Introduction to Numerical and Analytical Methods with MATLAB® for Engineers and Scientists

544 Pages·2013·8.38 MB·English

by  Bober

MATHEMATICS FOR ENGINEERS
Introduction to
Bober
Numerical and
Introduction to
Numerical and Analytical Methods with MATLAB®
Analytical Methods
for Engineers and Scientists N
u
m
Introduction to Numerical and Analytical Methods with MATLAB® for Engineers and e with MATLAB®
r
i
Scientists provides the basic concepts of programming in MATLAB for engineering c
a
applications.
l

a for Engineers and
f
• Teaches engineering students how to write computer programs on the on
rd
MATLAB platform
E
A
• Examines the selection and use of numerical and analytical methods through n Scientists
gn
examples and case studies ia In
n
l t
• Demonstrates mathematical concepts that can be used to help solve engineering ey r
o
et
problems, including matrices, roots of equations, integration, ordinary differential ric d
equations, curve fitting, algebraic linear equations, and more s aal uc
t
nM i
The text covers useful numerical methods, including interpolation, Simpson’s rule on o
d
e n
integration, the Gauss elimination method for solving systems of linear algebraic
St
h t
equations, the Runge–Kutta method for solving ordinary differential equations, and co o
i
the search method in combination with the bisection method for obtaining the roots of ed
ns
transcendental and polynomial equations. It also highlights MATLAB’s built-in functions. t
iw
These include interp1 function, the quad and dblquad functions, the inv function, the s
ti
st
ode45 function, the fzero function, and many others. The second half of the text covers h

more advanced topics, including the iteration method for solving pipe flow problems, M
the Hardy–Cross method for solving flow rates in a pipe network, separation of A
variables for solving partial differential equations, and the use of Laplace transforms T
L
to solve both ordinary and partial differential equations. A
B
This book serves as a textbook for a first course in numerical methods using MATLAB ®
to solve problems in mechanical, civil, aeronautical, and electrical engineering. It can
also be used as a textbook or as a reference book in higher level courses.
K16725
William Bober
6000 Broken Sound Parkway, NW
Suite 300, Boca Raton, FL 33487
711 Third Avenue
New York, NY 10017
an informa business
2 Park Square, Milton Park
www.crcpress.com Abingdon, Oxon OX14 4RN, UK www.crcpress.com
Introduction to
Numerical and
Analytical Methods
with MATLAB®
for Engineers and
Scientists

