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Tensor calculus is applied to the formulation of mathematical models of diverse phenomena. Aeronautics, fluid dynamics, and cosmology are among the areas of application. The feasibility of combining tensor methods and computer capability to formulate problems is demonstrated. The techniques described are an attempt to simplify the formulation of mathematical models by reducing the modeling process to a series of routine operations, which can be performed either manually or by computer.
Book Synopsis Mathematical Modeling of Diverse Phenomena by : James Carson Howard
Download or read book Mathematical Modeling of Diverse Phenomena written by James Carson Howard and published by . This book was released on 1979 with total page 408 pages. Available in PDF, EPUB and Kindle. Book excerpt: Tensor calculus is applied to the formulation of mathematical models of diverse phenomena. Aeronautics, fluid dynamics, and cosmology are among the areas of application. The feasibility of combining tensor methods and computer capability to formulate problems is demonstrated. The techniques described are an attempt to simplify the formulation of mathematical models by reducing the modeling process to a series of routine operations, which can be performed either manually or by computer.
This book is intended for those students, engineers, scientists, and applied mathematicians who find it necessary to formulate models of diverse phenomena. To facilitate the formulation of such models, some aspects of the tensor calculus will be introduced. However, no knowledge of tensors is assumed. The chief aim of this calculus is the investigation of relations that remain valid in going from one coordinate system to another. The invariance of tensor quantities with respect to coordinate transformations can be used to advantage in formulating mathematical models. As a consequence of the geometrical simplification inherent in the tensor method, the formulation of problems in curvilinear coordinate systems can be reduced to series of routine operations involving only summation and differentiation. When conventional methods are used, the form which the equations of mathematical physics assumes depends on the coordinate system used to describe the problem being studied. This dependence, which is due to the practice of expressing vectors in terms of their physical components, can be removed by the simple expedient of expressing all vectors in terms of their tensor components.
Book Synopsis Mathematical Modeling of Diverse Phenomena by : James C. Howard
Download or read book Mathematical Modeling of Diverse Phenomena written by James C. Howard and published by . This book was released on 2004-12-01 with total page 408 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is intended for those students, engineers, scientists, and applied mathematicians who find it necessary to formulate models of diverse phenomena. To facilitate the formulation of such models, some aspects of the tensor calculus will be introduced. However, no knowledge of tensors is assumed. The chief aim of this calculus is the investigation of relations that remain valid in going from one coordinate system to another. The invariance of tensor quantities with respect to coordinate transformations can be used to advantage in formulating mathematical models. As a consequence of the geometrical simplification inherent in the tensor method, the formulation of problems in curvilinear coordinate systems can be reduced to series of routine operations involving only summation and differentiation. When conventional methods are used, the form which the equations of mathematical physics assumes depends on the coordinate system used to describe the problem being studied. This dependence, which is due to the practice of expressing vectors in terms of their physical components, can be removed by the simple expedient of expressing all vectors in terms of their tensor components.
This book is intended for those students, engineers, scientists, and applied mathematicians who find it necessary to formulate models of diverse phenomena.
Book Synopsis Mathematical Modeling of Diverse Phenomena by : James Howard
Download or read book Mathematical Modeling of Diverse Phenomena written by James Howard and published by CreateSpace. This book was released on 2014-04-15 with total page 402 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is intended for those students, engineers, scientists, and applied mathematicians who find it necessary to formulate models of diverse phenomena.
