# Numerical Analysis Using R Solutions to ODEs and PDEs

Author: Graham W. Griffiths

Publisher: Cambridge University Press

ISBN: 131665415X

Category: Mathematics

Page: N.A

View: 3022

This book presents the latest numerical solutions to initial value problems and boundary value problems described by ODEs and PDEs. The author offers practical methods that can be adapted to solve wide ranges of problems and illustrates them in the increasingly popular open source computer language R, allowing integration with more statistically based methods. The book begins with standard techniques, followed by an overview of 'high resolution' flux limiters and WENO to solve problems with solutions exhibiting high gradient phenomena. Meshless methods using radial basis functions are then discussed in the context of scattered data interpolation and the solution of PDEs on irregular grids. Three detailed case studies demonstrate how numerical methods can be used to tackle very different complex problems. With its focus on practical solutions to real-world problems, this book will be useful to students and practitioners in all areas of science and engineering, especially those using R.

# Partielle Differentialgleichungen und numerische Methoden Author: Stig Larsson,Vidar Thomee

Publisher: Springer-Verlag

ISBN: 3540274227

Category: Mathematics

Page: 272

View: 3003

Das Buch ist für Studenten der angewandten Mathematik und der Ingenieurwissenschaften auf Vordiplomniveau geeignet. Der Schwerpunkt liegt auf der Verbindung der Theorie linearer partieller Differentialgleichungen mit der Theorie finiter Differenzenverfahren und der Theorie der Methoden finiter Elemente. Für jede Klasse partieller Differentialgleichungen, d.h. elliptische, parabolische und hyperbolische, enthält der Text jeweils ein Kapitel zur mathematischen Theorie der Differentialgleichung gefolgt von einem Kapitel zu finiten Differenzenverfahren sowie einem zu Methoden der finiten Elemente. Den Kapiteln zu elliptischen Gleichungen geht ein Kapitel zum Zweipunkt-Randwertproblem für gewöhnliche Differentialgleichungen voran. Ebenso ist den Kapiteln zu zeitabhängigen Problemen ein Kapitel zum Anfangswertproblem für gewöhnliche Differentialgleichungen vorangestellt. Zudem gibt es ein Kapitel zum elliptischen Eigenwertproblem und zur Entwicklung nach Eigenfunktionen. Die Darstellung setzt keine tiefer gehenden Kenntnisse in Analysis und Funktionalanalysis voraus. Das erforderliche Grundwissen über lineare Funktionalanalysis und Sobolev-Räume wird im Anhang im Überblick besprochen.

# 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: 971

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

# Using R for Numerical Analysis in Science and Engineering Author: Victor A. Bloomfield

Publisher: CRC Press

ISBN: 1315360497

Category: Mathematics

Page: 359

View: 6499

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.

# Solving Differential Equations in R Author: Karline Soetaert,Jeff Cash,Francesca Mazzia

Publisher: Springer Science & Business Media

ISBN: 3642280706

Category: Computers

Page: 248

View: 3824

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.

# Method of Lines PDE Analysis in Biomedical Science and Engineering Author: William E. Schiesser

Publisher: John Wiley & Sons

ISBN: 1119130484

Category: Mathematics

Page: 370

View: 1251

Presents the methodology and applications of ODE and PDE models within biomedical science and engineering With an emphasis on the method of lines (MOL) for partial differential equation (PDE) numerical integration, Method of Lines PDE Analysis in Biomedical Science and Engineering demonstrates the use of numerical methods for the computer solution of PDEs as applied to biomedical science and engineering (BMSE). Written by a well-known researcher in the field, the book provides an introduction to basic numerical methods for initial/boundary value PDEs before moving on to specific BMSE applications of PDEs. Featuring a straightforward approach, the book’s chapters follow a consistent and comprehensive format. First, each chapter begins by presenting the model as an ordinary differential equation (ODE)/PDE system, including the initial and boundary conditions. Next, the programming of the model equations is introduced through a series of R routines that primarily implement MOL for PDEs. Subsequently, the resulting numerical and graphical solution is discussed and interpreted with respect to the model equations. Finally, each chapter concludes with a review of the numerical algorithm performance, general observations and results, and possible extensions of the model. Method of Lines PDE Analysis in Biomedical Science and Engineering also includes: Examples of MOL analysis of PDEs, including BMSE applications in wave front resolution in chromatography, VEGF angiogenesis, thermographic tumor location, blood-tissue transport, two fluid and membrane mass transfer, artificial liver support system, cross diffusion epidemiology, oncolytic virotherapy, tumor cell density in glioblastomas, and variable grids Discussions on the use of R software, which facilitates immediate solutions to differential equation problems without having to first learn the basic concepts of numerical analysis for PDEs and the programming of PDE algorithms A companion website that provides source code for the R routines Method of Lines PDE Analysis in Biomedical Science and Engineering is an introductory reference for researchers, scientists, clinicians, medical researchers, mathematicians, statisticians, chemical engineers, epidemiologists, and pharmacokineticists as well as anyone interested in clinical applications and the interpretation of experimental data with differential equation models. The book is also an ideal textbook for graduate-level courses in applied mathematics, BMSE, biology, biophysics, biochemistry, medicine, and engineering.

