A Compendium of Partial Differential Equation Models

Method of Lines Analysis with Matlab

Author: William E. Schiesser,Graham W. Griffiths

Publisher: Cambridge University Press

ISBN: 1139477854

Category: Mathematics

Page: N.A

View: 8455

Mathematical modelling of physical and chemical systems is used extensively throughout science, engineering, and applied mathematics. To use mathematical models, one needs solutions to the model equations; this generally requires numerical methods. This book presents numerical methods and associated computer code in Matlab for the solution of a spectrum of models expressed as partial differential equations (PDEs). The authors focus on the method of lines (MOL), a well-established procedure for all major classes of PDEs, where the boundary value partial derivatives are approximated algebraically by finite differences. This reduces the PDEs to ordinary differential equations (ODEs) and makes the computer code easy to understand, implement, and modify. Also, the ODEs (via MOL) can be combined with any other ODEs that are part of the model (so that MOL naturally accommodates ODE/PDE models). This book uniquely includes a detailed line-by-line discussion of computer code related to the associated PDE model.

Traveling Wave Analysis of Partial Differential Equations

Numerical and Analytical Methods with Matlab and Maple

Author: Graham Griffiths,William E. Schiesser

Publisher: Academic Press

ISBN: 9780123846532

Category: Mathematics

Page: 461

View: 4528

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

The Numerical Method of Lines

Integration of Partial Differential Equations

Author: William E. Schiesser

Publisher: Elsevier

ISBN: 0128015519

Category: Mathematics

Page: 326

View: 9697

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."

Numerical Analysis Using R

Author: Graham W. Griffiths

Publisher: Cambridge University Press

ISBN: 1107115612

Category: Mathematics

Page: 500

View: 8248

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.

Partial Differential Equations

Modeling, Analysis, Computation

Author: R. M. M. Mattheij,S. W. Rienstra,J. H. M. ten Thije Boonkkamp

Publisher: SIAM

ISBN: 0898715946

Category: Mathematics

Page: 665

View: 9899

Textbook with a unique approach that integrates analysis and numerical methods and includes modelling to address real-life problems.

Advanced Engineering Mathematics

Author: Dennis G. Zill,Michael R. Cullen

Publisher: Jones & Bartlett Learning

ISBN: 9780763745912

Category: Technology & Engineering

Page: 929

View: 3687

Part of the new Digital Filmmaker Series! Digital Filmmaking: An Introductionis the first book in the newDigital Filmmaker Series. Designed for an introductory level course in digital filmmaking, it is intended for anyone who has an interest in telling stories with pictures and sound and won't assume any familiarity with equipment or concepts on the part of the student. In addition to the basics of shooting and editing, different story forms are introduced from documentary and live events through fictional narratives. Each of the topics is covered in enough depth to allow anyone with a camera and a computer to begin creating visual projects of quality.

Traveling Wave Analysis of Partial Differential Equations

Numerical and Analytical Methods with Matlab and Maple

Author: Graham Griffiths,William E. Schiesser

Publisher: Academic Press

ISBN: 9780123846532

Category: Mathematics

Page: 461

View: 625

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

The Neumann Compendium

Author: F Bródy,T Vámos

Publisher: World Scientific

ISBN: 9814500682

Category: Mathematics

Page: 760

View: 8379

After three decades since the first nearly complete edition of John von Neumann's papers, this book is a valuable selection of those papers and excerpts of his books that are most characteristic of his activity, and reveal that of his continuous influence. The results receiving the 1994 Nobel Prizes in economy deeply rooted in Neumann's game theory are only minor traces of his exceptionally broad spectrum of creativity and stimulation. The book is organized by the specific subjects-quantum mechanics, ergodic theory, operator algebra, hydrodynamics, economics, computers, science and society. In addition, one paper which was written in German will be translated and published in English for the first time. The sections are introduced by short explanatory notes with an emphasis on recent developments based on von Neumann's contributions. An overall picture is provided by Ulam's, one of his most intimate partners in thinking, 1958 memorial lecture. Facsimilae and translations of some of his personal letters and a newly completed bibliography based on von Neumann's own careful compilation are added. Contents:Quantum Mechanics:Mathematical Foundations of Quantum MechanicsThe Logic of Quantum Mechanics (with G Birkhoff)Ergodic Theory:Proof of the Quasi-Ergodic HypothesisOperator Methods in Classical Mechanics, II (with P R Halmos)Operator Algebra:Algebra of Functional Operations and Theory of Normal OperatorsOn Rings of Operators I–IVUse of Variational Methods in HydrodynamicsEconomics:Theory of Games and Economic Behavior (with O Morgenstern)Computers:On the Principles of Large Scale Computing Machines (with H H Goldstine)Science and Society:The MathematicianMethod in the Physical SciencesThe Role of Mathematics in the Sciences and in Societyand other papers Readership: Mathematicians. keywords:Mathematics;Science History;Computer Science;J V Neumann;Science and Society;Game Theory;Quantum Mechanics;Operator Algebra;Hydrodynamics;Ergodic Theory“The collection bears testimony to the lasting influence of John von Neumann's work on the course of modern mathematics.”R Siegmund-Schultze Mathematical Abstracts “This collection is a fascinating introduction to the work of John von Neumann … it has much to offer even to the casual browser and will also be relevant and interesting to those working today in the fields on which von Neumann had such enormous influence.”Mathematical Reviews