Mathematical modeling in the form of differential equations is a branch of applied mathematics that includes topics from physics, engineering, environmental and computer science. The mathematical model is an approximate description of real processes. Mathematical modeling can be thought of as a three step process: 1) Physical situation; 2) Mathematical formulation; 3) Solution by purely operations of the mathematical problem; 4) Physical interpretation of the mathematical solution. Over the centuries, Step 2 took on a life of its own. Mathematics was studied on its own, devoid of any contact with a physical problem; this is known as pure mathematics. Applied mathematics and mathematical modeling deals with all three steps. Improvements of approximations or their extensions to more general situations may increase the complexity of mathematical models significantly. Before the 18th century, applied mathematics and its methods received the close attention of the best mathematicians who were driven by a desire to develop approximate descriptions of natural phenomena. The goal of asymptotic and perturbation methods is to find useful, approximate solutions to difficult problems that arise from the desire to understand a physical process. Exact solutions are usually either impossible to obtain or too complicated to be useful. Approximate, useful solutions are often tested by comparison with experiments or observations rather than by rigorous mathematical methods. Hence, the authors will not be concerned with rigorous proofs in this book. The derivation of approximate solutions can be done in two different ways. First, one can find an approximate set of equations that can be solved, or, one can find an approximate solution of a set of equations. Usually one must do both. Models of natural science show that the possibilities of applying differential equations for solving problems in the disciplines of the natural scientific cycle are quite wide. This book represents a unique blend of the traditional analytical and numerical methods enriched by the authors developments and applications to ocean and atmospheric sciences. The overall viewpoint taken is a theoretical, unified approach to the study of both the atmosphere and the oceans. One of the key features in this book is the combination of approximate forms of the basic mathematical equations of mathematical modeling with careful and precise analysis. The approximations are required to make any progress possible, while precision is needed to make the progress meaningful. This combination is often the most elusive for student to appreciate. This book aims to highlight this issue by means of accurate derivation of mathematical models with precise analysis and MATLAB applications. This book is meant for undergraduate and graduate students interested in applied mathematics, differential equations and mathematical modeling of real world problems. This book might also be interested in experts working in the field of physics concerning the ocean and atmosphere.
Book Synopsis Mathematical Modeling of Natural Phenomena by : Ranis Ibragimov
Download or read book Mathematical Modeling of Natural Phenomena written by Ranis Ibragimov and published by . This book was released on 2017-12 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematical modeling in the form of differential equations is a branch of applied mathematics that includes topics from physics, engineering, environmental and computer science. The mathematical model is an approximate description of real processes. Mathematical modeling can be thought of as a three step process: 1) Physical situation; 2) Mathematical formulation; 3) Solution by purely operations of the mathematical problem; 4) Physical interpretation of the mathematical solution. Over the centuries, Step 2 took on a life of its own. Mathematics was studied on its own, devoid of any contact with a physical problem; this is known as pure mathematics. Applied mathematics and mathematical modeling deals with all three steps. Improvements of approximations or their extensions to more general situations may increase the complexity of mathematical models significantly. Before the 18th century, applied mathematics and its methods received the close attention of the best mathematicians who were driven by a desire to develop approximate descriptions of natural phenomena. The goal of asymptotic and perturbation methods is to find useful, approximate solutions to difficult problems that arise from the desire to understand a physical process. Exact solutions are usually either impossible to obtain or too complicated to be useful. Approximate, useful solutions are often tested by comparison with experiments or observations rather than by rigorous mathematical methods. Hence, the authors will not be concerned with rigorous proofs in this book. The derivation of approximate solutions can be done in two different ways. First, one can find an approximate set of equations that can be solved, or, one can find an approximate solution of a set of equations. Usually one must do both. Models of natural science show that the possibilities of applying differential equations for solving problems in the disciplines of the natural scientific cycle are quite wide. This book represents a unique blend of the traditional analytical and numerical methods enriched by the authors developments and applications to ocean and atmospheric sciences. The overall viewpoint taken is a theoretical, unified approach to the study of both the atmosphere and the oceans. One of the key features in this book is the combination of approximate forms of the basic mathematical equations of mathematical modeling with careful and precise analysis. The approximations are required to make any progress possible, while precision is needed to make the progress meaningful. This combination is often the most elusive for student to appreciate. This book aims to highlight this issue by means of accurate derivation of mathematical models with precise analysis and MATLAB applications. This book is meant for undergraduate and graduate students interested in applied mathematics, differential equations and mathematical modeling of real world problems. This book might also be interested in experts working in the field of physics concerning the ocean and atmosphere.
This is a book about the nature of mathematical modeling, and about the kinds of techniques that are useful for modeling. The text is in four sections. The first covers exact and approximate analytical techniques; the second, numerical methods; the third, model inference based on observations; and the last, the special role of time in modeling. Each of the topics in the book would be the worthy subject of a dedicated text, but only by presenting the material in this way is it possible to make so much material accessible to so many people. Each chapter presents a concise summary of the core results in an area. The text is complemented by extensive worked problems.
