Applied Numerical Methods for Engineers and Scientists

by Rao, Singiresu S.

• ISBN13: 9780130894809
• ISBN10: 013089480X
• Edition: 1st
• Format: Paperback
• Copyright: 2001-09-19
• Publisher: Pearson

Summary

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.

Table of Contents

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

Excerpts

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.