Numerical Methods For Engineers

Features include:which are based on exciting new areas such as bioengineering.and differential equations.students using this text will be able to apply their new skills to their chosen field.Electronic Textbook Optionsan online resource where students can purchase the complete text in a digital format at almosthalf the cost of the traditional textbook. Students can access the text online for one year.learning, which include full text search, notes and highlighting, and email tools for sharingcontact your salesrepresentative or visit www.CourseSmart.com.Sixth EditionNumericalMethodsfor EngineersChapraCanaleSteven C. ChapraRaymond P. CanaleMD DALIM #1009815 03/12/09 CYAN MAG YELO BLKFor more information, please visit www.mhhe.com/chaprafor Engineersadaptive quadrature.SixthEditionNumerical MethodsThe sixth edition of Numerical Methods for Engineers offers an innovative and accessiblepresentation of numerical methods; the book has earned the Meriam-Wiley award, which isgiven by the American Society for Engineering Education for the best textbook. Because software packages are now regularly used for numerical analysis, this eagerly anticipated revisionmaintains its strong focus on appropriate use of computational tools.

cha01064 fm.qxd3/25/0910:51 AMPage iNumerical Methodsfor EngineersSIXTH EDITIONSteven C. ChapraBerger Chair in Computing and EngineeringTufts UniversityRaymond P. CanaleProfessor Emeritus of Civil EngineeringUniversity of Michigan

cha01064 fm.qxd3/25/0910:51 AMPage iiNUMERICAL METHODS FOR ENGINEERS, SIXTH EDITIONPublished by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020.Copyright 2010 by The McGraw-Hill Companies, Inc. All rights reserved. Previous editions 2006, 2002, and 1998. No part of thispublication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the priorwritten consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage ortransmission, or broadcast for distance learning.Some ancillaries, including electronic and print components, may not be available to customers outside the United States.This book is printed on acid-free paper.1 2 3 4 5 6 7 8 9 0 VNH/VNH 0 9ISBN 978–0–07–340106–5MHID 0–07–340106–4Global Publisher: Raghothaman SrinivasanSponsoring Editor: Debra B. HashDirector of Development: Kristine TibbettsDevelopmental Editor: Lorraine K. BuczekSenior Marketing Manager: Curt ReynoldsProject Manager: Joyce WattersLead Production Supervisor: Sandy LudovissyAssociate Design Coordinator: Brenda A. RolwesCover Designer: Studio Montage, St. Louis, Missouri(USE) Cover Image: BrandX/JupiterImagesCompositor: Macmillan Publishing SolutionsTypeface: 10/12 Times RomanPrinter: R. R. Donnelley Jefferson City, MOAll credits appearing on page or at the end of the book are considered to be an extension of the copyright page.MATLAB is a registered trademark of The MathWorks, Inc.Library of Congress Cataloging-in-Publication DataChapra, Steven C.Numerical methods for engineers / Steven C. Chapra, Raymond P. Canale. — 6th ed.p. cm.Includes bibliographical references and index.ISBN 978–0–07–340106–5 — ISBN 0–07–340106–4 (hard copy : alk. paper)1. Engineering mathematics—Data processing. 2. Numerical calculations—Data processing 3. Microcomputers—Programming. I. Canale, Raymond P. II. Title.TA345.C47 2010518.02462—dc222008054296www.mhhe.com

cha01064 p08.qxd3/25/091:33 PMPage 843PARTIAL DIFFERENTIALEQUATIONSPT8.1MOTIVATIONGiven a function u that depends on both x and y, the partial derivative of u with respect tox at an arbitrary point (x, y) is defined asu(x "x, y) u(x, y) u lim"x 0 x"x(PT8.1)Similarly, the partial derivative with respect to y is defined asu(x, y "y) u(x, y) u lim"y 0 y"y(PT8.2)An equation involving partial derivatives of an unknown function of two or more independent variables is called a partial differential equation, or PDE. For example, 2u 2u 2xy u 1 x2 y2 3u 2u x 2 8u 5y2 x y y! 2 "3 u 3u 6 x x2 x y 2 2u u xu x2 x y(PT8.3)(PT8.4)(PT8.5)(PT8.6)The order of a PDE is that of the highest-order partial derivative appearing in the equation.For example, Eqs. (PT8.3) and (PT8.4) are second- and third-order, respectively.A partial differential equation is said to be linear if it is linear in the unknown functionand all its derivatives, with coefficients depending only on the independent variables. Forexample, Eqs. (PT8.3) and (PT8.4) are linear, whereas Eqs. (PT8.5) and (PT8.6) are not.Because of their widespread application in engineering, our treatment of PDEs willfocus on linear, second-order equations. For two independent variables, such equations canbe expressed in the following general form:A 2u 2u 2u B C 2 D 02 x x y y(PT8.7)where A, B, and C are functions of x and y and D is a function of x, y, u, u/ x , and u/ y.Depending on the values of the coefficients of the second-derivative terms—A, B, C—843