# Traveling Wave Analysis of Partial Differential Equations Numerical and Analytical Methods with Matlab and Maple

Author: Graham Griffiths,William E. Schiesser

ISBN: 9780123846532

Category: Mathematics

Page: 461

View: 7105

Although the Partial Differential Equations (PDE) models that are now studied are usually beyond traditional mathematical analysis, the numerical methods that are being developed and used require testing and validation. This is often done with PDEs that have known, exact, analytical solutions. The development of analytical solutions is also an active area of research, with many advances being reported recently, particularly traveling wave solutions for nonlinear evolutionary PDEs. Thus, the current development of analytical solutions directly supports the development of numerical methods by providing a spectrum of test problems that can be used to evaluate numerical methods. This book surveys some of these new developments in analytical and numerical methods, and relates the two through a series of PDE examples. The PDEs that have been selected are largely "named'' since they carry the names of their original contributors. These names usually signify that the PDEs are widely recognized and used in many application areas. The authors’ intention is to provide a set of numerical and analytical methods based on the concept of a traveling wave, with a central feature of conversion of the PDEs to ODEs. The Matlab and Maple software will be available for download from this website shortly. www.pdecomp.net Includes a spectrum of applications in science, engineering, applied mathematics Presents a combination of numerical and analytical methods Provides transportable computer codes in Matlab and Maple

# Numerical Methods for Differential Equations A Computational Approach

Author: J.R. Dormand

Publisher: CRC Press

ISBN: 9780849394331

Category: Mathematics

Page: 384

View: 5957

With emphasis on modern techniques, Numerical Methods for Differential Equations: A Computational Approach covers the development and application of methods for the numerical solution of ordinary differential equations. Some of the methods are extended to cover partial differential equations. All techniques covered in the text are on a program disk included with the book, and are written in Fortran 90. These programs are ideal for students, researchers, and practitioners because they allow for straightforward application of the numerical methods described in the text. The code is easily modified to solve new systems of equations. Numerical Methods for Differential Equations: A Computational Approach also contains a reliable and inexpensive global error code for those interested in global error estimation. This is a valuable text for students, who will find the derivations of the numerical methods extremely helpful and the programs themselves easy to use. It is also an excellent reference and source of software for researchers and practitioners who need computer solutions to differential equations.

# Computer-Lösung gewöhnlicher Differentialgleichungen Das Anfangswertproblem

Author: Lawrence F. Shampine,Marilyn K. Gordon

Publisher: Springer-Verlag

ISBN: 3322938018

Category: Mathematics

Page: 259

View: 1855

# Numerical Methods for Fluid Dynamics With Applications to Geophysics

Author: Dale R. Durran

Publisher: Springer Science & Business Media

ISBN: 9781441964120

Category: Mathematics

Page: 516

View: 9724

This scholarly text provides an introduction to the numerical methods used to model partial differential equations, with focus on atmospheric and oceanic flows. The book covers both the essentials of building a numerical model and the more sophisticated techniques that are now available. Finite difference methods, spectral methods, finite element method, flux-corrected methods and TVC schemes are all discussed. Throughout, the author keeps to a middle ground between the theorem-proof formalism of a mathematical text and the highly empirical approach found in some engineering publications. The book establishes a concrete link between theory and practice using an extensive range of test problems to illustrate the theoretically derived properties of various methods. From the reviews: "...the books unquestionable advantage is the clarity and simplicity in presenting virtually all basic ideas and methods of numerical analysis currently actively used in geophysical fluid dynamics." Physics of Atmosphere and Ocean

# 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: 9631

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.

# Methods in Neuronal Modeling From Ions to Networks

Author: Christof Koch,Idan Segev

Publisher: MIT Press

ISBN: 9780262112314

Category: Medical

Page: 671

View: 8644

This book serves as a handbook of computational methods and techniques for modeling the functional properties of single and groups of nerve cells.

# Numerical Solution of Ordinary Differential Equations Author: L.F. Shampine

Publisher: CRC Press

ISBN: 9780412051517

Category: Mathematics

Page: 632

View: 8334

# 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: 5010

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.

# Special Volume: Foundations of Computational Mathematics Author: Phillipe G. Ciarlet

Publisher: Gulf Professional Publishing

ISBN: 9780444512475

Category: Mathematics

Page: 536

View: 6079

From geometric integration and its applications, and linear programming and condition numbers under the real number computational model, to chaos in finite difference schemes, these essays explore the foundational issues of computational mathematics.

# Numerical Methods for Delay Differential Equations Author: Alfredo Bellen,Marino Zennaro

Publisher: Oxford University Press

ISBN: 0199671370

Category: Business & Economics

Page: 410

View: 5799

This unique book describes, analyses, and improves various approaches and techniques for the numerical solution of delay differential equations. It includes a list of available codes and also aids the reader in writing his or her own.