Book Synopsis The Nature of Mathematical Modeling by : Neil A. Gershenfeld
Download or read book The Nature of Mathematical Modeling written by Neil A. Gershenfeld and published by Cambridge University Press. This book was released on 1999 with total page 268 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a book about the nature of mathematical modeling, and about the kinds of techniques that are useful for modeling. The text is in four sections. The first covers exact and approximate analytical techniques; the second, numerical methods; the third, model inference based on observations; and the last, the special role of time in modeling. Each of the topics in the book would be the worthy subject of a dedicated text, but only by presenting the material in this way is it possible to make so much material accessible to so many people. Each chapter presents a concise summary of the core results in an area. The text is complemented by extensive worked problems.
Broadly speaking, there are two general approaches to teaching mathematical modeling: 1) the case study approach, and 2) the method based approach (that teaches mathematical techniques with applications to relevant mathematical models). This text emphasizes instead the scientific issues for modeling different phenomena. For the natural or harvested growth of a fish population, we may be interested in the evolution of the population, whether it reaches a steady state (equilibrium or cycle), stable or unstable with respect to a small perturbation from equilibrium, or whether a small change in the environment would cause a catastrophic change, etc. Each scientific issue requires an appropriate model and a different set of mathematical tools to extract information from the model. Models examined are chosen to help explain or justify empirical observations such as cocktail drug treatments are more effective and regenerations after injuries or illness are fast-tracked (compared to original developments). Volume I of this three-volume set limits its scope to phenomena and scientific issues that are modeled by ordinary differential equations (ODE). Scientific issues such as signal and wave propagation, diffusion, and shock formation involving spatial dynamics to be modeled by partial differential equations (PDE) will be treated in Vol. II. Scientific issues involving randomness and uncertainty are examined in Vol. III. Request Inspection Copy Contents: Mathematical Models and the Modeling CycleGrowth of a Population:Evolution and EquilibriumStability and BifurcationInteracting Populations:Linear InteractionsNonlinear Autonomous InteractionsHIV Dynamics and Drug TreatmentsIndex Theory, Bistability and FeedbackOptimization:The Economics of GrowthOptimization over a Planning PeriodModifications of the Basic ProblemBoundary Value Problems are More ComplexConstraints and Control:"Do Your Best" and the Maximum PrincipleChlamydia TrachomatisGenetic Instability and CarcinogenesisMathematical Modeling RevisitedAppendices:First Order ODEBasic Numerical MethodsAssignments Readership: Undergraduates in mathematical biology, mathematical modeling of dynamical systems, optimization and control, viral dynamics (infectious diseases), oncology.
Book Synopsis Dynamical System Models in the Life Sciences and Their Underlying Scientific Issues by : Frederic Y M Wan
Download or read book Dynamical System Models in the Life Sciences and Their Underlying Scientific Issues written by Frederic Y M Wan and published by World Scientific Publishing Company. This book was released on 2017-08-16 with total page 400 pages. Available in PDF, EPUB and Kindle. Book excerpt: Broadly speaking, there are two general approaches to teaching mathematical modeling: 1) the case study approach, and 2) the method based approach (that teaches mathematical techniques with applications to relevant mathematical models). This text emphasizes instead the scientific issues for modeling different phenomena. For the natural or harvested growth of a fish population, we may be interested in the evolution of the population, whether it reaches a steady state (equilibrium or cycle), stable or unstable with respect to a small perturbation from equilibrium, or whether a small change in the environment would cause a catastrophic change, etc. Each scientific issue requires an appropriate model and a different set of mathematical tools to extract information from the model. Models examined are chosen to help explain or justify empirical observations such as cocktail drug treatments are more effective and regenerations after injuries or illness are fast-tracked (compared to original developments). Volume I of this three-volume set limits its scope to phenomena and scientific issues that are modeled by ordinary differential equations (ODE). Scientific issues such as signal and wave propagation, diffusion, and shock formation involving spatial dynamics to be modeled by partial differential equations (PDE) will be treated in Vol. II. Scientific issues involving randomness and uncertainty are examined in Vol. III. Request Inspection Copy Contents: Mathematical Models and the Modeling CycleGrowth of a Population:Evolution and EquilibriumStability and BifurcationInteracting Populations:Linear InteractionsNonlinear Autonomous InteractionsHIV Dynamics and Drug TreatmentsIndex Theory, Bistability and FeedbackOptimization:The Economics of GrowthOptimization over a Planning PeriodModifications of the Basic ProblemBoundary Value Problems are More ComplexConstraints and Control:"Do Your Best" and the Maximum PrincipleChlamydia TrachomatisGenetic Instability and CarcinogenesisMathematical Modeling RevisitedAppendices:First Order ODEBasic Numerical MethodsAssignments Readership: Undergraduates in mathematical biology, mathematical modeling of dynamical systems, optimization and control, viral dynamics (infectious diseases), oncology.
