Engineering Mathematics with Examples PDF

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Engineering Mathematics with Examples and Applications

Xin-She Yang Middlesex University School of Science and Technology London, United Kingdom

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1800, San Diego, CA 92101-4495, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright © 2017 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-809730-4 For information on all Academic Press publications visit our website at https://www.elsevier.com

Publisher: Nikki Levy Acquisition Editor: Graham Nisbet Editorial Project Manager: Susan Ikeda Production Project Manager: Mohanapriyan Rajendran Designer: Matthew Limbert Typeset by VTeX

Contents About the Author Preface Acknowledgment

ix xi xiii

Part I Fundamentals 1. Equations and Functions 1.1. Numbers and Real Numbers 1.1.1. Notes on Notations and Conventions 1.1.2. Rounding Numbers and Significant Digits 1.1.3. Concept of Sets 1.1.4. Special Sets 1.2. Equations 1.2.1. Modular Arithmetic 1.3. Functions 1.3.1. Domain and Range 1.3.2. Linear Function 1.3.3. Modulus Function 1.3.4. Power Function 1.4. Quadratic Equations 1.5. Simultaneous Equations Exercises

3 5 6 8 8 10 11 12 12 14 15 16 19 20

21 23 24 25 28

3. Binomial Theorem and Expansions 3.1. Binomial Expansions 3.2. Factorials 3.3. Binomial Theorem and Pascal’s Triangle Exercises

4.1. Simple Sequences 4.1.1. Arithmetic Sequence 4.1.2. Geometric Sequence 4.2. Fibonacci Sequence 4.3. Sum of a Series 4.4. Infinite Series Exercises

37 38 39 40 41 44 45

5. Exponentials and Logarithms 3

2. Polynomials and Roots 2.1. Index Notation 2.2. Floating Point Numbers 2.3. Polynomials 2.4. Roots Exercises

4. Sequences

31 32 33 35

5.1. Exponential Function 5.2. Logarithm 5.3. Change of Base for Logarithm Exercises

47 48 53 54

6. Trigonometry 6.1. Angle 6.2. Trigonometrical Functions 6.2.1. Identities 6.2.2. Inverse 6.2.3. Trigonometrical Functions of Two Angles 6.3. Sine Rule 6.4. Cosine Rule Exercises

55 57 58 59 59 62 62 63

Part II Complex Numbers 7. Complex Numbers 7.1. 7.2. 7.3. 7.4. 7.5.

Why Do Need Complex Numbers? Complex Numbers Complex Algebra Euler’s Formula Hyperbolic Functions 7.5.1. Hyperbolic Sine and Cosine 7.5.2. Hyperbolic Identities 7.5.3. Inverse Hyperbolic Functions Exercises

67 67 68 71 72 72 74 74 75 v

vi Contents

14. Multiple Integrals and Special Integrals

Part III Vectors and Matrices 8. Vectors and Vector Algebra 8.1. Vectors 8.2. Vector Algebra 8.3. Vector Products 8.4. Triple Product of Vectors Exercises

79 80 83 85 86

9. Matrices 9.1. 9.2. 9.3. 9.4. 9.5.

Matrices Matrix Addition and Multiplication Transformation and Inverse System of Linear Equations Eigenvalues and Eigenvectors 9.5.1. Distribution of Eigenvalues 9.5.2. Definiteness of a Matrix Exercises

111 114 117 120

11. Integration 121 125 128 130

12. Ordinary Differential Equations 12.1. Differential Equations 12.2. First-Order Equations 12.3. Second-Order Equations 12.4. Higher-Order ODEs 12.5. System of Linear ODEs Exercises

131 132 136 142 143 144

13. Partial Differentiation 13.1. Partial Differentiation 13.2. Differentiation of Vectors 13.3. Polar Coordinates 13.4. Three Basic Operators Exercises

153 153 155 157 157 158 159 161

15.1. Analytic Functions 15.2. Complex Integrals 15.2.1. Cauchy’s Integral Theorem 15.2.2. Residue Theorem Exercises

163 165 166 168 169

Part V Fourier and Laplace Transforms 16. Fourier Series and Transform

10. Differentiation

11.1. Integration 11.2. Integration by Parts 11.3. Integration by Substitution Exercises

Line Integral Multiple Integrals Jacobian Special Integrals 14.4.1. Asymptotic Series 14.4.2. Gaussian Integrals 14.4.3. Error Functions Exercises

15. Complex Integrals 87 90 93 98 99 104 107 108

Part IV Calculus 10.1. Gradient and Derivative 10.2. Differentiation Rules 10.3. Series Expansions and Taylor Series Exercises

14.1. 14.2. 14.3. 14.4.

