Solving Differential Equations in R

Author: Karline Soetaert,Jeff Cash,Francesca Mazzia

Publisher: Springer Science & Business Media

ISBN: 3642280706

Category: Computers

Page: 248

View: 3767

Mathematics plays an important role in many scientific and engineering disciplines. This book deals with the numerical solution of differential equations, a very important branch of mathematics. Our aim is to give a practical and theoretical account of how to solve a large variety of differential equations, comprising ordinary differential equations, initial value problems and boundary value problems, differential algebraic equations, partial differential equations and delay differential equations. The solution of differential equations using R is the main focus of this book. It is therefore intended for the practitioner, the student and the scientist, who wants to know how to use R for solving differential equations. However, it has been our goal that non-mathematicians should at least understand the basics of the methods, while obtaining entrance into the relevant literature that provides more mathematical background. Therefore, each chapter that deals with R examples is preceded by a chapter where the theory behind the numerical methods being used is introduced. In the sections that deal with the use of R for solving differential equations, we have taken examples from a variety of disciplines, including biology, chemistry, physics, pharmacokinetics. Many examples are well-known test examples, used frequently in the field of numerical analysis.

Gewöhnliche Differentialgleichungen

Eine Einführung

Author: Wolfgang Walter

Publisher: Springer-Verlag

ISBN: 364296317X

Category: Mathematics

Page: 232

View: 6168

Computer-Lösung gewöhnlicher Differentialgleichungen

Das Anfangswertproblem

Author: Lawrence F. Shampine,Marilyn K. Gordon

Publisher: Springer-Verlag

ISBN: 3322938018

Category: Mathematics

Page: 259

View: 8698

Vorlesungen über partielle Differentialgleichungen

Publisher: Springer-Verlag

ISBN: 3540350314

Category: Mathematics

Page: 174

View: 2791

Nach seinem bekannten und viel verwendeten Buch über gewöhnliche Differentialgleichungen widmet sich der berühmte Mathematiker Vladimir Arnold nun den partiellen Differentialgleichungen in einem neuen Lehrbuch. In seiner unnachahmlich eleganten Art führt er über einen geometrischen, anschaulichen Weg in das Thema ein, und ermöglicht den Lesern so ein vertieftes Verständnis der Natur der partiellen Differentialgleichungen. Für Studierende der Mathematik und Physik ist dieses Buch ein Muss. Wie alle Bücher Vladimir Arnolds ist dieses Buch voller geometrischer Erkenntnisse. Arnold illustriert jeden Grundsatz mit einer Abbildung. Das Buch behandelt die elementarsten Teile des Fachgebiets and beschränkt sich hauptsächlich auf das Cauchy-Problem und das Neumann-Problems für die klassischen Lineargleichungen der mathematischen Physik, insbesondere auf die Laplace-Gleichung und die Wellengleichung, wobei die Wärmeleitungsgleichung und die Korteweg-de-Vries-Gleichung aber ebenfalls diskutiert werden. Die physikalische Intuition wird besonders hervorgehoben. Eine große Anzahl von Problemen ist übers ganze Buch verteilt, und ein ganzer Satz von Aufgaben findet sich am Ende. Was dieses Buch so einzigartig macht, ist das besondere Talent Arnolds, ein Thema aus einer neuen, frischen Perspektive zu beleuchten. Er lüftet gerne den Schleier der Verallgemeinerung, der so viele mathematische Texte umgibt, und enthüllt die im wesentlichen einfachen, intuitiven Ideen, die dem Thema zugrunde liegen. Das kann er besser als jeder andere mathematische Autor.

