9780130894809

This comprehensive book includes over 800 problems including open ended, project type and design problems.Chapter topics include Introduction to Numerical Methods; Solution of Nonlinear Equations; Simultaneous Linear Algebraic Equations; Solution of Matrix Eigenvalue Problem; Curve Fitting and Interpolation; Statistical Methods; Numerical Differentiation; Numerical Integration; Numerical Solution of Ordinary Differential Equations: Initial Value Problems; Numerical Solution of Ordinary Differential Equations: Boundary Value Problems; Numerical Solution of Partial Differential Equations; Numerical Methods of Optimization ;Finite Element Method.This book is intended as a reference for numerical methods in engineering.

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Preface

xvii

Introduction to Numerical Methods

1

(47)

Importance of Numerical Methods in Engineering

1

(1)

Computers

2

(3)

Computer Programming Languages

5

(2)

Data Representation

7

(5)

Programming Structure

12

(1)

Errors

13

(9)

Numerical Methods Considered

22

(8)

Software for Numerical Analysis

30

(1)

Use of Software Packages

31

(2)

Computer Programs

33

(15)

References and Bibliography

37

(1)

Review Questions

38

(3)

Problems

41

(7)

Solution of Nonlinear Equations

48

(93)

Introduction

48

(5)

Engineering Applications

53

(5)

Incremental Search Method

58

(3)

Bisection Method

61

(3)

Newton-Raphson Method

64

(6)

Secant Method

70

(3)

Regula Falsi Method

73

(2)

Fixed Point Interation or Successive Substitution Method

75

(2)

Determination of Multiple Roots

77

(2)

Bairstow's Method

79

(5)

Muller's Method

84

(4)

Newton-Raphson Method for Simultaneous Nonlinear Equations

88

(4)

Unconstrained Minimization

92

(1)

Convergence of Methods

93

(4)

Choice of the Method

97

(1)

Use of Software Packages

98

(9)

Computer Programs

107

(34)

References and Bibliography

113

(2)

Review Questions

115

(4)

Problems

119

(22)

Solution of Simultaneous Linear Algebraic Equations

141

(129)

Introduction

142

(1)

Engineering Applications

143

(6)

Vector and Matrix Norms

149

(3)

Basic Concepts of Solution

152

(1)

Linearly Independent Equations

152

(1)

Ill-Conditioned Equations

153

(4)

Graphical Interpretation of the Solution

157

(2)

Solution Using Cramer's Rule

159

(3)

Gauss Elimination Method

162

(7)

Gauss-Jordan Elimination Procedure

169

(3)

LU Decomposition Method

172

(10)

Jacobi Iteration Method

182

(3)

Gauss-Seidel Iteration Method

185

(2)

Relaxation Methods

187

(1)

Simultaneous Linear Equations with Complex Coefficients and Constants

188

(3)

Matrix Inversion

191

(2)

Equations with Special Form of Coefficient Matrix

193

(7)

Overdetermined, Underdetermined, and Homogeneous Equations

200

(9)

Comparative Efficiencies of Various Methods and Recommendations

209

(5)

Choice of the Method

214

(1)

Use of Software Packages

214

(9)

Computer Programs

223

(47)

References and Bibliography

226

(2)

Review Questions

228

(5)

Problems

233

(37)

Solution of Matrix Eigenvalue Problem

270

(94)

Introduction

270

(4)

Engineering Applications

274

(6)

Conversion of General Eigenvalue Problem to Standard Form

280

(5)

Methods of Solving Eigenvalue Problems

285

(1)

Solution of the Characteristic Polynomial Equation

286

(5)

Jacobi Method

291

(6)

Given's Method

297

(4)

Householder's Method

301

(9)

Eigenvalues of a Tridiagonal Matrix

310

(3)

Eigenvectors of a Tridiagonal Matrix

313

(2)

Power Method

315

(9)

Choice of Method

324

(1)

Use of Software Packages

324

(9)

Computer Programs

333

(31)

References and Bibliography

337

(1)