Spontaneous Phenomena: A Mathematical Analysis covers certain aspects in the teaching of mathematics, including historical perspective, model-building, and the inner nature of mathematics. This book is organized into 12 chapters beginning with the development of the relevant mathematics and physics. This topic is followed by considerable chapters on the theoretical and statistical principles of mathematical analysis, with an emphasis on a model for a radioactive decay. Other chapters discuss various phenomena within biology, medicine, statistics of medicine, determination of age, traffic analysis, and other fields. The concluding chapters present the fundamentals of the Poisson approximation to the binomial distribution and the chi-square test for goodness of fit. This book is an ideal source for mathematics and physics pre-college and early college students.
Book Synopsis Spontaneous Phenomena by :
Download or read book Spontaneous Phenomena written by and published by Elsevier. This book was released on 2012-12-02 with total page 195 pages. Available in PDF, EPUB and Kindle. Book excerpt: Spontaneous Phenomena: A Mathematical Analysis covers certain aspects in the teaching of mathematics, including historical perspective, model-building, and the inner nature of mathematics. This book is organized into 12 chapters beginning with the development of the relevant mathematics and physics. This topic is followed by considerable chapters on the theoretical and statistical principles of mathematical analysis, with an emphasis on a model for a radioactive decay. Other chapters discuss various phenomena within biology, medicine, statistics of medicine, determination of age, traffic analysis, and other fields. The concluding chapters present the fundamentals of the Poisson approximation to the binomial distribution and the chi-square test for goodness of fit. This book is an ideal source for mathematics and physics pre-college and early college students.
This conference series intends to illuminate the relationship between different types of waves. This second conference focused primarily on classical wave modeling of acoustic waves in solids and fluids, electromagnetic waves, as well as elastic wave modeling, and both direct and inverse problems are addressed. Topics included are: (1) Classical linear wave propagation modeling, analysis and computation: general, electromagnetic applications, acoustics of fluids, acoustics of solids; (2) classical nonlinear wave propagation modeling, analysis, and computation; (3) inverse scattering modeling: gneral and electromagnetic imaging, wood imaging, seismic imaging; (4) quantum and statistical mechanics; (5) signal processing and analysis.
Book Synopsis Mathematical Modelling of Wave Phenomena by : Börje Nilsson
Download or read book Mathematical Modelling of Wave Phenomena written by Börje Nilsson and published by American Institute of Physics. This book was released on 2006-05-12 with total page 406 pages. Available in PDF, EPUB and Kindle. Book excerpt: This conference series intends to illuminate the relationship between different types of waves. This second conference focused primarily on classical wave modeling of acoustic waves in solids and fluids, electromagnetic waves, as well as elastic wave modeling, and both direct and inverse problems are addressed. Topics included are: (1) Classical linear wave propagation modeling, analysis and computation: general, electromagnetic applications, acoustics of fluids, acoustics of solids; (2) classical nonlinear wave propagation modeling, analysis, and computation; (3) inverse scattering modeling: gneral and electromagnetic imaging, wood imaging, seismic imaging; (4) quantum and statistical mechanics; (5) signal processing and analysis.
