Overview

Treating numerical methods as a tool for use by the engineer or applied scientist, this introductory text is concerned with the application of such methods to the solution of algebraic, transcendental and differential equations. With a minimum of mathematical theory needed for understanding, the book concentrates on the methods likely to be needed by students in training and later in their careers. The emphasis, as far as differential equations are concerned, is towards finite difference methods, since they form the basis of most introductory courses on numerical techniques. However, an introduction to integral methods is also given. For the same reason, the depth of coverage given to ordinary differential equations is rather greater than that to partial differential equations (especially hyperbolic equations). Nevertheless, the material included on PDEs would be suitable for leading on to more advanced courses, such as one in computational fluid dynamics. Worked examples and problems are provided. Many of the methods can be used with a simple electronic calculator, others involve so much computation that a programmable device is required, and some will need a digital computer. This text is intended for undergraduate engineers and applied scientists. It will service a standard introductory course of approximately 40 hours duration, and equip students to tackle more advanced courses on specialized topics. Worked examples and problems are provided. Many of the methods can be used with a simple electronic calculator, others involve so much computation that a programme is required, and some will need a digital computer. This book should be of interest to undergraduate engineers and applied scientists.

Full Product Details

Author:   Graham de Vahl Davis
Publisher:   Chapman and Hall
Imprint:   Chapman and Hall
Edition:   New edition
Dimensions:   Width: 15.50cm , Height: 1.60cm , Length: 23.50cm
Weight:   0.468kg
ISBN:  

9780412438806

ISBN 10:   0412438801
Pages:   286
Publication Date:   20 November 1986
Audience:   College/higher education ,  Professional and scholarly ,  Undergraduate ,  Postgraduate, Research & Scholarly
Format:   Paperback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

1 Introduction.- 1.1 What are numerical methods?.- 1.2 Numerical methods versus numerical analysis.- 1.3 Why use numerical methods?.- 1.4 Approximate equations and approximate solutions.- 1.5 The use of numerical methods.- 1.6 Errors.- 1.7 Non-dimensional equations.- 1.8 The use of computers.- 2 The solution of equations.- 2.1 Introduction.- 2.2 Location of initial estimates.- 2.3 Interval halving.- 2.4 Simple iteration.- 2.5 Convergence.- 2.6 Aitken's extrapolation.- 2.7 Damped simple iteration.- 2.8 Newton-Raphson method.- 2.9 Extended Newton's method.- 2.10 Other iterative methods.- 2.11 Polynomial equations.- 2.12 Bairstow's method 56 Worked examples 58 Problems.- 3 Simultaneous equations.- 3.1 Introduction.- 3.2 Elimination methods.- 3.3 Gaussian elimination.- 3.4 Extensions to the basic algorithm.- 3.5 Operation count for the basic algorithm.- 3.6 Tridiagonal systems.- 3.7 Extensions to the Thomas algorithm.- 3.8 Iterative methods for linear systems.- 3.9 Matrix inversion.- 3.10 The method of least squares.- 3.11 The method of differential correction.- 3.12 Simple iteration for non-linear systems.- 3.13 Newton's method for non-linear systems.- Worked examples.- Problems.- 4 Interpolation, differentiation and integration.- 4.1 Introduction.- 4.2 Finite difference operators.- 4.3 Difference tables.- 4.4 Interpolation.- 4.5 Newton's forward formula.- 4.6 Newton's backward formula.- 4.7 Stirling's central difference formula.- 4.8 Numerical differentiation.- 4.9 Truncation errors.- 4.10 Summary of differentiation formulae.- 4.11 Differentiation at non-tabular points: maxima and minima.- 4.12 Numerical integration.- 4.13 Error estimation.- 4.14 Integration using backward differences.- 4.15 Summary of integration formulae.- 4.16 Reducing the truncation error 146 Worked examples 149 Problems.- 5 Ordinary differential equations.- 5.1 Introduction.- 5.2 Euler's method.- 5.3 Solution using Taylor's series.- 5.4 The modified Euler method.- 5.5 Predictor-corrector methods.- 5.6 Milne's method, Adams' method, and Hamming's method.- 5.7 Starting procedure for predictor-corrector methods.- 5.8 Estimation of error of predictor-corrector methods.- 5.9 Runge-Kutta methods.- 5.10 Runge-Kutta-Merson method.- 5.11 Application to higher-order equations and to systems.- 5.12 Two-point boundary value problems.- 5.13 Non-linear two-point boundary value problems 198 Worked examples 199 Problems.- 6 Partial differential equations I - elliptic equations.- 6.1 Introduction.- 6.2 The approximation of elliptic equations.- 6.3 Boundary conditions.- 6.4 Non-dimensional equations again.- 6.5 Method of solution.- 6.6 The accuracy of the solution.- 6.7 Use of Richardson's extrapolation.- 6.8 Other boundary conditions.- 6.9 Relaxation by hand-calculation.- 6.10 Non-rectangular solution regions.- 6.11 Higher-order equations 238 Problems.- 7 Partial differential equations II - parabolic equations.- 7.1 Introduction.- 7.2 The conduction equation.- 7.3 Non-dimensional equations yet again.- 7.4 Notation.- 7.5 An explicit method.- 7.6 Consistency.- 7.7 The Dufort-Frankel method.- 7.8 Convergence.- 7.9 Stability.- 7.10 An unstable finite difference approximation.- 7.11 Richardson's extrapolation 261 Worked examples 262 Problems.- 8 Integral methods for the solution of boundary value problems.- 8.1 Introduction.- 8.2 Integral methods.- 8.3 Implementation of integral methods 271 Worked examples 278 Problems.- Suggestions for further reading.

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