Handbook of Nonlinear Partial Differential Equations, Second Edition

Author: Andrei D. Polyanin,Valentin F. Zaitsev

Publisher: CRC Press

ISBN: 142008724X

Category: Mathematics

Page: 1912

View: 603

New to the Second Edition More than 1,000 pages with over 1,500 new first-, second-, third-, fourth-, and higher-order nonlinear equations with solutions Parabolic, hyperbolic, elliptic, and other systems of equations with solutions Some exact methods and transformations Symbolic and numerical methods for solving nonlinear PDEs with MapleTM, Mathematica®, and MATLAB® Many new illustrative examples and tables A large list of references consisting of over 1,300 sources To accommodate different mathematical backgrounds, the authors avoid wherever possible the use of special terminology. They outline the methods in a schematic, simplified manner and arrange the material in increasing order of complexity.

Numerical and Computer Methods in Structural Mechanics

Author: Steven J. Fenves,Nicholas Perrone,Arthur R. Robinson

Publisher: Elsevier

ISBN: 1483272540

Category: Technology & Engineering

Page: 698

View: 3027

Numerical and Computer Methods in Structural Mechanics is a compendium of papers that deals with the numerical methods in structural mechanics, computer techniques, and computer capabilities. Some papers discus the analytical basis of the computer technique most widely used in software, that is, the finite element method. This method includes the convergence (in terms of variation principles) isoparametrics, hybrid models, and incompatible displacement models. Other papers explain the storage or retrieval of data, as well as equation-solving algorithms. Other papers describe general-purpose structural mechanics programs, alternatives to, and extension of the usual finite element approaches. Another paper explores nonlinear, dynamic finite element problems, and a direct physical approach to determine finite difference models. Special papers explain structural mechanics used in computing, particularly, those related to integrated data bases, such as in the Structures Oriented Exchange System of the Office of Naval Research and the integrated design of tanker structures. Other papers describe software and hardware capabilities, for example, in ship design, fracture mechanics, biomechanics, and crash safety. The text is suitable for programmers, computer engineers, researchers, and scientists involved in materials and industrial design.

Applied Numerical Methods Using MATLAB

Author: Won Y. Yang,Wenwu Cao,Tae-Sang Chung,John Morris

Publisher: John Wiley & Sons

ISBN: 0471705187

Category: Mathematics

Page: 400

View: 3830

In recent years, with the introduction of new media products, there has been a shift in the use of programming languages from FORTRAN or C to MATLAB for implementing numerical methods. This book makes use of the powerful MATLAB software to avoid complex derivations, and to teach the fundamental concepts using the software to solve practical problems. Over the years, many textbooks have been written on the subject of numerical methods. Based on their course experience, the authors use a more practical approach and link every method to real engineering and/or science problems. The main benefit is that engineers don't have to know the mathematical theory in order to apply the numerical methods for solving their real-life problems. An Instructor's Manual presenting detailed solutions to all the problems in the book is available online.

Applied Partial Differential Equations

Author: J. R. Ockendon,Sam Howison,Andrew Lacey,Alexander Movchan

Publisher: Oxford University Press on Demand

ISBN: 9780198527718

Category: Mathematics

Page: 449

View: 4963

Partial differential equations are used in mathematical models of a huge range of real-world phenomena, from electromagnetism to financial markets. This revised edition of Applied Partial Differential Equations contains many new sections and exercises including transform methods, free surface flows, linear elasticity and complex characteristics.

Essentials of Stochastic Finance

Facts, Models, Theory

Author: Albert N. Shiryaev

Publisher: World Scientific

ISBN: 9810236050

Category: Business & Economics

Page: 834

View: 8854

Readership: Undergraduates and researchers in probability and statistics; applied, pure and financial mathematics; economics; chaos.

