A Modern Introduction to Differential Equations


Author: Henry J. Ricardo
Publisher: Academic Press
ISBN: 0080886035
Category: Mathematics
Page: 536
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A Modern Introduction to Differential Equations, Second Edition, provides an introduction to the basic concepts of differential equations. The book begins by introducing the basic concepts of differential equations, focusing on the analytical, graphical, and numerical aspects of first-order equations, including slope fields and phase lines. The discussions then cover methods of solving second-order homogeneous and nonhomogeneous linear equations with constant coefficients; systems of linear differential equations; the Laplace transform and its applications to the solution of differential equations and systems of differential equations; and systems of nonlinear equations. Each chapter concludes with a summary of the important concepts in the chapter. Figures and tables are provided within sections to help students visualize or summarize concepts. The book also includes examples and exercises drawn from biology, chemistry, and economics, as well as from traditional pure mathematics, physics, and engineering. This book is designed for undergraduate students majoring in mathematics, the natural sciences, and engineering. However, students in economics, business, and the social sciences with the necessary background will also find the text useful. Student friendly readability- assessible to the average student Early introduction of qualitative and numerical methods Large number of exercises taken from biology, chemistry, economics, physics and engineering Exercises are labeled depending on difficulty/sophistication End of chapter summaries Group projects

A Modern Introduction to Differential Equations

Mathematics, Differential equations
Author: CTI Reviews
Publisher: Cram101 Textbook Reviews
ISBN: 1478422653
Category: Education
Page: 20
View: 1378

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Facts101 is your complete guide to A Modern Introduction to Differential Equations. In this book, you will learn topics such as The Numerical Approximation of Solutions, Second- and Higher-Order Equations, Systems of Linear Differential Equations, and The Laplace Transform plus much more. With key features such as key terms, people and places, Facts101 gives you all the information you need to prepare for your next exam. Our practice tests are specific to the textbook and we have designed tools to make the most of your limited study time.

Differential Equations

An Introduction to Basic Concepts, Results and Applications Third Edition
Author: Ioan I Vrabie
Publisher: World Scientific Publishing Company
ISBN: 981474980X
Category: Mathematics
Page: 528
View: 908

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This book presents, in a unitary frame and from a new perspective, the main concepts and results of one of the most fascinating branches of modern mathematics, namely differential equations, and offers the reader another point of view concerning a possible way to approach the problems of existence, uniqueness, approximation, and continuation of the solutions to a Cauchy problem. In addition, it contains simple introductions to some topics which are not usually included in classical textbooks: the exponential formula, conservation laws, generalized solutions, Caratheodory solutions, differential inclusions, variational inequalities, viability, invariance, and gradient systems. In this new edition, some typos have been corrected and two new topics have been added: Delay differential equations and differential equations subjected to nonlocal initial conditions. The bibliography has also been updated and expanded.

Introduction to Partial Differential Equations


Author: G. B. Folland
Publisher: Princeton University Press
ISBN: 9780691043616
Category: Mathematics
Page: 324
View: 7545

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The aim of this text is to aquaint the student with the fundamental classical results of partial differential equations and to guide them into some of the modern theory, enabling them to read more advanced works on the subject

Ordinary Differential Equations

Introduction to the Theory of Ordinary Differential Equations in the Real Domain
Author: J. Kurzweil
Publisher: Elsevier
ISBN: 1483297659
Category: Mathematics
Page: 440
View: 2055

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The author, Professor Kurzweil, is one of the world's top experts in the area of ordinary differential equations - a fact fully reflected in this book. Unlike many classical texts which concentrate primarily on methods of integration of differential equations, this book pursues a modern approach: the topic is discussed in full generality which, at the same time, permits us to gain a deep insight into the theory and to develop a fruitful intuition. The basic framework of the theory is expanded by considering further important topics like stability, dependence of a solution on a parameter, Carathéodory's theory and differential relations. The book is very well written, and the prerequisites needed are minimal - some basics of analysis and linear algebra. As such, it is accessible to a wide circle of readers, in particular to non-mathematicians.