Spline Collocation Methods for Partial Differential Equations

With Applications in R

Author: William E. Schiesser

Publisher: John Wiley & Sons

ISBN: 1119301033

Category: MATHEMATICS

Page: 576

View: 1996

One-dimensional PDEs -- Multidimensional PDEs -- Navier-Stokes, Burgers equations -- Korteweg-deVries equation -- Maxwell equations -- Poisson-Nernst-Planck equations -- Fokker-Planck equation -- Fisher-Kolmogorov equation -- Klein-Gordon equation -- Boussinesq equation -- Cahn-Hilliard equation -- Camassa-Holm equation -- Burgers-Huxley equation -- Gierer-Meinhardt equations -- Keller-Segel equations -- Fitzhugh-Nagumo equations -- Euler-Poisson-Darboux equation -- Kuramoto-Sivashinsky equation -- Einstein-Maxwell equations

Differentialgleichungen und ihre Anwendungen

Author: Martin Braun

Publisher: Springer-Verlag

ISBN: 3642973418

Category: Mathematics

Page: 596

View: 337

Dieses richtungsweisende Lehrbuch für die Anwendung der Mathematik in anderen Wissenschaftszweigen gibt eine Einführung in die Theorie der gewöhnlichen Differentialgleichungen. Fortran und APL-Programme geben den Studenten die Möglichkeit, verschiedene numerische Näherungsverfahren an ihrem PC selbst durchzurechnen. Aus den Besprechungen: "Die Darstellung ist überall mathematisch streng und zudem ungemein anregend. Abgesehen von manchen historischen Bemerkungen ... tragen dazu die vielen mit ausführlichem Hintergrund sehr eingehend entwickelten praktischen Anwendungen bei. ... Besondere Aufmerksamkeit wird der physikalisch und technisch so wichtigen Frage nach Stabilität von Lösungen eines Systems von Differentialgleichungen gewidmet. Das Buch ist wegen seiner geringen Voraussetzungen und vorzüglichen Didaktik schon für alle Studenten des 3. Semesters geeignet; seine eminent praktische Haltung empfiehlt es aber auch für alle Physiker, die mit Differentialgleichungen und ihren Anwendungen umzugehen haben." #Physikalische Blätter#

Partielle Differentialgleichungen

Eine Einführung

Author: Walter A. Strauss

Publisher: Springer-Verlag

ISBN: 366312486X

Category: Mathematics

Page: 458

View: 4676

Dieses Buch ist eine umfassende Einführung in die klassischen Lösungsmethoden partieller Differentialgleichungen. Es wendet sich an Leser mit Kenntnissen aus einem viersemestrigen Grundstudium der Mathematik (und Physik) und legt seinen Schwerpunkt auf die explizite Darstellung der Lösungen. Es ist deshalb besonders auch für Anwender (Physiker, Ingenieure) sowie für Nichtspezialisten, die die Methoden der mathematischen Physik kennenlernen wollen, interessant. Durch die große Anzahl von Beispielen und Übungsaufgaben eignet es sich gut zum Gebrauch neben Vorlesungen sowie zum Selbststudium.

Using R for Numerical Analysis in Science and Engineering

Author: Victor A. Bloomfield

Publisher: CRC Press

ISBN: 1315360497

Category: Mathematics

Page: 359

View: 9952

Instead of presenting the standard theoretical treatments that underlie the various numerical methods used by scientists and engineers, Using R for Numerical Analysis in Science and Engineering shows how to use R and its add-on packages to obtain numerical solutions to the complex mathematical problems commonly faced by scientists and engineers. This practical guide to the capabilities of R demonstrates Monte Carlo, stochastic, deterministic, and other numerical methods through an abundance of worked examples and code, covering the solution of systems of linear algebraic equations and nonlinear equations as well as ordinary differential equations and partial differential equations. It not only shows how to use R’s powerful graphic tools to construct the types of plots most useful in scientific and engineering work, but also: Explains how to statistically analyze and fit data to linear and nonlinear models Explores numerical differentiation, integration, and optimization Describes how to find eigenvalues and eigenfunctions Discusses interpolation and curve fitting Considers the analysis of time series Using R for Numerical Analysis in Science and Engineering provides a solid introduction to the most useful numerical methods for scientific and engineering data analysis using R.