145 146 147 149 152

16.1. Fourier Series 16.1.1. Fourier Series 16.1.2. Orthogonality 16.1.3. Determining the Coefficients 16.2. Fourier Transforms 16.3. Solving Differential Equations Using Fourier Transforms 16.4. Discrete and Fast Fourier Transforms Exercises

173 173 175 176 179 182 183 185

17. Laplace Transforms 17.1. Laplace Transform 17.1.1. Laplace Transform Pairs 17.1.2. Scalings and Properties 17.1.3. Derivatives and Integrals 17.2. Transfer Function 17.3. Solving ODE via Laplace Transform 17.4. Z-Transform 17.5. Relationships between Fourier, Laplace and Z-transforms Exercises

187 189 189 191 192 194 196 197 197

Part VI Statistics and Curve Fitting 18. Probability and Statistics 18.1. Random Variables

201

Contents

18.2. Mean and Variance 18.3. Binomial and Poisson Distributions 18.4. Gaussian Distribution 18.5. Other Distributions 18.6. The Central Limit Theorem 18.7. Weibull Distribution Exercises

202 203 207 209 211 212 214

19. Regression and Curve Fitting 19.1. Sample Mean and Variance 19.2. Method of Least Squares 19.2.1. Maximum Likelihood 19.2.2. Linear Regression 19.3. Correlation Coefficient 19.4. Linearization 19.5. Generalized Linear Regression 19.6. Hypothesis Testing 19.6.1. Confidence Interval 19.6.2. Student’s t -Distribution 19.6.3. Student’s t -Test Exercises

215 217 217 217 219 221 222 225 225 226 227 228

Part VII Numerical Methods 20. Numerical Methods 20.1. 20.2. 20.3. 20.4. 20.5.

Finding Roots Bisection Method Newton-Raphson Method Numerical Integration Numerical Solutions of ODEs 20.5.1. Euler Scheme 20.5.2. Runge-Kutta Method Exercises

231 232 233 234 237 237 237 241

21. Computational Linear Algebra 21.1. 21.2. 21.3. 21.4.

System of Linear Equations Gauss Elimination LU Factorization Iteration Methods 21.4.1. Jacobi Iteration Method 21.4.2. Gauss-Seidel Iteration 21.4.3. Relaxation Method 21.5. Newton-Raphson Method 21.6. Conjugate Gradient Method Exercises

243 244 247 249 250 253 253 254 254 255

vii

Part VIII Optimization 22. Linear Programming 22.1. Linear Programming 22.2. Simplex Method 22.2.1. Basic Procedure 22.2.2. Augmented Form 22.3. A Worked Example Exercises

259 260 261 262 263 265

23. Optimization 23.1. Optimization 23.2. Optimality Criteria 23.2.1. Feasible Solution 23.2.2. Optimality Criteria 23.3. Unconstrained Optimization 23.3.1. Univariate Functions 23.3.2. Multivariate Functions 23.4. Gradient-Based Methods 23.4.1. Newton’s Method 23.4.2. Steepest Descent Method 23.5. Nonlinear Optimization 23.5.1. Penalty Method 23.5.2. Lagrange Multipliers 23.6. Karush-Kuhn-Tucker Conditions 23.7. Sequential Quadratic Programming 23.7.1. Quadratic Programming 23.7.2. Sequential Quadratic Programming Exercises

267 269 269 269 270 270 271 275 275 276 278 278 279 280 281 281 282 283

Part IX Advanced Topics 24. Partial Differential Equations 24.1. 24.2. 24.3. 24.4.

Introduction First-Order PDEs Classification of Second-Order PDEs Classic Mathematical Models: Some Examples 24.4.1. Laplace’s and Poisson’s Equation 24.4.2. Parabolic Equation 24.4.3. Hyperbolic Equation 24.5. Solution Techniques 24.5.1. Separation of Variables 24.5.2. Laplace Transform 24.5.3. Fourier Transform 24.5.4. Similarity Solution 24.5.5. Change of Variables Exercises

287 288 290 291 291 292 293 294 294 296 296 297 298 299

viii Contents

25. Tensors 25.1. Summation Notations 25.2. Tensors 25.2.1. Rank of a Tensor 25.2.2. Contraction 25.2.3. Symmetric and Antisymmetric Tensors 25.2.4. Tensor Differentiation 25.3. Hooke’s Law and Elasticity Exercises

301 302 303 303 304 305 306 307

309 309 311 314 316 317

319 319 320 321 321 322 324

28. Mathematical Modeling 28.1. 28.2. 28.3. 28.4. 28.5.

338 340

A.1. A.2. A.3. A.4. A.5. A.6.

Differentiation and Integration Complex Numbers Vectors and Matrices Fourier Series and Transform Asymptotics Special Integrals

341 341 341 342 343 343

B. Mathematical Software Packages

27. Integral Equations 27.1. Integral Equations 27.1.1. Fredholm Integral Equations 27.1.2. Volterra Integral Equation 27.2. Solution of Integral Equations 27.2.1. Separable Kernels 27.2.2. Volterra Equation Exercises

335 336 337 337 338

A. Mathematical Formulas

26. Calculus of Variations 26.1. Euler-Lagrange Equation 26.1.1. Curvature 26.1.2. Euler-Lagrange Equation 26.2. Variations with Constraints 26.3. Variations for Multiple Variables Exercises

28.5.4. Functional and Integral Equations 28.5.5. Statistical Models 28.5.6. Fuzzy Models 28.5.7. Learned Models 28.5.8. Data-Driven Models 28.6. Brownian Motion and Diffusion: A Worked Example Exercises