A Modern Introduction to Linear Algebra


Author: Henry Ricardo
Publisher: CRC Press
ISBN: 1439894612
Category: Mathematics
Page: 670
View: 1787

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Useful Concepts and Results at the Heart of Linear Algebra A one- or two-semester course for a wide variety of students at the sophomore/junior undergraduate level A Modern Introduction to Linear Algebra provides a rigorous yet accessible matrix-oriented introduction to the essential concepts of linear algebra. Concrete, easy-to-understand examples motivate the theory. The book first discusses vectors, Gaussian elimination, and reduced row echelon forms. It then offers a thorough introduction to matrix algebra, including defining the determinant naturally from the PA=LU factorization of a matrix. The author goes on to cover finite-dimensional real vector spaces, infinite-dimensional spaces, linear transformations, and complex vector spaces. The final chapter presents Hermitian and normal matrices as well as quadratic forms. Taking a computational, algebraic, and geometric approach to the subject, this book provides the foundation for later courses in higher mathematics. It also shows how linear algebra can be used in various areas of application. Although written in a "pencil and paper" manner, the text offers ample opportunities to enhance learning with calculators or computer usage. Solutions manual available for qualifying instructors

Modern Differential Equations


Author: Martha L. Abell,James P. Braselton
Publisher: Brooks/Cole Publishing Company
ISBN: N.A
Category: Mathematics
Page: 700
View: 2507

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1. Introduction to Differential Equations. Introduction. A Graphical Approach to Solutions: Slope Fields and Direction Fields. Summary. Review Exercises. 2. First Order Equations. Separable Equations. First-Order Linear Equations. Substitution Methods and Special Equations. Exact Equations. Theory of First-Order-Equations. Numerical Methods for First-Order Equations. Summary. Review Exercises. Differential Equations at Work. Modeling the Spread of a Disease. Linear Population Model with Harvesting. Logistic Model with Harvesting. Logistic Model with Predation. 3. Applications of First Order Equations. Population Growth and Decay. Newton's Law of Cooling and Related Problems. Free-Falling Bodies. Summary. Review Exercises. Chapter 3 Differential Equations at Work. Mathematics of Finance. Algae Growth. Dialysis. Antibiotic Production. 4. Higher Order Equations. Second-Order Equations: An Introduction. Solutions of Second-Order Linear Homogeneous Equations with Constant Coefficients. Higher Order Equations: An Introduction. Solutions to Higher Order Linear Homogeneous Equations with Constant Coefficients. Introduction to Solving Nonhomogeneous Equations with Constant Coefficients: Method of Undetermined Coefficients. Nonhomogeneous Equations with Constant Coefficients: Variation of Parameters. Cauchy-Euler Equations. Series Solutions of Ordinary Differential Equations. Summary. Review Exercises. Differential Equations at Work. Testing for Diabetes. Modeling the Motion of a Skier. The Schröinger Equation. 5. Applications of Higher Order Equations. Simple Harmonic Motion. Damped Motion. Forced Motion. Other Applications. The Pendulum Problem. Summary. Review Exercises. Differential Equations at Work. Rack-and-Gear Systems. Soft Springs. Hard Springs. Aging Springs. Bodé Plots. 6. Systems of First Order Equations. Introduction. Review of Matrix Algebra and Calculus. Preliminary Definitions and Notation. First-Order Linear Homogeneous Systems with Constant Coefficients. First-Order Linear Nonhomogeneous Systems: Undetermined Coefficients and Variation of Parameters. Phase Portraits. Nonlinear Systems. Numerical Methods. Summary. Review Exercises. Differential Equations at Work. Modeling a Fox Population in Which Rabies is Present. Controlling the Spread of Disease. FitzHugh-Nagumo Model. 7. Applications of First-Order Systems. Mechanical and Electrical Problems with First-Order Linear Systems. Diffusion and Population Problems with First-Order Linear Systems. Nonlinear Systems of Equations. Summary. Review Exercises. Differential Equations at Work. Competing Species. Food Chains. Chemical Reactor. 8. Laplace Transforms. The Laplace Transform: Preliminary Definitions and Notation. Solving Initial-Value Problems with the Laplace Transform. Laplace Transforms of Several Important Functions. The Convolution Theorem. Laplace Transform Methods for Solving Systems. Applications Using Laplace Transforms. Summary. Review Exercises. Differential Equations at Work. The Tautochrone. Vibration Absorbers. Airplane Wing. Free Vibration of a Three-Story Building. Control Systems. 9. Fourier Series. Boundary-Value Problems, Eigenvalue Problems, Sturm-Liouville Problems. Fourier Sine Series and Cosine Series. Fourier Series. Generalized Fourier Series. Summary. Review Exercises. Differential Equations at Work. Free Vibration of a Three-Story Building. Forced Damped Spring-Mass System. Approximations with Fourier Series. 10. Partial Differential Equations. Introduction to Partial Differential Equations and Separation of Variables. The One-Dimensional Heat Equation. The One-Dimensional Wave Equation. Problems in Two Dimensions: Laplace's Equation. Two-Dimensional Problems in a Circular Region. Summary. Review Exercises. Differential Equations at Work. Laplace Transforms. Waves in a Steel Rod. Media Sterilization. Numerical Methods for Solving Partial Differential Equations. Answers to Selected Questions. Index.