Differential Equation Analysis in Biomedical Science and Engineering

Partial Differential Equation Applications with R

Author: William E. Schiesser

Publisher: John Wiley & Sons

ISBN: 1118705165

Category: Mathematics

Page: 344

View: 6735

Features a solid foundation of mathematical and computationaltools to formulate and solve real-world PDE problems across variousfields With a step-by-step approach to solving partial differentialequations (PDEs), Differential Equation Analysis in BiomedicalScience and Engineering: Partial Differential Equation Applicationswith R successfully applies computational techniques forsolving real-world PDE problems that are found in a variety offields, including chemistry, physics, biology, and physiology. Thebook provides readers with the necessary knowledge to reproduce andextend the computed numerical solutions and is a valuable resourcefor dealing with a broad class of linear and nonlinear partialdifferential equations. The author’s primary focus is on models expressed assystems of PDEs, which generally result from including spatialeffects so that the PDE dependent variables are functions of bothspace and time, unlike ordinary differential equation (ODE) systemsthat pertain to time only. As such, the book emphasizes details ofthe numerical algorithms and how the solutions were computed.Featuring computer-based mathematical models for solving real-worldproblems in the biological and biomedical sciences and engineering,the book also includes: R routines to facilitate the immediate use of computation forsolving differential equation problems without having to firstlearn the basic concepts of numerical analysis and programming forPDEs Models as systems of PDEs and associated initial and boundaryconditions with explanations of the associated chemistry, physics,biology, and physiology Numerical solutions of the presented model equations with adiscussion of the important features of the solutions Aspects of general PDE computation through various biomedicalscience and engineering applications Differential Equation Analysis in Biomedical Science andEngineering: Partial Differential Equation Applications with Ris an excellent reference for researchers, scientists, clinicians,medical researchers, engineers, statisticians, epidemiologists, andpharmacokineticists who are interested in both clinicalapplications and interpretation of experimental data withmathematical models in order to efficiently solve the associateddifferential equations. The book is also useful as a textbook forgraduate-level courses in mathematics, biomedical science andengineering, biology, biophysics, biochemistry, medicine, andengineering.

Advanced Spatial Modeling with Stochastic Partial Differential Equations Using R and INLA

Author: Elias T. Krainski,Virgilio Gómez-Rubio,Haakon Bakka,Amanda Lenzi,Daniela Castro-Camilo,Daniel Simpson,Finn Lindgren,Håvard Rue

Publisher: CRC Press

ISBN: 0429628218

Category: Mathematics

Page: 284

View: 8001

Modeling spatial and spatio-temporal continuous processes is an important and challenging problem in spatial statistics. Advanced Spatial Modeling with Stochastic Partial Differential Equations Using R and INLA describes in detail the stochastic partial differential equations (SPDE) approach for modeling continuous spatial processes with a Matérn covariance, which has been implemented using the integrated nested Laplace approximation (INLA) in the R-INLA package. Key concepts about modeling spatial processes and the SPDE approach are explained with examples using simulated data and real applications. This book has been authored by leading experts in spatial statistics, including the main developers of the INLA and SPDE methodologies and the R-INLA package. It also includes a wide range of applications: * Spatial and spatio-temporal models for continuous outcomes * Analysis of spatial and spatio-temporal point patterns * Coregionalization spatial and spatio-temporal models * Measurement error spatial models * Modeling preferential sampling * Spatial and spatio-temporal models with physical barriers * Survival analysis with spatial effects * Dynamic space-time regression * Spatial and spatio-temporal models for extremes * Hurdle models with spatial effects * Penalized Complexity priors for spatial models All the examples in the book are fully reproducible. Further information about this book, as well as the R code and datasets used, is available from the book website at http://www.r-inla.org/spde-book. The tools described in this book will be useful to researchers in many fields such as biostatistics, spatial statistics, environmental sciences, epidemiology, ecology and others. Graduate and Ph.D. students will also find this book and associated files a valuable resource to learn INLA and the SPDE approach for spatial modeling.

Methoden der Mathematischen Physik

Author: Richard Courant,David Hilbert

Publisher: Springer-Verlag

ISBN: 3642474349

Category: Mathematics

Page: N.A

View: 4778

Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.