Models of Science Dynamics aims to capture the structure and evolution of science, the emerging arena in which scholars, science and the communication of science become themselves the basic objects of research. In order to capture the essence of phenomena as diverse as the structure of co-authorship networks or the evolution of citation diffusion patterns, such models can be represented by conceptual models based on historical and ethnographic observations, mathematical descriptions of measurable phenomena, or computational algorithms. Despite its evident importance, the mathematical modeling of science still lacks a unifying framework and a comprehensive study of the topic. This volume fills this gap, reviewing and describing major threads in the mathematical modeling of science dynamics for a wider academic and professional audience. The model classes presented cover stochastic and statistical models, system-dynamics approaches, agent-based simulations, population-dynamics models, and complex-network models. The book comprises an introduction and a foundational chapter that defines and operationalizes terminology used in the study of science, as well as a review chapter that discusses the history of mathematical approaches to modeling science from an algorithmic-historiography perspective. It concludes with a survey of remaining challenges for future science models and their relevance for science and science policy.
Book Synopsis Models of Science Dynamics by : Andrea Scharnhorst
Download or read book Models of Science Dynamics written by Andrea Scharnhorst and published by Springer Science & Business Media. This book was released on 2012-01-23 with total page 292 pages. Available in PDF, EPUB and Kindle. Book excerpt: Models of Science Dynamics aims to capture the structure and evolution of science, the emerging arena in which scholars, science and the communication of science become themselves the basic objects of research. In order to capture the essence of phenomena as diverse as the structure of co-authorship networks or the evolution of citation diffusion patterns, such models can be represented by conceptual models based on historical and ethnographic observations, mathematical descriptions of measurable phenomena, or computational algorithms. Despite its evident importance, the mathematical modeling of science still lacks a unifying framework and a comprehensive study of the topic. This volume fills this gap, reviewing and describing major threads in the mathematical modeling of science dynamics for a wider academic and professional audience. The model classes presented cover stochastic and statistical models, system-dynamics approaches, agent-based simulations, population-dynamics models, and complex-network models. The book comprises an introduction and a foundational chapter that defines and operationalizes terminology used in the study of science, as well as a review chapter that discusses the history of mathematical approaches to modeling science from an algorithmic-historiography perspective. It concludes with a survey of remaining challenges for future science models and their relevance for science and science policy.
This book highlights the remarkable importance of special functions, operational calculus, and variational methods. A considerable portion of the book is dedicated to second-order partial differential equations, as they offer mathematical models of various phenomena in physics and engineering. The book provides students and researchers with essential help on key mathematical topics, which are applied to a range of practical problems. These topics were chosen because, after teaching university courses for many years, the authors have found them to be essential, especially in the contexts of technology, engineering and economics. Given the diversity topics included in the book, the presentation of each is limited to the basic notions and results of the respective mathematical domain. Chapter 1 is devoted to complex functions. Here, much emphasis is placed on the theory of holomorphic functions, which facilitate the understanding of the role that the theory of functions of a complex variable plays in mathematical physics, especially in the modeling of plane problems. In addition, the book demonstrates the importance of the theories of special functions, operational calculus, and variational calculus. In the last chapter, the authors discuss the basic elements of one of the most modern areas of mathematics, namely the theory of optimal control.
Book Synopsis Complements of Higher Mathematics by : Marin Marin
Download or read book Complements of Higher Mathematics written by Marin Marin and published by Springer. This book was released on 2018-02-13 with total page 353 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book highlights the remarkable importance of special functions, operational calculus, and variational methods. A considerable portion of the book is dedicated to second-order partial differential equations, as they offer mathematical models of various phenomena in physics and engineering. The book provides students and researchers with essential help on key mathematical topics, which are applied to a range of practical problems. These topics were chosen because, after teaching university courses for many years, the authors have found them to be essential, especially in the contexts of technology, engineering and economics. Given the diversity topics included in the book, the presentation of each is limited to the basic notions and results of the respective mathematical domain. Chapter 1 is devoted to complex functions. Here, much emphasis is placed on the theory of holomorphic functions, which facilitate the understanding of the role that the theory of functions of a complex variable plays in mathematical physics, especially in the modeling of plane problems. In addition, the book demonstrates the importance of the theories of special functions, operational calculus, and variational calculus. In the last chapter, the authors discuss the basic elements of one of the most modern areas of mathematics, namely the theory of optimal control.
