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- Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow
- Math 142: Mathematical Modeling
- On Solution to Traffic Flow Problem by Method of Characteristics
Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow
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Protter, Charles B. Morrey - Intermediate Calculus , Springer. Ullrich D. Jump to Page. Search inside document. Includes bibliographies and index, 1.
Mathematics 2. Mathematical models. Vibration—Mathematical models. Traffic ow Mathematical models. H2 Englewood Cliffs, New Jersey All rights reserved.
No part of this book may be reproduced in any form or by any means without permission in writing from the publisher. Method of Elimination, B. General Remarks, Saddle Points, Nodes, Spirals, By studying three diverse areas in which mathematics has been applied, this text attempts to introduce to the reader some of the fundamental concepts and techniques of applied mathematics.
In each area, relevant observations and experiments are discussed. In this way a mathematical model is carefully formulated.
The resulting mathematical problem is solved, requiring at times the introduction of new mathematical methods. The solution is then inter- preted, and the validity of the mathematical model is questioned.
Often the mathematical model must be modified and the process of formulation, solu- tion, and interpretation continued. Thus we will be illustrating the relation- ships between each area and the appropriate mathematics.
Since one area at a time is investigated in depth in this way, the reader has the opportunity to understand each topic, not just the mathematical techniques. Mechanical vibrations, population dynamics, and traffic flow are chosen as areas to investigate in an introduction to applied mathematics for similar reasons.
In each, the experiments and common observations necessary to formulate and understand the mathematical models are relatively well known to the average reader.
We will not find it necessary to refer to exceedingly technical research results. Furthermore these three topics were chosen for inclusion in this text because each serves as an introduction to more special- ized investigations.
Here we attempt only to introduce these various topics and leave the reader to pursue those of most interest. In addition, it is hoped that the reader will find these three topics as interesting as the author does. A previous exposure to physics will aid the reader in the part on mechan- ical vibrations, but the text is readily accessible to those without this back- ground. The topics discussed supplement rather than substitute for an introductory physics course.
The material on population dynamics requires no background in biology; experimental motivation is self-contained. Similarly, there is sufficient familarity with traffic situations to enable the reader to thoroughly understand the traffic models that are developed.
This text has been written with the assumption that the reader has had the equivalent of the usual first two years of college mathematics calculus and some elementary ordinary differential equations.
Many critical aspects of these prerequisites are briefly reviewed. More specifically, a knowledge of calculus including partial derivatives is required, but vector integral calculus for example, the divergence theorem is never used nor is it needed. Linear algebra and probability are also not required although they are briefly utilized in a few sections which the reader may skip.
Although some knowl- edge of differential equations is required, it is mostly restricted to first and second order constant coefficient equations. A background in more advanced techniques is not necessary, as they are fully explained where needed. Mathematically, the discussion of mechanical vibrations and population dynamics proceed in similar ways. In both, emphasis is placed on the non- linear aspects of ordinary differential equations. The concepts of equilibrium solutions and their stability are developed, considered by many to be one of the fundamental unifying themes of applied mathematics.
Phase plane methods are introduced and linearization procedures are explained in both parts. On the other hand, the mathematical models of traffic flow involve first-order nonlinear partial differential equations, and hence is relatively independent of the previous material.
The method of characteristics is slowly and carefully explained, resulting in the concept of traffic density wave propagation. Throughout, mathematical techniques are developed, but equal emphasis is placed on the mathematical formulation of the problem and the interpretation of the results. I believe, in order to learn mathematics, the reader must take an active part.
This is best accomplished by attempting a significant number of the included exercises. Many more problems are included than are reasonable for the average reader to do. The exercises have been designed such that their difficulty varies. Most are word problems, enabling the reader to consider the relationships between the mathematics and the models.
Each major part is divided into many subsections. However, these sections are not of equal length. Few correspond to as much as a single lecture. Usually more than one and occasionally, depending on the background of the reader, many of the sections can be covered in an amount of time equal to that of a single lecture. In this way the book has been designed to be substantially covered in one semester. However, a longer treatment of these subjects will be beneficial for some.
This text is a reflection of my own philosophy of applied mathematics. For my own education, the applied mathematics group at the Massachusetts Institute of Technology must be sincerely thanked.