Nonlinear Ordinary Differential Equations

Author: R. Grimshaw

Publisher: CRC Press

ISBN: 9780849386077

Category: Mathematics

Page: 336

View: 9465

Ordinary differential equations have long been an important area of study because of their wide application in physics, engineering, biology, chemistry, ecology, and economics. Based on a series of lectures given at the Universities of Melbourne and New South Wales in Australia, Nonlinear Ordinary Differential Equations takes the reader from basic elementary notions to the point where the exciting and fascinating developments in the theory of nonlinear differential equations can be understood and appreciated. Each chapter is self-contained, and includes a selection of problems together with some detailed workings within the main text. Nonlinear Ordinary Differential Equations helps develop an understanding of the subtle and sometimes unexpected properties of nonlinear systems and simultaneously introduces practical analytical techniques to analyze nonlinear phenomena. This excellent book gives a structured, systematic, and rigorous development of the basic theory from elementary concepts to a point where readers can utilize ideas in nonlinear differential equations.

Partielle Differentialgleichungen der Geometrie und der Physik 1

Grundlagen und Integraldarstellungen

Author: Friedrich Sauvigny

Publisher: Springer-Verlag

ISBN: 3540350276

Category: Mathematics

Page: 418

View: 8721

Mathematical Methods in Engineering and Physics

Author: Gary N. Felder,Kenny M. Felder

Publisher: John Wiley & Sons

ISBN: 1118449606

Category: Science

Page: 830

View: 1891

This text is intended for the undergraduate course in math methods, with an audience of physics and engineering majors. As a required course in most departments, the text relies heavily on explained examples, real-world applications and student engagement. Supporting the use of active learning, a strong focus is placed upon physical motivation combined with a versatile coverage of topics that can be used as a reference after students complete the course. Each chapter begins with an overview that includes a list of prerequisite knowledge, a list of skills that will be covered in the chapter, and an outline of the sections. Next comes the motivating exercise, which steps the students through a real-world physical problem that requires the techniques taught in each chapter.

Grundlagen der Photonik

Author: Bahaa E. A. Saleh,Malvin Carl Teich

Publisher: John Wiley & Sons

ISBN: 9783527406777

Category: Optoelectronics

Page: 1406

View: 9410

Schon die erste Auflage des englischen Lehrbuchs 'Fundamentals of Photonics' zeichnete sich durch seine ausgewogene Mischung von Theorie und Praxis aus, und deckte in detaillierter Darstellung die grundlegenden Theorien des Lichts ab. Es umfasste sowohl die Themen Strahlenoptik, Wellenoptik, elektromagnetische Optik, Photonenoptik, sowie die Wechselwirkung von Licht und Materie, als auch die Theorie der optischen Eigenschaften von Halbleitern. Die Photonik-Technologie hat eine rasante Entwicklung genommen seit der Publikation der ersten Ausgabe von 'Fundamentals of Photonics' vor 15 Jahren. Die nun vorliegende Zweite Auflage des Marksteins auf dem Gebiet der Photonik trägt mit zwei neuen und zusätzlichen Kapiteln den neuesten technologischen Fortschritten Rechnung: Photonische Kristalle sowie Ultrakurzpuls-Optik. Zudem wurden alle Kapitel gründlich überarbeitet und viele Abschnitte hinzugefügt, so z.B. über Laguerre-Gauss Strahlen, die Sellmeier-Gleichung, Photonenkristall-Wellenleiter, photonische Kristallfasern, Mikrosphären-Resonatoren, Optische Kohärenz Tomographie, Bahndrehimpuls des Photons, Bohrsche Theorie, Raman-Verstärker, rauscharme Avalanche-Photodioden, Abstimmkurven und Dispersions-Management.