Review Questions

338

(3)

Problems

341

(23)

Curve Fitting and Interpolation

364

(88)

Introduction

364

(2)

Engineering Applications

366

(2)

Collocation-Polynomial Fit

368

(2)

Interpolation

370

(1)

Lagrange Interpolation Formula

370

(3)

Newton's Divided-Difference Interpolating Polynomials

373

(10)

Interpolation Using Chebyshev Polynomials

383

(4)

Interpolation Using Splines

387

(12)

Least-Squares Regression

399

(11)

Curve Fitting with Multiple Variables

410

(2)

Choice of Method

412

(2)

Use of Software Packages

414

(20)

Computer Programs

434

(18)

References and Bibliography

436

(1)

Review Questions

437

(4)

Problems

441

(11)

Statistical Methods

452

(48)

Introduction

452

(1)

Engineering Applications

452

(2)

Basic Definitions

454

(1)

Histogram and Probability Density Function

455

(5)

Statistical Characteristics

460

(4)

Normal Distribution

464

(4)

Statistical Tests

468

(6)

Chi-Square Test for Distribution

474

(2)

Choice of Method

476

(1)

Use of Software Packages

477

(6)

Computer Programs

483

(17)

References and Bibliography

487

(1)

Review Questions

488

(3)

Problems

491

(9)

Numerical Differentiation

500

(60)

Introduction

500

(1)

Engineering Applications

501

(2)

Definition of the Derivative

503

(1)

Basic Finite-Difference Approximations

503

(1)

Using Taylor's Series Expansions

504

(15)

Using Difference Operators

519

(1)

Approximation of Derivatives Using Difference Operators

520

(2)

Using Differentiation of Interpolating Polynomials

522

(3)

Finite-Difference Approximations for Partial Derivatives

525

(4)

Choice of Method

529

(1)

Use of Software Packages

529

(11)

Computer Programs

540

(20)

References and Bibliography

543

(2)

Review Questions

545

(2)

Problems

547

(13)

Numerical Integration

560

(71)

Introduction

560

(2)

Engineering Applications

562

(3)

Newton--Cotes Formulas

565

(9)

Simpson's Rule

574

(9)

General Newton-Cotes Formulas

583

(1)

Richardson's Extrapolation

584

(3)

Romberg Integration

587

(1)

Gauss Quadrature

588

(9)

Integration with Unequal Segments

597

(1)

Numerical Integration of Improper Integrals

598

(4)

Numerical Integration in Two- and Three-Dimensional Domains

602

(3)

Choice of Method

605

(1)

Use of Software Packages

605

(5)

Computer Programs

610

(21)

References and Bibliography

614

(1)

Review Questions

615

(3)

Problems

618

(13)

Ordinary Differential Equations: Initial-Value Problems

631

(92)

Introduction

631

(2)

Engineering Applications

633

(3)

Simultaneous Differential Equations

636

(2)

Solution Concept

638

(1)

Euler's Method

639

(8)

Improvements and Modifications of Euler's Method

647

(7)

Runge-Kutta Methods

654

(11)

Multistep Methods

665

(1)

Adams Methods

665

(7)

Predictor--Corrector Methods

672

(11)

Simultaneous Differential Equations

683

(1)

Stiff Equations

684

(2)

Choice of Method

686

(1)

Use of Software Packages

687

(12)

Computer Programs

699

(24)

References and Bibliography

703

(2)

Review Questions

705

(6)

Problems

711

(12)

Ordinary Differential Equations: Boundary--Value Problems

723

(71)

Introduction

723

(1)

Engineering Applications

724

(3)

Shooting Methods

727

(5)

Generalization to n Equations

732

(8)

Finite-Difference Methods

740

(12)

Solution of Nonlinear Boundary-Value Problems

752

(2)

Solution of Eigenvalue Problems

754

(7)

Choice of Method

761

(1)

Use of Software Packages

762

(15)

Computer Programs

777

(17)

References and Bibliography

781

(1)

Review Questions

782

(3)