cha01064 p08.qxd8443/25/091:33 PMPage 844PARTIAL DIFFERENTIAL EQUATIONSTABLE PT8.1 Categories into which linear, second-order partial differential equations intwo variables can be classified.B2 ! 4AC 0CategoryExampleEllipticLaplace equation (steady state with two spatial dimensions) 2T 2T! !2 0 x 2 y 0ParabolicHeat conduction equation (time variable with one spatialdimension) T 2T! k ′ !2 t x 0HyperbolicWave equation (time variable with one spatial dimension) 2y1 2y!2 !2 ! xc t 2Eq. (PT8.7) can be classified into one of three categories (Table PT8.1). This classification,which is based on the method of characteristics (for example, see Vichnevetsky, 1981, orLapidus and Pinder, 1981), is useful because each category relates to specific and distinctengineering problem contexts that demand special solution techniques. It should be notedthat for cases where A, B, and C depend on x and y, the equation may actually fall into a different category, depending on the location in the domain for which the equation holds. Forsimplicity, we will limit the present discussion to PDEs that remain exclusively in one ofthe categories.PT8.1.1 PDEs and Engineering PracticeEach of the categories of partial differential equations in Table PT8.1 conforms to specifickinds of engineering problems. The initial sections of the following chapters will be devoted to deriving each type of equation for a particular engineering problem context. Forthe time being, we will discuss their general properties and applications and show how theycan be employed in different physical contexts.Elliptic equations are typically used to characterize steady-state systems. As in theLaplace equation in Table PT8.1, this is indicated by the absence of a time derivative.Thus, these equations are typically employed to determine the steady-state distribution ofan unknown in two spatial dimensions.A simple example is the heated plate in Fig. PT8.1a. For this case, the boundaries ofthe plate are held at different temperatures. Because heat flows from regions of high to lowtemperature, the boundary conditions set up a potential that leads to heat flow from the hotto the cool boundaries. If sufficient time elapses, such a system will eventually reach thestable or steady-state distribution of temperature depicted in Fig. PT8.1a. The Laplaceequation, along with appropriate boundary conditions, provides a means to determine thisdistribution. By analogy, the same approach can be employed to tackle other problems involving potentials, such as seepage of water under a dam (Fig. PT8.1b) or the distributionof an electric field (Fig. PT8.1c).

cha01064 p08.qxd3/25/091:33 PMPage 845845PT8.1 MOTIVATIONHotDamHotCoolFlow lineoructdnCoEquipotentiallineCoolImpermeable rock(a)(b)(c)FIGURE PT8.1Three steady-state distribution problems that can be characterized by elliptic PDEs. (a) Temperature distribution on a heated plate, (b) seepage of water under a dam, and (c) the electricfield near the point of a conductor.HotFIGURE PT8.2(a) A long, thin rod that isinsulated everywhere but at itsend. The dynamics of the onedimensional distribution oftemperature along the rod’slength can be described by aparabolic PDE. (b) The solution,consisting of distributionscorresponding to the state of therod at various times.Cool(a)Tt 3" t(b)t 2t " "ttt 0xIn contrast to the elliptic category, parabolic equations determine how an unknownvaries in both space and time. This is manifested by the presence of both spatial andtemporal derivatives in the heat conduction equation from Table PT8.1. Such cases arereferred to as propagation problems because the solution “propagates,’’ or changes, intime.A simple example is a long, thin rod that is insulated everywhere except at its end(Fig. PT8.2a). The insulation is employed to avoid complications due to heat loss along the

cha01064 p08.qxd3/25/091:33 PM846Page 846PARTIAL DIFFERENTIAL EQUATIONSFIGURE PT8.3A taut string vibrating at a lowamplitude is a simple physicalsystem that can be characterized by a hyperbolic PDE.rod’s length. As was the case for the heated plate in Fig. PT8.1a, the ends of the rod are setat fixed temperatures. However, in contrast to Fig. PT8.1a, the rod’s thinness allows us toassume that heat is distributed evenly over its cross section—that is, laterally. Consequently,lateral heat flow is not an issue, and the problem reduces to studying the conduction of heatalong the rod’s longitudinal axis. Rather than focusing on the steady-state distribution intwo spatial dimensions, the problem shifts to determining how the one-dimensional spatialdistribution changes as a function of time (Fig. PT8.2b). Thus, the solution consists of aseries of spatial distributions corresponding to the state of the rod at various times. Using ananalogy from photography, the elliptic case yields a portrait of a system’s stable state,whereas the parabolic case provides a motion picture of how it changes from one state to another. As with the other types of PDEs described herein, parabolic equations can be used tocharacterize a wide variety of other engineering problem contexts by analogy.The final class of PDEs, the hyperbolic category, also deals with propagation problems. However, an important distinction manifested by the wave equation in Table PT8.1 isthat the unknown is characterized by a second derivative with respect to time. As a consequence, the solution oscillates.The vibrating string in Fig. PT8.3 is a simple physical model that can be describedwith the wave equation. The solution consists of a number of characteristic states withwhich the string oscillates. A variety of engineering systems such as vibrations of rods andbeams, motion of fluid waves, and transmission of sound and electrical signals can be characterized by this model.PT8.1.2 Precomputer Methods for Solving PDEsPrior to the advent of digital computers, engineers relied on analytical or exact solutions ofpartial differential equations. Aside from the simplest cases, these solutions often requireda great deal of effort and mathematical sophistication. In addition, many physical systemscould not be solved directly but had to be simplified using linearizations, simple geometric representations, and other idealizations. Although these solutions are elegant andyield insight, they are limited with respect to how faithfully they represent real systems—especially those that are highly nonlinear and irregularly shaped.PT8.2ORIENTATIONBefore we proceed to the numerical methods for solving partial differential equations,some orientation might be helpful. The following material is intended to provide you withan overview of the material discussed in Part Eight. In addition, we have formulated objectives to focus your studies in the subject area.