Computational Methods in Chemical Engineering with Maple

Author: Ralph E. White,Venkat R. Subramanian

Publisher: Springer Science & Business Media

ISBN: 9783642043116

Category: Science

Page: 860

View: 8506

This book presents Maple solutions to a wide range of problems relevant to chemical engineers and others. Many of these solutions use Maple’s symbolic capability to help bridge the gap between analytical and numerical solutions. The readers are strongly encouraged to refer to the references included in the book for a better understanding of the physics involved, and for the mathematical analysis. This book was written for a senior undergraduate or a first year graduate student course in chemical engineering. Most of the examples in this book were done in Maple 10. However, the codes should run in the most recent version of Maple. We strongly encourage the readers to use the classic worksheet (*. mws) option in Maple as we believe it is more user-friendly and robust. In chapter one you will find an introduction to Maple which includes simple basics as a convenience for the reader such as plotting, solving linear and nonlinear equations, Laplace transformations, matrix operations, ‘do loop,’ and ‘while loop. ’ Chapter two presents linear ordinary differential equations in section 1 to include homogeneous and nonhomogeneous ODEs, solving systems of ODEs using the matrix exponential and Laplace transform method. In section two of chapter two, nonlinear ordinary differential equations are presented and include simultaneous series reactions, solving nonlinear ODEs with Maple’s ‘dsolve’ command, stop conditions, differential algebraic equations, and steady state solutions. Chapter three addresses boundary value problems.

Introduction to Partial Differential Equations with Applications

Author: E. C. Zachmanoglou,Dale W. Thoe

Publisher: Courier Corporation

ISBN: 048613217X

Category: Mathematics

Page: 432

View: 9486

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.

Mathematik für Ökonomen

Grundlagen, Methoden und Anwendungen

Author: Alpha C. Chiang,Kevin Wainwright,Harald Nitsch

Publisher: Vahlen

ISBN: 3800645122

Page: 476

View: 6923

Klar und verständlich: Mathematik für Ökonomen. Für viele Studierende der BWL und VWL hat die Mathematik eine ähnliche Anziehungskraft wie bittere Medizin notwendig, aber extrem unangenehm. Das muss nicht sein. Mit diesem Buch gelingt es jedem, die Methoden zu erlernen. Anhand konkreter ökonomischer Anwendungen wird die Mathematik sehr anschaulich erklärt. Schnelle Lernerfolge Von der Wiederholung des Abiturwissens bis zum Niveau aktueller ökonomischer Lehrbücher wird Schritt für Schritt vorgegangen und alle wichtigen Bereiche der Mathematik systematisch erklärt. Der Lernerfolg stellt sich schnell ein: die klare und ausführliche Darstellung sowie die graphische Unterstützung machen es möglich.

Partial Differential Equations

Author: Lawrence C. Evans

Publisher: American Mathematical Soc.

ISBN: 0821849743

Category: Mathematics

Page: 749

View: 634

This is the second edition of the now definitive text on partial differential equations (PDE). It offers a comprehensive survey of modern techniques in the theoretical study of PDE with particular emphasis on nonlinear equations. Its wide scope and clear exposition make it a great text for a graduate course in PDE. For this edition, the author has made numerous changes, including a new chapter on nonlinear wave equations, more than 80 new exercises, several new sections, a significantly expanded bibliography. About the First Edition: I have used this book for both regular PDE and topics courses. It has a wonderful combination of insight and technical detail. ... Evans' book is evidence of his mastering of the field and the clarity of presentation. --Luis Caffarelli, University of Texas It is fun to teach from Evans' book. It explains many of the essential ideas and techniques of partial differential equations ... Every graduate student in analysis should read it. --David Jerison, MIT I use Partial Differential Equations to prepare my students for their Topic exam, which is a requirement before starting working on their dissertation. The book provides an excellent account of PDE's ... I am very happy with the preparation it provides my students. --Carlos Kenig, University of Chicago Evans' book has already attained the status of a classic. It is a clear choice for students just learning the subject, as well as for experts who wish to broaden their knowledge ... An outstanding reference for many aspects of the field. --Rafe Mazzeo, Stanford University

Partial Differential Equations for Scientists and Engineers

Author: Stanley J. Farlow

Publisher: Courier Corporation

ISBN: 048667620X

Category: Mathematics

Page: 414

View: 8257