case studies are included throughout. This is a useful course text, and basic calculus and fundamental computing skills are required.

Focusing on growth and decay processes, interacting populations, and heating/cooling problems, Mathematical Modelling with Case Studies: A Differential Equations Approach using MapleTM and MATLAB®, Second Edition presents mathematical techniques applicable to models involving differential equations that describe rates of change. Although the authors concentrate on models involving differential equations, the ideas used can be applied to many other areas. The book carefully details the process of constructing a model, including the conversion of a seemingly complex problem into a much simpler one. It uses flow diagrams and word equations to aid in the model building process and to develop the mathematical equations. Employing theoretical, graphical, and computational tools, the authors analyze the behavior of the models under changing conditions. They discuss the validation of the models and suggest extensions to the models with an emphasis on recognizing the strengths and limitations of each model. Through applications and the tools of MapleTM and MATLAB®, this textbook provides hands-on model building skills. It develops, extends, and improves simple models as well as interprets the results.

Assuming virtually no prior knowledge, Modular Mathematics encourages the reader to develop and solve real models, as well as looking at traditional examples. Accessible and concise, it contains tutorial problems, case studies and exercises.

The book presents a new scientific approach to the problem of biomechanical systems description.This approach is based on development of a universal anthropomorphic model and employment of methodology of imitational dynamic modeling (IDM). The novelty of this approach is that there appears a possibility to operate with a whole class of models, derived from the universal model on the basis of motion separation principle. This is followed by utilization of iterational procedures realizing the method of successive approximations and resulting in description of the real motion with the pre-set accuracy level. By use of the IDM there has been for the first time ascertained certain laws governing human locomotions: presence of so-called controlling and stabilizing interlink moments, wavelike speeding of forces extremums along the kinematic chain, adaptation of control functions for astronauts motion coordination preservation. The book includes new theoretical conceptions explaining the deterioration of functional state of skeletal-muscular apparatus of astronauts due to zero-gravity influence.

This reference offers both a basic introduction and advanced technical details of available mathematical and computing methods for modeling sustainable development, closing an exisiting gap in this field, as well as illustrating their use through case studies and examples. The methods and case studies presented here are targetted at sustainable development, although they have a wide range of other applications, including economics, medicine and control systems.

Mathematical modelling is increasingly applicable to the practical sciences. Here, mathematical approaches are applied to the study of mechanisms of digestion and metabolism in primary animal species. It also explores common themes between species, and provides an integrated approach to mathematical modelling in animal nutrition.

This book takes stock of the state of affairs of the teaching and learning of mathematical modelling with regard to research, development and practice. It provides a conceptual framework for mathematical modelling in mathematics education at all education levels, as well as the background and resources for teachers to acquire the knowledge and competencies that will allow them to successfully include modelling in their teaching, with an emphasis on the secondary school level. Mathematics teachers, mathematics education researchers and developers will benefit from this book. Expertly written and researched, this book includes a comprehensive overview of research results in the field, an exposition of the educational goals associated with modelling, the essential components of modelling competency and an extensive discussion of didacticopedagogical challenges in modelling. Moreover, it offers a wide variety of illuminating cases and best-practice examples in addition to insights into the focal points for future research and practice. The Learning and Teaching of Mathematical Modelling is an invaluable resource for teachers, researchers, textbook authors, secondary school mathematics teachers, undergraduate and graduate students of mathematics as well as student teachers.

This book provides a thorough introduction to the challenge of applying mathematics in real-world scenarios. Modelling tasks rarely involve well-defined categories, and they often require multidisciplinary input from mathematics, physics, computer sciences, or engineering. In keeping with this spirit of modelling, the book includes a wealth of cross-references between the chapters and frequently points to the real-world context. The book combines classical approaches to modelling with novel areas such as soft computing methods, inverse problems, and model uncertainty. Attention is also paid to the interaction between models, data and the use of mathematical software. The reader will find a broad selection of theoretical tools for practicing industrial mathematics, including the analysis of continuum models, probabilistic and discrete phenomena, and asymptotic and sensitivity analysis.