Mathematical Physics

A Modern Introduction to Its Foundations
Author: Sadri Hassani
Publisher: Springer Science & Business Media
ISBN: 9780387985794
Category: Science
Page: 1026
View: 9969

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For physics students interested in the mathematics they use, and for math students interested in seeing how some of the ideas of their discipline find realization in an applied setting. The presentation strikes a balance between formalism and application, between abstract and concrete. The interconnections among the various topics are clarified both by the use of vector spaces as a central unifying theme, recurring throughout the book, and by putting ideas into their historical context. Enough of the essential formalism is included to make the presentation self-contained.

Differential Equations

An Introduction to Modern Methods and Applications
Author: William E. Boyce
Publisher: John Wiley & Sons
ISBN: 0470458240
Category: Mathematics
Page: 704
View: 6487

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Unlike other books in the market, this second edition presents differential equations consistent with the way scientists and engineers use modern methods in their work. Technology is used freely, with more emphasis on modeling, graphical representation, qualitative concepts, and geometric intuition than on theoretical issues. It also refers to larger-scale computations that computer algebra systems and DE solvers make possible. And more exercises and examples involving working with data and devising the model provide scientists and engineers with the tools needed to model complex real-world situations.

Scientific Computing and Differential Equations

An Introduction to Numerical Methods
Author: Gene H. Golub,James M. Ortega
Publisher: Elsevier
ISBN: 0080516696
Category: Mathematics
Page: 344
View: 1247

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Scientific Computing and Differential Equations: An Introduction to Numerical Methods, is an excellent complement to Introduction to Numerical Methods by Ortega and Poole. The book emphasizes the importance of solving differential equations on a computer, which comprises a large part of what has come to be called scientific computing. It reviews modern scientific computing, outlines its applications, and places the subject in a larger context. This book is appropriate for upper undergraduate courses in mathematics, electrical engineering, and computer science; it is also well-suited to serve as a textbook for numerical differential equations courses at the graduate level. An introductory chapter gives an overview of scientific computing, indicating its important role in solving differential equations, and placing the subject in the larger environment Contains an introduction to numerical methods for both ordinary and partial differential equations Concentrates on ordinary differential equations, especially boundary-value problems Contains most of the main topics for a first course in numerical methods, and can serve as a text for this course Uses material for junior/senior level undergraduate courses in math and computer science plus material for numerical differential equations courses for engineering/science students at the graduate level