Introduction to
Numerical and
Analytical Methods
with MATLAB®
for Engineers and
Scientists
William Bober
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not
warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® soft-
ware or related products does not constitute endorsement or sponsorship by The MathWorks of a particular
pedagogical approach or particular use of the MATLAB® software.
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2014 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Version Date: 20130827
International Standard Book Number-13: 978-1-4665-7609-4 (eBook - PDF)
This book contains information obtained from authentic and highly regarded sources. Reasonable efforts
have been made to publish reliable data and information, but the author and publisher cannot assume
responsibility for the validity of all materials or the consequences of their use. The authors and publishers
have attempted to trace the copyright holders of all material reproduced in this publication and apologize to
copyright holders if permission to publish in this form has not been obtained. If any copyright material has
not been acknowledged please write and let us know so we may rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmit-
ted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented,
including photocopying, microfilming, and recording, or in any information storage or retrieval system,
without written permission from the publishers.
For permission to photocopy or use material electronically from this work, please access www.copyright.
com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood
Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and
registration for a variety of users. For organizations that have been granted a photocopy license by the CCC,
a separate system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used
only for identification and explanation without intent to infringe.
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
Contents
Preface ............................................................................................................xi
Acknowledgments .........................................................................................xv
Author .........................................................................................................xvii
1 Computer Programming with MATLAB• for Engineers ......................1
1.1 Introduction ......................................................................................1
1.2 C omputer Usage in Engineering ........................................................1
1.3 Mathematical Model .........................................................................2
1.4 Computer Programming ...................................................................3
1.5 Why MATLAB•? .............................................................................3
1.6 Programming Methodologies ............................................................4
1.7 MATLAB• Programming Language .................................................4
1.8 B uilding Blocks in Writing a Program ...............................................5
1.9 C onventions in This Book .................................................................5
1.10 Example Programs .............................................................................6
2 MATLAB• Fundamentals ......................................................................7
2.1 Introduction ......................................................................................7
2.2 MATLAB• Desktop .........................................................................8
2.3 Constructing a Program in MATLAB• ..........................................11
2.4 MATLAB• Fundamentals ..............................................................14
2.5 MATLAB• Input/Output ...............................................................23
2.5.1 Output ...............................................................................23
2.5.2 Input ..................................................................................26
2.6 Loops ...............................................................................................28
2.7 MATLAB• Graphics .......................................................................43
2.8 Conditional Operators and Alternate Paths .....................................58
2.9 W orking with Built-In Functions with Vector Arguments ...............70
2.10 More on MATLAB• Graphics ........................................................72
2.11 Debugging a Program......................................................................75
Projects .....................................................................................................77
References ..................................................................................................97
v
vi  ◾  Contents
3 Taylor Series, Self-Written Functions and MATLAB•’s interp1
Function ...............................................................................................99
3.1 Introduction ....................................................................................99
3.2 F unctions Expressed as a Series ......................................................100
3.3 Self-Written Functions ...................................................................105
3.4 Anonymous Functions ...................................................................110
3.5 MATLAB•’s interp1 Function .................................................112
3.6 Working with Characters and Strings ............................................115
Projects ....................................................................................................121
References ................................................................................................125
4 Matrices ..............................................................................................127
4.1 Introduction ..................................................................................127
4.2 Matrix Operations .........................................................................127
4.3 System of Linear Equations ...........................................................133
4.4 Statics Truss Problem ....................................................................136
4.5 Resistive Circuit Problem ..............................................................143
4.6 Gauss Elimination .........................................................................147
4.7 Gauss–Jordan Method...................................................................152
4.8 Number of Solutions .....................................................................154
4.9 Inverse Matrix ...............................................................................155
4.10 Eigenvalue Problem in Mechanical Vibrations...............................160
4.11 Eigenvalue Problem in Electrical Circuits ......................................164
Projects ....................................................................................................168
Reference .................................................................................................173
5 Roots of Algebraic and Transcendental Equations .............................175
5.1 Introduction ..................................................................................175
5.2 The Search Method .......................................................................175
5.3 Bisection Method ..........................................................................176
5.4 Newton–Raphson Method ............................................................178
5.5 MATLAB•’s Root-Finding Functions ...........................................179
5.5.1 fzero Function ..............................................................179
5.5.2 roots Function ..............................................................186
Projects ....................................................................................................189
Reference .................................................................................................199
6 Numerical Integration ........................................................................201
6.1 Introduction ..................................................................................201
6.2 Numerical Integration with the Trapezoidal Rule .........................201
6.3 Numerical Integration and Simpson’s Rule ....................................203