# Wissenschaftliches Rechnen mit MATLAB Author: Alfio Quarteroni,Fausto Saleri

Publisher: Springer-Verlag

ISBN: 3540293078

Category: Mathematics

Page: 269

View: 6611

Aus den Rezensionen der englischen Auflage: Dieses Lehrbuch ist eine Einführung in das Wissenschaftliche Rechnen und diskutiert Algorithmen und deren mathematischen Hintergrund. Angesprochen werden im Detail nichtlineare Gleichungen, Approximationsverfahren, numerische Integration und Differentiation, numerische Lineare Algebra, gewöhnliche Differentialgleichungen und Randwertprobleme. Zu den einzelnen Themen werden viele Beispiele und Übungsaufgaben sowie deren Lösung präsentiert, die durchweg in MATLAB formuliert sind. Der Leser findet daher nicht nur die graue Theorie sondern auch deren Umsetzung in numerischen, in MATLAB formulierten Code. MATLAB select 2003, Issue 2, p. 50. [Die Autoren] haben ein ausgezeichnetes Werk vorgelegt, das MATLAB vorstellt und eine sehr nützliche Sammlung von MATLAB Funktionen für die Lösung fortgeschrittener mathematischer und naturwissenschaftlicher Probleme bietet. [...] Die Präsentation des Stoffs ist durchgängig gut und leicht verständlich und beinhaltet Lösungen für die Übungen am Ende jedes Kapitels. Als exzellenter Neuzugang für Universitätsbibliotheken- und Buchhandlungen wird dieses Buch sowohl beim Selbststudium als auch als Ergänzung zu anderen MATLAB-basierten Büchern von großem Nutzen sein. Alles in allem: Sehr empfehlenswert. Für Studenten im Erstsemester wie für Experten gleichermassen. S.T. Karris, University of California, Berkeley, Choice 2003.

# Numerical Integration of Space Fractional Partial Differential Equations Vol 1 - Introduction to Algorithms and Computer Coding in R

Author: Younes Salehi,William E. Schiesser

Publisher: Morgan & Claypool Publishers

ISBN: 1681732084

Category: Mathematics

Page: 201

View: 6410

Partial differential equations (PDEs) are one of the most used widely forms of mathematics in science and engineering. PDEs can have partial derivatives with respect to (1) an initial value variable, typically time, and (2) boundary value variables, typically spatial variables. Therefore, two fractional PDEs can be considered, (1) fractional in time (TFPDEs), and (2) fractional in space (SFPDEs). The two volumes are directed to the development and use of SFPDEs, with the discussion divided as: Vol 1: Introduction to Algorithms and Computer Coding in R Vol 2: Applications from Classical Integer PDEs. Various definitions of space fractional derivatives have been proposed. We focus on the Caputo derivative, with occasional reference to the Riemann-Liouville derivative. Partial differential equations (PDEs) are one of the most used widely forms of mathematics in science and engineering. PDEs can have partial derivatives with respect to (1) an initial value variable, typically time, and (2) boundary value variables, typically spatial variables. Therefore, two fractional PDEs can be considered, (1) fractional in time (TFPDEs), and (2) fractional in space (SFPDEs). The two volumes are directed to the development and use of SFPDEs, with the discussion divided as: Vol 1: Introduction to Algorithms and Computer Coding in R Vol 2: Applications from Classical Integer PDEs. Various definitions of space fractional derivatives have been proposed. We focus on the Caputo derivative, with occasional reference to the Riemann-Liouville derivative. The Caputo derivative is defined as a convolution integral. Thus, rather than being local (with a value at a particular point in space), the Caputo derivative is non-local (it is based on an integration in space), which is one of the reasons that it has properties not shared by integer derivatives. A principal objective of the two volumes is to provide the reader with a set of documented R routines that are discussed in detail, and can be downloaded and executed without having to first study the details of the relevant numerical analysis and then code a set of routines. In the first volume, the emphasis is on basic concepts of SFPDEs and the associated numerical algorithms. The presentation is not as formal mathematics, e.g., theorems and proofs. Rather, the presentation is by examples of SFPDEs, including a detailed discussion of the algorithms for computing numerical solutions to SFPDEs and a detailed explanation of the associated source code.

# Lie-Theoretic Ode Numerical Analysis, Mechanics and Differential Systems Author: Robert Hermann

Publisher: Math-Sci Press

ISBN: 9780915692453

Category: Mathematics

Page: 286

View: 9698

# The Numerical Method of Lines Integration of Partial Differential Equations

Author: William E. Schiesser

Publisher: Elsevier

ISBN: 0128015519

Category: Mathematics

Page: 326

View: 8836

This is the first book on the numerical method of lines, a relatively new method for solving partial differential equations. The Numerical Method of Lines is also the first book to accommodate all major classes of partial differential equations. This is essentially an applications book for computer scientists. The author will separately offer a disk of FORTRAN 77 programs with 250 specific applications, ranging from "Shuttle Launch Simulation" to "Temperature Control of a Nuclear Fuel Rod."