A Modern Introduction to Dynamical Systems


Author: Richard Brown
Publisher: Oxford University Press
ISBN: 0198743289
Category: Mathematics
Page: 432
View: 2693

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This text is a high-level introduction to the modern theory of dynamical systems; an analysis-based, pure mathematics course textbook in the basic tools, techniques, theory and development of both the abstract and the practical notions of mathematical modelling, using both discrete and continuous concepts and examples comprising what may be called the modern theory of dynamics. Prerequisite knowledge is restricted to calculus, linear algebra and basic differential equations, and all higher-level analysis, geometry and algebra is introduced as needed within the text. Following this text from start to finish will provide the careful reader with the tools, vocabulary and conceptual foundation necessary to continue in further self-study and begin to explore current areas of active research in dynamical systems.

Introduction to Partial Differential Equations with MATLAB


Author: Jeffery M. Cooper,Jeffery Cooper
Publisher: Springer Science & Business Media
ISBN: 9780817639679
Category: Mathematics
Page: 540
View: 9436

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The subject of partial differential equations has an unchanging core of material but is constantly expanding and evolving. Introduction to Partial Differential Equations with MATLAB is a careful integration of traditional core topics with modern topics, taking full advantage of the computational power of MATLAB to enhance the learning experience. This advanced text/reference is an introduction to partial differential equations covering the traditional topics within a modern context. To provide an up-to-date treatment, techniques of numerical computation have been included with carefully selected nonlinear topics, including nonlinear first order equations. Each equation studied is placed in the appropriate physical context. The analytical aspects of solutions are discussed in an integrated fashion with extensive examples and exercises, both analytical and computational. The book is excellent for classroom use and can be used for self-study purposes. Topic and Features: • Nonlinear equations including nonlinear conservation laws; • Dispersive wave equations and the Schrodinger equation; • Numerical methods for each core equation including finite difference methods, finite element methods, and the fast Fourier transform; • Extensive use of MATLAB programs in exercise sets. MATLAB m files for numerical and graphics programs available by ftp from this web site. This text/reference is an excellent resources designed to introduce advanced students in mathematics, engineering and sciences to partial differential equations. It is also suitable as a self-study resource for professionals and practitioners.

Introduction to Partial Differential Equations with Applications


Author: E. C. Zachmanoglou,Dale W. Thoe
Publisher: Courier Corporation
ISBN: 048613217X
Category: Mathematics
Page: 432
View: 1922

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This text explores the essentials of partial differential equations as applied to engineering and the physical sciences. Discusses ordinary differential equations, integral curves and surfaces of vector fields, the Cauchy-Kovalevsky theory, more. Problems and answers.

Partial Differential Equations and the Finite Element Method


Author: Pavel Ŝolín
Publisher: John Wiley & Sons
ISBN: 0471764094
Category: Mathematics
Page: 512
View: 3005

