Applied Numerical Methods for Engineers and Scientists, 1st edition

  • Singiresu S. Rao

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The availability of inexpensive high-speed computing power makes the numerical solution of even complex engineering problems economically feasible. In the face of ever-increasing demands on the engineering profession to perform better, students preparing to face the challenges of the twenty-first century should learn not only the theory behind the numerical methods, but also acquire the skills needed to implement the methods for computer solution. In addition, students should be aware of the many commercial software systems available and their appropriate uses in the solution of engineering problems. The student should be in a position to intelligently select and use suitable numerical methods and software systems as the need arises in practice. He or she should solve a given problem using different approaches with a variety of software systems and experiment with the various parameters of the problem. This book is intended for courses on numerical methods at the junior/senior level as well as at the beginning graduate level. The book also serves as a reference for numerical methods in engineering.

Special Features:

  • In derivations and developments, steps needed for continuity of understanding have been included to aid the reader at the introductory level.
  • A variety of engineering applications are presented. More than 800 problems are included.
  • Review questions are given 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.
  • Several open-ended project- and design-type problems are included at the end of each chapter.
  • Examples and problems based on MATLAB, Maple, and Mathcad are included in each chapter.
  • Representative Fortran 90 and C programs are given in the book and several additional programs can be found at the course website.
  • Helpful appendices describing the basics of Fortran 90, C Language, MATLAB, Maple, Mathcad, and matrix algebra are included.
  • Brief biographical information and photographs of scientists and mathematicians who contributed to the development of numerical methods are given in the book.

Table of contents

(NOTE: Each chapter concludes with References and Bibliography, Review Questions, and Problems.)

 1. Introduction to Numerical Methods.

Importance of Numerical Methods in Engineering. Computers. Computer Programming Languages. Data Representation. Programming Structure. Errors. Numerical Methods Considered. Software for Numerical Analysis. Use of Software Packages. Computer Programs.

 2. Solution of Nonlinear Equations.

Introduction. Engineering Applications. Incremental Search Method. Bisection Method. Newton-Raphson Method. Secant Method. Regula Falsi Method. Fixed Point Iteration or Successive Substitution Method. Determination of Multiple Roots. Bairstow's Method. Muller's Method. Newton-Raphson Method for Simultaneous Nonlinear Equations. Unconstrained Minimization. Convergence of Methods. Choice of Method. Use of Software Packages. Computer Programs.

 3. Solution of Simultaneous Linear Algebraic Equations.

Introduction. Engineering Applications. Vector and Matrix Norms. Basic Concepts of Solution. Linearly Independent Equations. Ill-Conditioned Equations. Graphical Interpretation of the Solution. Solution Using Cramer's Rule. Gauss Elimination Method. Gauss-Jordan Elimination Procedure. LU Decomposition Method. Jacobi Iteration Method. Gauss-Seidel Iteration Method. Relaxation Methods. Simultaneous Linear Equations with Complex Coefficients and Constants. Matrix Inversion. Equations with Special Form of Coefficient Matrix. Overdetermined, Underdetermined, and Homogeneous Equations. Comparative Efficiencies of Various Methods and Recommendations. Choice of the Method. Use of Software Packages. Computer Programs.

 4. Solution of Matrix Eigenvalue Problem.

Introduction. Engineering Applications. Conversion of General Eigenvalue Problem to Standard Form. Methods of Solving Eigenvalue Problems. Solution of the Characteristic Polynomial Equations. Jacobi Method. Given's Method. Householder's Method. Eigenvalues of a Tridiagonal Matrix. Eigenvectors of a Tridiagonal Matrix. Power Method. Choice of Method. Use of Software Packages. Computer Programs.

 5. Curve Fitting and Interpolation.

Introduction. Engineering Applications. Collocation-Polynomial Fit. Interpolation. Lagrange Interpolation Formula. Newton's Divided-Difference Interpolating Polynomials. Interpolation Using Chebysev Polynomials. Interpolation Using Splines. Least-Squares Regression. Curve Fitting with Multiple Variables. Choice of Method. Use of Software Packages. Computer Programs.

 6. Statistical Methods.

Introduction. Engineering Applications. Basic Definitions. Histogram and Probability Density Function. Statistical Characteristics. Normal Distributions. Statistical Tests. Chi-Square Test for Distribution. Choice of Method. Use of Software Packages. Computer Programs.

 7. Numerical Differentiation.

Introduction. Engineering Applications. Definition of the Derivative. Basic Finite-Difference Approximations. Using Taylor's Series Expansions. Using Difference Operators. Approximation of Derivatives Using Difference Operators. Using Differentiation of Interpolating Polynomials. Finite-Difference Approximations for Partial Derivatives. Choice of Method. Use of Software Packages. Computer Programs.

 8. Numerical Integration.

Introduction. Engineering Applications. Newton-Cotes Formulas. Simpson's Rule. General Newton-Cotes Formulas. Richardson's Extrapolation. Romberg Integration. Gauss Quadrature. Integration with Unequal Segments. Numerical Integration of Improper Integrals. Numerical Integration in Two- and Three-Dimensional Domains. Choice of Method. Use of Software Packages. Computer Programs.

 9. Ordinary Differential Equations: Initial-Value Problems.

Introduction. Engineering Applications. Simultaneous Differential Equations. Solution Concept. Euler's Method. Improvements and Modifications of Euler's Method. Runge-Kutta Methods. Multistep Methods. Adams Methods. Predictor-Corrector Methods. Simultaneous Differential Equations. Stiff Equations. Choice of Method. Use of Software Packages. Computer Programs.

10. Ordinary Differential Equations: Boundary-Value Problems.

Introduction. Engineering Applications. Shooting Methods. Generalization to n Equations. Finite-Difference Methods. Solution of Nonlinear Boundary-Value Problems. Solution of Eigenvalue Problems. Choice of Method. Use of Software Packages. Computer Programs.

11. Partial Differential Equations.

Introduction. Engineering Applications. Initial and Boundary Conditions. Elliptic Partial Differential Equations. Parabolic Partial Differential Equations. Crank-Nicholson Method. Method of Lines. Two-Dimensional Parabolic Problems. Hyperbolic Partial Differential Equations. Method of Characteristics. Finite-Difference Formulas in Polar Coordinate System. Choice of Method. Use of Software Packages. Computer Programs.

12. Optimization.

Introduction. Types of Optimization Problems. Engineering Applications. Optimization Methods from Differential Calculus. Linear-Programming Problem. Simplex Method. Search Methods for Nonlinear Optimization. Optimization of a Function of a Single Variable. Unconstrained Minimization of a Function of Several Variables. Constrained Minimization of a Function of Several Variables. Choice of Method. Use of Software Packages. Computer Programs.

13. Finite-Element Method.

Introduction. Engineering Applications. Discretization of the Domain. Interpolation Functions. Derivation of Element Characteristic Matrices and Vectors. Assemblage of Element Characteristics Matrices and Vectors. Solution of System Equations. Choice of Method. Use of Software Packages. Computer Programs.

Appendix A: Basics of Fortran 90.

Appendix B: Basics of C Language.

Appendix C: Basics of MAPLE.

Appendix D: Basics of MATLAB.

Appendix E: Basics of MathCAD.

Appendix F: Review of Matrix Algebra.

Appendix G: Statistical Tables.


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Published by Pearson (September 19th 2001) - Copyright © 2002