David Betounes

Partial Differential Equations for Computational Science

With Maple® and Vector Analysis

• Gebundenes Buch

Produktbeschreibung

This is a classroom-tested and developed textbook designed for use in either one- or two term courses in Partial Differential Equations taught at the advanced undergradute and beginning graduate levels of instruction. In addition to covering the standard PDE topics taught in such courses, there is a heavy focus on vector analysis. Maple examples and exercises for the students are presented strategically within the book, and an Appendix on Maple facilitates the use of this computer algebra language, acting as a quick reference source for readers. In addition, a cross-platform CD-ROM which is packaged with the text contains many analytical examples for students in the use of Maple. However, despite its general computational orientation, this book is designed so that it can also be readily used by instructors not wishing to bring the computational aspects of a computer algebra system into the classroom. This book will have strong appeal to interdisciplinary audiences of engineers and scientists, particulary in regard to its treatments of fluid mechanics, heat equations, and continuum mechanics, inclusion of the latter being a unique feature of the presentation.

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This book provides an extensive introduction to the methods and theory of PDEs without relying on the background in functional analysis and differential geometry. In addition to the heat, wave, and Poisson equations, the text also treats some particular systems of particular differential equations: Maxwell`s equations, the Navier-Stokes equations, and Navier`s equations. It also discusses existence and uniqueness results, Dirichlet forms, weak solutions, and Ritz-Galerkin methods for the heat equation. Appendices in vector analysis, continuum mechanics, and the Maple computer algebra system augment the work.
While primarily intended for students taking courses at the upper-division and graduate levels of instruction, the book will also be of interest to many professional readers. Maple is set up within the book in such a way that it can be omitted, depending upon the instructor`s approach to teaching the course. However, the incorporation of this important computational tool into this work has significant advantages in helping students deal with the subject matter at hand. Therefore, the use of Maple is strongly recommended.
A multiplatform CD-ROM accompanies the printed text, and is designed to work an Windows 3.1, Windows 95, Macintosh 680x0, Macintosh Power PCs, and most UNIX machines. This disc contains most of the worked examples from the book as Maple Worksheets, delivered in both Release 3 and Release 4 of the Maple software system. The CD also includes a Maple demo which runs on all computer platforms indicated above, and explains the Maple system in some detail.

Produktdetails

• TELOS, The Electronic Library of Science
• Verlag: Springer / Springer New York / Springer, Berlin
• Artikelnr. des Verlages: 978-0-387-98300-4
• 1998.
• Seitenzahl: 540
• Erscheinungstermin: 15. Mai 1998
• Englisch
• Abmessung: 241mm x 183mm x 34mm
• Gewicht: 1114g
• ISBN-13: 9780387983004
• ISBN-10: 0387983007
• Artikelnr.: 07508593

Inhaltsangabe

1 Introduction.- 1.1 Vector Analysis: Some Basic Notions.- 1.2 General Systems of PDEs.- 1.3 The Main Examples of PDEs.- 1.4 The Flow Generated by a Vector Field.- 2 Derivation of the Heat Equation.- 2.1 Heat Flow: Fourier's Law.- 2.2 The Heat Equation (Without Convection).- 2.3 Initial Conditions and Boundary Conditions.- 2.4 Initial-Boundary Value Problems.- 3 The 1-D Heat Equation.- 3.1 The Homogeneous Problem.- 3.2 Separation of Variables.- 3.3 Sturm-Liouville Problems.- 4 Solution of the 1-D Heat Problem.- 4.1 Solution of the Homogeneous Heat IBVP.- 4.2 Transforming to Homogeneous BCs.- 4.3 The Semi-Homogeneous Heat Problem.- 5 Computational Analysis.- 5.1 Plotting Solutions of Heat Problems.- 5.2 Sturm-Liouville Problems.- 5.3 Fourier Analysis.- 6 Two-Dimensional Heat Flow.- 6.1 The 2-D Heat Equation.- 6.2 Rectangular Regions R.- 6.3 The Homogeneous Case.- 6.4 The Semi-Homogeneous Problem.- 6.5 Transforming to Homogeneous BCs.- 7 Boundary Value Problems.- 7.1 The Dirichlet Problem for the Unit Square.- 7.2 The Neumann Problem for the Unit Square.- 7.3 Mixed BVPs.- 8 3-D Heat Flow.- 8.1 The Heat Equation for the Unit Cube.- 8.2 The Semi-Homogeneous Problem.- 8.3 Transforming to Homogeneous BCs.- 8.4 Examples of BVPs for the Unit Cube.- 8.5 Heat Problems for the Cube.- 9 Maxwell's Equations.- 9.1 Maxwell's Equations in Empty Space.- 9.2 Electrostatics and Magnetostatics.- 9.3 Existence of Potentials.- 9.4 Potentials and Gauge Transformations.- 10 Fluid Mechanics.- 10.1 The Stress Tensor for a Fluid.- 10.2 The Fluid Equations.- 10.3 Special Solutions of the Euler Equations.- 10.4 Special Solutions of the Navier-Stokes Equations.- 10.5 Solution of the Navier-Stokes Equations.- 11 Waves in Elastic Materials.- 11.1 Strings: 1-D Elastic Materials.- 11.2 Membranes: 2-Dimensional Elastic Materials.- 11.3 Solids: 3-D Elastic Materials.- 12 The Heat IBVP in Polar Coordinates.- 12.1 The Polar Coordinate Map.- 12.2 The Laplacian in Polar Coordinates.- 12.3 The BCs and IC in Polar Coordinates.- 12.4 The Heat IBVP for a Disk.- 12.5 Solution of the Homogeneous Problem.- 12.6 The Heat Equation for an Annulus.- 12.7 The Homogeneous Problem for an Annulus.- 12.8 BVPs in Polar Coordinates.- 13 Solution of the Heat IBVP in General.- 13.1 Weak Solutions of Poisson's Equation.- 13.2 Solution of the Heat IBVP.- Appendix A Vector Analysis.- A.1 Curves and Line Integrals.- A.2 Surfaces and Surface Integrals.- A.3 Regions and Volume Integrals.- A.4 Boundaries.- A.5 Closed Curves and Surfaces.- A.6 The Theorems of Gauss, Green, and Stokes.- A.7 Curvatures and Fundamental Forms.- Appendix B Continuum Mechanics.- B.1 The Eulerian Description.- B.2 The Lagrangian Description.- B.3 Two-Dimensional Materials.- B.4 One-Dimensional Materials.- Appendix C Maple Reference Guide.- C.1 Mathematical Expressions and Functions.- C.2 Packages.- C.3 Plotting and Visualization.- C.4 Plotting Flows.- C.5 Programming.- Appendix D Symbols and Tables.- References.