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A systematic introduction to partial differential equations and modern finite element methods for their efficientnumerical solution Partial Differential Equations and the Finite Element Methodprovides a much-needed, clear, and systematic introduction tomodern theory of partial differential equations (PDEs) and finiteelement methods (FEM). Both nodal and hierachic concepts of the FEMare examined. Reflecting the growing complexity and multiscalenature of current engineering and scientific problems, the authoremphasizes higher-order finite element methods such as the spectralor hp-FEM. A solid introduction to the theory of PDEs and FEM contained inChapters 1-4 serves as the core and foundation of the publication.Chapter 5 is devoted to modern higher-order methods for thenumerical solution of ordinary differential equations (ODEs) thatarise in the semidiscretization of time-dependent PDEs by theMethod of Lines (MOL). Chapter 6 discusses fourth-order PDEs rootedin the bending of elastic beams and plates and approximates theirsolution by means of higher-order Hermite and Argyris elements.Finally, Chapter 7 introduces the reader to various PDEs governingcomputational electromagnetics and describes their finite elementapproximation, including modern higher-order edge elements forMaxwell's equations. The understanding of many theoretical and practical aspects of bothPDEs and FEM requires a solid knowledge of linear algebra andelementary functional analysis, such as functions and linearoperators in the Lebesgue, Hilbert, and Sobolev spaces. Thesetopics are discussed with the help of many illustrative examples inAppendix A, which is provided as a service for those readers whoneed to gain the necessary background or require a refreshertutorial. Appendix B presents several finite element computationsrooted in practical engineering problems and demonstrates thebenefits of using higher-order FEM. Numerous finite element algorithms are written out in detailalongside implementation discussions. Exercises, including manythat involve programming the FEM, are designed to assist the readerin solving typical problems in engineering and science. Specifically designed as a coursebook, this student-testedpublication is geared to upper-level undergraduates and graduatestudents in all disciplines of computational engineeringandscience. It is also a practical problem-solving reference forresearchers, engineers, and physicists.

Introduction to Partial Differential Equations

A Computational Approach
Author: Aslak Tveito,Ragnar Winther
Publisher: Springer Science & Business Media
ISBN: 0387227733
Category: Mathematics
Page: 392
View: 4730

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Combining both the classical theory and numerical techniques for partial differential equations, this thoroughly modern approach shows the significance of computations in PDEs and illustrates the strong interaction between mathematical theory and the development of numerical methods. Great care has been taken throughout the book to seek a sound balance between these techniques. The authors present the material at an easy pace and exercises ranging from the straightforward to the challenging have been included. In addition there are some "projects" suggested, either to refresh the students memory of results needed in this course, or to extend the theories developed in the text. Suitable for undergraduate and graduate students in mathematics and engineering.

Differential Equations with Boundary Value Problems

An Introduction to Modern Methods & Applications
Author: James R. Brannan
Publisher: John Wiley & Sons
ISBN: 0470595353
Category: Mathematics
Page: 976
View: 3735

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Unlike other books in the market, this second edition presents differential equations consistent with the way scientists and engineers use modern methods in their work. Technology is used freely, with more emphasis on modeling, graphical representation, qualitative concepts, and geometric intuition than on theoretical issues. It also refers to larger-scale computations that computer algebra systems and DE solvers make possible. And more exercises and examples involving working with data and devising the model provide scientists and engineers with the tools needed to model complex real-world situations.

A First Course in Ordinary Differential Equations

Analytical and Numerical Methods
Author: Martin Hermann,Masoud Saravi
Publisher: Springer Science & Business
ISBN: 8132218353
Category: Mathematics
Page: 288
View: 4597

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This book presents a modern introduction to analytical and numerical techniques for solving ordinary differential equations (ODEs). Contrary to the traditional format—the theorem-and-proof format—the book is focusing on analytical and numerical methods. The book supplies a variety of problems and examples, ranging from the elementary to the advanced level, to introduce and study the mathematics of ODEs. The analytical part of the book deals with solution techniques for scalar first-order and second-order linear ODEs, and systems of linear ODEs—with a special focus on the Laplace transform, operator techniques and power series solutions. In the numerical part, theoretical and practical aspects of Runge-Kutta methods for solving initial-value problems and shooting methods for linear two-point boundary-value problems are considered. The book is intended as a primary text for courses on the theory of ODEs and numerical treatment of ODEs for advanced undergraduate and early graduate students. It is assumed that the reader has a basic grasp of elementary calculus, in particular methods of integration, and of numerical analysis. Physicists, chemists, biologists, computer scientists and engineers whose work involves solving ODEs will also find the book useful as a reference work and tool for independent study. The book has been prepared within the framework of a German–Iranian research project on mathematical methods for ODEs, which was started in early 2012.