6.4 Improper Integrals .........................................................................207
Contents  ◾  vii
6.5 MATLAB•’s quad Function ........................................................210
6.6 MATLAB•’s dblquad Function .................................................215
Projects ....................................................................................................219
7 Numerical Integration of Ordinary Differential Equations ...............231
7.1 Introduction ..................................................................................231
7.2 Initial Value Problem .....................................................................232
7.3 Euler Algorithm ............................................................................232
7.4 M odified Euler Method with Predictor–Corrector Algorithm.......233
7.5 Fourth-Order Runge–Kutta Method ............................................241
7.6 System of Two First-Order Differential Equations ........................244
7.7 Single Second-Order Equation ......................................................248
7.8 MATLAB•’s ODE Function .........................................................256
Projects ....................................................................................................259
8 Boundary Value Problems of Ordinary Differential Equations .........277
8.1 Introduction ..................................................................................277
8.2 Difference Formulas ......................................................................277
8.3 S olution of a Tri-Diagonal System of Linear Equations .................289
Projects ....................................................................................................293
9 Curve Fitting ......................................................................................303
9.1 Introduction ..................................................................................303
9.2 Method of Least Squares ...............................................................303
9.2.1 Best-Fit Straight Line ........................................................303
9.2.2 Best-Fit mth-Degree Polynomial .......................................305
9.3 Curve Fitting with the Exponential Function ...............................306
9.4 MATLAB•’s Curve Fitting Functions ...........................................309
9.5 Cubic Splines .................................................................................314
9.6 MATLAB•’s Cubic Spline Curve Fitting Function .......................316
9.7 Curve Fitting with Fourier Series ..................................................319
Projects ....................................................................................................324
10 Simulink• ...........................................................................................329
10.1 Introduction ..................................................................................329
10.2 Creating a Model in Simulink• .....................................................329
10.3 Typical Building Blocks in Constructing a Model .........................332
10.4 Tips for Constructing and Running Models .................................335
10.5 Constructing a Subsystem .............................................................337
10.6 Using the Mux and Fcn Blocks .....................................................340
10.7 Using the Transfer Fcn Block .......................................................340
10.8 Using the Relay and Switch Blocks ................................................341
10.9 Trigonometric Function Blocks ....................................................344
viii  ◾  Contents
10.10 To Workspace Block ......................................................................345
Projects ....................................................................................................351
Reference .................................................................................................355
11 Optimization ......................................................................................357
11.1 Introduction ..................................................................................357
11.2 Unconstrained Optimization Problems .........................................358
11.3 Method of Steepest Descent ..........................................................361
11.4 MATLAB•’s fminbnd and fminsearch Functions ................365
11.5 Optimization with Constraints .....................................................368
11.6 Lagrange Multipliers .....................................................................371
11.7 MATLAB•’s fmincon Function .................................................373
Projects ....................................................................................................383
Reference .................................................................................................387
12 Iteration Method ................................................................................389
12.1 Introduction ..................................................................................389
12.2 Iteration in Pipe Flow Analysis ......................................................389
12.3 Hardy–Cross Method ....................................................................394
Projects ...................................................................................................406
References ................................................................................................413
13 Partial Differential Equations ............................................................415
13.1 Classification of Partial Differential Equations ..............................415
13.2 Solution by Separation of Variables................................................416
13.2.1 Vibrating String ................................................................416
13.2.2 Unsteady Heat Transfer I .................................................420
13.2.3 Unsteady Heat Transfer in 2-D ........................................425
13.3 Review of Finite-Difference Formulas ..........................................430
13.4 Finite-Difference Methods Applied to Partial Differential
Equations .....................................................................................430
13.4.1 Explicit Method ................................................................431
13.4.2 Implicit Method ...............................................................433
13.5 The Gauss–Seidel Method ............................................................434
Projects ....................................................................................................437
14 Laplace Transforms ............................................................................449
14.1 Introduction ..................................................................................449
14.2 Laplace Transform and Inverse Transform ....................................449
14.3 Transforms of Derivatives ..............................................................452
14.4 Ordinary Differential Equations, Initial Value Problem ................453
14.5 MATLAB•’s residue Function .................................................457
14.6 Unit Step Function ........................................................................459
14.7 Convolution .................................................................................464
Contents  ◾  ix
14.8 Laplace Transforms Applied to Circuits ........................................466
14.9 Delta Function ..............................................................................471
14.10 Laplace Transforms Applied to Partial Differential Equations .......474
Projects ....................................................................................................479
References ................................................................................................487
Review Answers ...........................................................................................489
Appendix A: Special Characters in MATLAB• Plots ..................................499
Appendix B: Derivation of the Heat Transfer Equation in Solids ...............503
Appendix C: Derivation of the Beam Deflection Equation .........................509
Appendix D: Proof of Orthogonal Relationship of the cos(λ x) Functions .........513
n
Appendix E: Getting Started with MATLAB• Version R2012a ............................517