Any credit for much of this book must be shared with them in some ill-defined way. Student comments have been most helpful as have been the insights given to me by Dr.
Eugene Speer and Dr. Richard Falk who have co-taught the material with me. Also I would like to express my appreciation to Dr. Mark Ablowitz for his many thoughtful and useful suggestions. For the opportunity and encouragement to develop an applied mathemat- ics course for which this text was written, I wish sincerely to thank Dr. Terry Butler.
Furthermore his interest in the needs of students reinforced my own attitudes and resulted in this text. Having no interest or knowledge in mathematics, this was an exceptionally difficult effort. My appreciation to the typists of the manuscript originally class notes , especially Mrs.
Annette Roselli whose accurate work was second only to her patience with the numerous revisions. To use mathematics, one needs to understand the physical context.
Here problems in mechanical vibrations, population dynamics, and traffic flow are developed from first principles. In the independent presentations of mechanical vibrations and population dynamics, nonlinear ordinary differential equations are analyzed by investigating equilibria solutions and their linearized stability.
The phase plane is introduced to discuss nonlinear phenomena, Discrete models for population growth are also presented, and when teaching in recent years I have supplemented the book with a discussion of iterations of the logistic map and the period doubling route to chaos.
Math 142: Mathematical Modeling
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Lecture, three hours; discussion, one hour. Prerequisites: courses 32B, 33B. Introduction to fundamental principles and spirit of applied mathematics. Emphasis on manner in which mathematical models are constructed for physical problems. Illustrations from many fields of endeavor, such as the physical sciences, biology, economics, and traffic dynamics. From catalog.
Skip to search form Skip to main content You are currently offline. Some features of the site may not work correctly. DOI: James and E. James , E.
Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow differential equations and develops the concepts of equilibrium solutions and Haberman develops the method of characteristics to analyze the nonlinear partial differential equations that describe traffic flow. Abstract | PDF ( KB).
On Solution to Traffic Flow Problem by Method of Characteristics
This is a good course for starting to use the program. The student version is available at the campus Computer Store. Tentative syllabus: Population dynamics and mathematical ecology. Introduction to traffic flow.
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Received 21 February ; accepted 24 April ; published 27 April
Здравствуйте, Это Сьюзан Флетчер. Извините, меня нет дома, но если вы оставите свое сообщение… Беккер выслушал все до конца. Где же. Наверняка Сьюзан уже начала волноваться. Уж не уехала ли она в Стоун-Мэнор без .
Правда открылась со всей очевидностью: Хейл столкнул Чатрукьяна. Нетвердой походкой Сьюзан подошла к главному выходу- двери, через которую она вошла сюда несколько часов. Отчаянное нажатие на кнопки неосвещенной панели ничего не дало: массивная дверь не поддалась. Они в ловушке, шифровалка превратилась в узилище. Купол здания, похожий на спутник, находился в ста девяти ярдах от основного здания АНБ, и попасть туда можно было только через главный вход. Поскольку в шифровалке имелось автономное энергоснабжение, на главный распределительный щит, наверное, даже не поступил сигнал, что здесь произошла авария. - Основное энергоснабжение вырубилось, - сказал Стратмор, возникший за спиной Сьюзан.
Тогда-то виновников компьютерных сбоев и стали называть вирусами.
Там не окажется никакого Клауса, но Беккер понимал, что клиенты далеко не всегда указывают свои подлинные имена. - Хм-м, извините, - произнесла женщина. - Не нахожу. Как, вы сказали, имя девушки, которую нанял ваш брат. - Рыжеволосая, - сказал Беккер, уклоняясь от ответа.
Ну видите, все не так страшно, правда? - Она села в кресло и скрестила ноги. - И сколько вы заплатите. Вздох облегчения вырвался из груди Беккера. Он сразу же перешел к делу: - Я могу заплатить вам семьсот пятьдесят тысяч песет.
Он относится к ТРАНСТЕКСТУ как к священной корове. Мидж кивнула. В глубине души она понимала, что абсурдно обвинять в нерадивости Стратмора, который был беззаветно предан своему делу и воспринимал все зло мира как свое личное .