Problems

785

(9)

Partial Differential Equations

794

(91)

Introduction

794

(2)

Engineering Applications

796

(3)

Initial and Boundary Conditions

799

(1)

Elliptic Partial Differential Equations

800

(14)

Parabolic Partial Differential Equations

814

(8)

Crank-Nicholson Method

822

(1)

Method of Lines

823

(3)

Two-Dimensional Parabolic Problems

826

(3)

Hyperbolic Partial Differential Equations

829

(11)

Method of Characteristics

840

(6)

Finite-Difference Formulas in Polar Coordinate System

846

(2)

Choice of Method

848

(1)

Use of Software Packages

849

(14)

Computer Programs

863

(22)

References and Bibliography

868

(2)

Review Questions

870

(3)

Problems

873

(12)

Optimization

885

(73)

Introduction

885

(4)

Types of Optimization Problems

889

(1)

Engineering Applications

890

(4)

Optimization Methods from Differential Calculus

894

(8)

Linear-Programming Problem

902

(4)

Simplex Method

906

(1)

Search Methods for Nonlinear Optimization

907

(1)

Optimization of a Function of a Single Variable

908

(10)

Unconstrained Minimization of a Function of Several Variables

918

(6)

Constrained Minimization of a Function of Several Variables

924

(4)

Choice of Method

928

(1)

Use of Software Packages

928

(8)

Computer Programs

936

(22)

References and Bibliography

944

(1)

Review Questions

945

(4)

Problems

949

(9)

Finite-Element Method

958

(61)

Introduction

958

(1)

Engineering Applications

959

(3)

Discretization of the Domain

962

(3)

Interpolation Functions

965

(10)

Derivation of Element Characteristics Matrices and Vectors

975

(6)

Assemblage of Element Characteristic Matrices and Vectors

981

(1)

Solution of System Equations

981

(11)

Choice of Method

992

(1)

Use of Software Packages

993

(8)

Computer Programs

1001

(18)

References and Bibliography

1005

(1)

Review Questions

1006

(4)

Problems

1010

(9)

Appendix A Basics of Fortran 90

1019

(7)

Appendix B Basics of C Language

1026

(4)

Appendix C Basics of Maple

1030

(4)

Appendix D Basics of Matlab

1034

(4)

Appendix E Basics of Mathcad

1038

(4)

Appendix F Review of Matrix Algebra

1042

(7)

Appendix G Statistical Tables

1049

(4)

Index

1053

What is included with this book?

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.