Solving Nonlinear Partial Differential Equations with Maple and Mathematica

Author: Inna Shingareva,Carlos Lizárraga-Celaya

Publisher: Springer Science & Business Media

ISBN: 370910517X

Category: Mathematics

Page: 357

View: 2538

The emphasis of the book is given in how to construct different types of solutions (exact, approximate analytical, numerical, graphical) of numerous nonlinear PDEs correctly, easily, and quickly. The reader can learn a wide variety of techniques and solve numerous nonlinear PDEs included and many other differential equations, simplifying and transforming the equations and solutions, arbitrary functions and parameters, presented in the book). Numerous comparisons and relationships between various types of solutions, different methods and approaches are provided, the results obtained in Maple and Mathematica, facilitates a deeper understanding of the subject. Among a big number of CAS, we choose the two systems, Maple and Mathematica, that are used worldwide by students, research mathematicians, scientists, and engineers. As in the our previous books, we propose the idea to use in parallel both systems, Maple and Mathematica, since in many research problems frequently it is required to compare independent results obtained by using different computer algebra systems, Maple and/or Mathematica, at all stages of the solution process. One of the main points (related to CAS) is based on the implementation of a whole solution method (e.g. starting from an analytical derivation of exact governing equations, constructing discretizations and analytical formulas of a numerical method, performing numerical procedure, obtaining various visualizations, and comparing the numerical solution obtained with other types of solutions considered in the book, e.g. with asymptotic solution).

Methods of Solving Sequence and Series Problems

Author: Ellina Grigorieva

Publisher: Birkhäuser

ISBN: 3319456865

Category: Mathematics

Page: 281

View: 4397

This book aims to dispel the mystery and fear experienced by students surrounding sequences, series, convergence, and their applications. The author, an accomplished female mathematician, achieves this by taking a problem solving approach, starting with fascinating problems and solving them step by step with clear explanations and illuminating diagrams. The reader will find the problems interesting, unusual, and fun, yet solved with the rigor expected in a competition. Some problems are taken directly from mathematics competitions, with the name and year of the exam provided for reference. Proof techniques are emphasized, with a variety of methods presented. The text aims to expand the mind of the reader by often presenting multiple ways to attack the same problem, as well as drawing connections with different fields of mathematics. Intuitive and visual arguments are presented alongside technical proofs to provide a well-rounded methodology. With nearly 300 problems including hints, answers, and solutions, Methods of Solving Sequences and Series Problems is an ideal resource for those learning calculus, preparing for mathematics competitions, or just looking for a worthwhile challenge. It can also be used by faculty who are looking for interesting and insightful problems that are not commonly found in other textbooks.

Adaptive Method of Lines

Author: A, Vande Wouwer,Ph. Saucez,W.E. Schiesser

Publisher: CRC Press

ISBN: 1420035614

Category: Mathematics

Page: 432

View: 7908

The general Method of Lines (MOL) procedure provides a flexible format for the solution of all the major classes of partial differential equations (PDEs) and is particularly well suited to evolutionary, nonlinear wave PDEs. Despite its utility, however, there are relatively few texts that explore it at a more advanced level and reflect the method's current state of development. Written by distinguished researchers in the field, Adaptive Method of Lines reflects the diversity of techniques and applications related to the MOL. Most of its chapters focus on a particular application but also provide a discussion of underlying philosophy and technique. Particular attention is paid to the concept of both temporal and spatial adaptivity in solving time-dependent PDEs. Many important ideas and methods are introduced, including moving grids and grid refinement, static and dynamic gridding, the equidistribution principle and the concept of a monitor function, the minimization of a functional, and the moving finite element method. Applications addressed include shallow water flow, combustion and flame propagation, transport in porous media, gas dynamics, chemical engineering processes, solitary waves, and magnetohydrodynamics. As the first advanced text to represent the modern era of the method of lines, this monograph offers an outstanding opportunity to discover new concepts, learn new techniques, and explore a wide range of applications.

A First Course in Analysis

Author: John B. Conway

Publisher: Cambridge University Press

ISBN: 1107173140

Category: Mathematics

Page: 375

View: 6976

This rigorous textbook is intended for a year-long analysis or advanced calculus course for advanced undergraduate or beginning graduate students. Starting with detailed, slow-paced proofs that allow students to acquire facility in reading and writing proofs, it clearly and concisely explains the basics of differentiation and integration of functions of one and several variables, and covers the theorems of Green, Gauss, and Stokes. Minimal prerequisites are assumed, and relevant linear algebra topics are reviewed right before they are needed, making the material accessible to students from diverse backgrounds. Abstract topics are preceded by concrete examples to facilitate understanding, for example, before introducing differential forms, the text examines low-dimensional examples. The meaning and importance of results are thoroughly discussed, and numerous exercises of varying difficulty give students ample opportunity to test and improve their knowledge of this difficult yet vital subject.