The use of numerical methods for the analysis, simulation, and design of engineering processes and systems has been increasing at a rapid rate in recent years. The availability of cheap high-speed computing power makes the numerical solution of even complex engineering problems economically feasible. In the face of ever increasing demands on engineering profession to perform better, the students who learn numerical methods in preparing to face the challenges of 21st century should learn not only the theory behind the methods, but also acquire skills to implement the methods for computer solution. In addition, the students should be aware of the many commercial software systems available and their use in the solution of engineering problems. Although a student may not learn all the numerical methods described in this book and use all the software systems available in any one course, he or she should be in a position to intelligently select and use suitable numerical methods and software systems as the need arises in practice. The use of numerical methods in engineering can be considered partly science and partly art. Thus, a cookbook-type procedure will not be effective in learning the methods. A student should solve a problem using different approaches and a variety of software systems and experiment with the various parameters of the problem. The different results obtained through this process will form an experience base for selecting a suitable method and interpreting the results for a new problem. It is always desirable to compare and verify the results with other available solutions based on engineering judgment and intuition. This book is intended for courses on numerical methods at the junior and senior level as well as at the beginning graduate level. The book also serves as a reference for numerical methods in engineering. Fortran and C programs, along with illustrative examples, are given in each chapter to implement many of the numerical methods discussed in that chapter. The use of commercial numerical softwares--MATLAB, MAPLE and MATHCAD--in the solution of practical problems is demonstrated in every chapter. Even when a program from a software package is used, we need to understand the basic principles, purpose, and limitations of the program. Often, in many engineering applications, an available standard program cannot be used directly; we need to adapt and modify it. This invariably requires a sound knowledge of the numerical method as well as some computational experience with the method. The book is aimed at presenting numerical methods along with their practical applications in a manner that helps students achieve the goals just outlined. Organization Applied Numerical Methods for Engineersis organized into 13 chapters and 6 appendices. Chapter 1 presents an overview of numerical methods, iterative- processes, numerical errors, software available for numerical methods, programming languages, and the various aspects of computer program development. The methods of solving nonlinear equations are given in Chapter 2. The solution of sets of linear algebraic equations is presented in Chapter 3. Both direct and iterative methods are considered. The matrix eigenvalue problem is the topic of Chapter 4. Chapter 5 deals with the methods of curve fitting and interpolation. The probabilistic and statistical methods are considered in Chapter 6. The numerical differentiation and numerical integration are the topics of Chapters 7 and 8, respectively. The numerical solution of ordinary differential equations is considered in Chapters 9 and 10. While Chapter 9 presents the methods of solving initial-value problems, Chapter 10 deals with the solution of boundary-value problems. The numerical solution of partial differential equations is considered in Chapter 11. The optimization and the finite-element methods are presented in Chapters 12 and 13, respectively. Appendices A and B provide the basics of Fortran and C languages while Appendices C, D, and E summarize the basics of MAPLE, MATLAB, and MATHCAD, respectively. A review of matrix algebra is given in Appendix F. Finally, Appendix G presents tables of statistical distributions. The material of the book provides flexible options for different types of numerical methods courses. A junior and senior level course may cover the basic techniques of Chapters 1, 2, 3, and 5 to 9. A first-level graduate course can cover Chapters 4, 10, 11, 12, and 13 as well. The prerequisites for using the text are elementary calculus, basic concepts of linear algebra, and an introduction to differential equations. Each topic for Applied Numerical Methods for Engineersis self-contained. In derivations and developments, steps needed for continuity of understanding have been included to aid the reader at the introductory level. Representative engineering applications are given at the beginning of each chapter so that the reader can appreciate the practical use and application of the numerical methods presented in that chapter. Many sample problems are solved by using several methods, and the results are compared, discussed, and general conclusions are drawn. Most of the algorithms described in the book are implemented in the form of Fortran and C codes and are made available at the Web site of the book. The use of different commercial software systems, as well as the programs available at the Web site of the book, is illustrated in each chapter. Features The specific features of the book include A variety of engineering applications at the beginning of each chapter to illustrate the practicality of the methods considered in that chapter. The presentation of the material in a simple and user-friendly form. Illustrative examples follow the presentation of the topics. A discussion of convergence rate, error, relative performance, and recommendations for the numerical methods. Review questions to help students in reviewing and testing their understanding of the text material. These include multiple choice questions, questions with brief answers, true-false questions, questions involving matching of related descriptions, and fill-in-the-blank type questions. Answers to review questions can be found at the Web site of the book. A summary of important algorithms in the instructor''s manual. Over 700 problems, with solutions in the instructor''s manual. The inclusion of several open ended, project type and design problems at the end of chapters. Fortran and C programs for many of the methods presented in the book can be found at the Web site of the book. The inclusion of examples and problems based on the use of MATLAB, MAPLE, and MATHCAD in every chapter. References to lead the reader to specialized and advanced literature. Brief biographical information and photographs of scientists and mathematicians who contributed to the development of numerical methods, found at the Web site of the book. Web site of the book The Fortran and C programs used in the book, answers to problems, solutions to review questions, and brief biographical information of scientists can be found at the web site of the book: http://www.prenhall.com/rao. Note that the programs and techniques presented in the book and at the web site are intended for use by students in learning the material. Although the material has been tested, no warranty is implied as to their accuracy. I would appreciate receiving any errors found in the book.

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Applied Numerical Methods for Engineers and Scientists

Applied Numerical Methods for Engineers and Scientists

013089480X

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