Mathematical Modeling 325 Model Formulation 326 Different Levels of Approximations 328 Parameter Estimation 330 Types of Mathematical Models 332 28.5.1. Algebraic Equations 332 28.5.2. Tensor Relationships 333 28.5.3. Differential Equations: ODE and PDEs 333

B.1. Matlab B.1.1. Matlab B.1.2. MuPAD B.2. Software Packages Similar to Matlab B.2.1. Octave B.2.2. Scilab B.3. Symbolic Computation Packages B.3.1. Mathematica B.3.2. Maple B.3.3. Maxima B.4. R and Python B.4.1. R B.4.2. Python

345 345 346 347 347 348 348 348 348 349 350 350 350

C. Answers to Exercises Bibliography Index

381 383

Chapter 28

Mathematical Modeling Chapter Points • Mathematical modeling is introduced with the basic modeling procedure, including mathematical model formulation based on physical laws, parameter estimation and normalization.

• Different levels of approximations are explained to discuss the assumptions, abstractions and the balance of accuracy and model complexity.

• Different types of models are explained with some examples relevant to science and engineering applications.

• A worked example is presented in detail to model Brownian motion and diffusion.

28.1

MATHEMATICAL MODELING

Mathematical modeling is the process of formulating an abstract model in terms of mathematical language to describe the complex behavior of a real system. Mathematical models are quantitative models and often expressed in terms of ordinary differential equations and partial differential equations. Mathematical models can also be statistical models, fuzzy logic models and empirical relationships. In fact, any model description using mathematical language can be called a mathematical model. Mathematical modeling is widely used in natural sciences, computing, engineering, meteorology, economics and finance. For example, theoretical physics is essentially all about the modeling of real-world processes using several basic principles (such as the conservation of energy and momentum) and a dozen important equations (such as the wave equation, the Schrödinger equation, the Einstein equation). Most of these equations are partial differential equations. An important feature of mathematical modeling is its interdisciplinary nature. It involves applied mathematics, computer sciences, physics, chemistry, engineering, biology and other disciplines such as economics, depending on the problem of interest. Mathematical modeling in combination with scientific computing is an emerging interdisciplinary technology. Many international companies use it to model physical processes, to design new products, to find solutions to challenging problems, and to increase their competitiveness in international markets. Example 28.1 One of the simplest models we learned in school is probably Newton’s second law that relates the force F acted on a body with a mass m to its acceleration a. That is F = ma, which is one of the most accurate models in science. This is a linear relationship and thus a linear model, but a very well-tested model. Apart from a simple mathematical formula, as a mathematical model, all the quantities involved such as force, mass and acceleration must have appropriate units. For example, the unit of F is Newton (N), the unit of mass is kilogram (kg), while the acceleration has a derived unit (a combination of units) of m/s2 . Therefore, a person of 80 kg has a weight (the force acted upon the person by the Earth) is W = mg where g = 9.8 m/s2 is the acceleration due to gravity. That is W = mg = 80 (kg) × 9.8 (m/s2 ) = 784 N. If the units are wrong, even a good model will give wrong values. This highlights the importance of units and the parameters (e.g., g here) in mathematical modeling.

Engineering Mathematics with Examples and Applications Copyright © 2017 Elsevier Inc. All rights reserved.

325

326

PART | IX Advanced Topics

FIGURE 28.1 Mathematical modeling.

Mathematical modeling is an iterative, multidisciplinary process with many steps from the abstraction of the processes in nature to the construction of the full mathematical models. The basic steps of mathematical modeling can be summarized as meta-steps shown in Fig. 28.1. The process typically starts with the analysis of a real world problem so as to extract the fundamental physical processes by idealization and various assumptions. Once an idealized physical model is formulated, it can then be translated into the corresponding mathematical model in terms of partial differential equations (PDEs), integral equations, and statistical models. Then, the mathematical model should be investigated in great detail by mathematical analysis (if possible), numerical simulations and other tools so as to make predictions under appropriate conditions. Then, these simulation results and predictions will be validated against the existing models, well-established benchmarks, and experimental data. If the results are satisfactory (but they rarely are at first), then the mathematical model can be accepted. If not, both the physical model and mathematical model will be modified based on the feedback, then the new simulations and prediction will be validated again. After a certain number of iterations of the whole process (often many), a good mathematical model can properly be formulated, which will provide great insight into the real world problem and may also predict the behavior of the process under study. For any physical problem in physics and engineering, for example, there are traditionally two ways to deal with it by either theoretical approaches or field observations and experiments. The theoretical approach in terms of mathematical modeling is an idealization and simplification of the real problem and the theoretical models often extract the essential or major characteristics of the problem. The mathematical equations obtained even for such over-simplified systems are usually very difficult for mathematical analysis. On the other hand, the field studies and experimental approach can be expensive if not impractical. Apart from financial and practical limitations, other constraining factors include the inaccessibility of the locations, the range of physical parameters, and time for carrying out various experiments. As the computing speed and power of computers have increased dramatically in the last few decades, a practical third way or approach is emerging, which is computational modeling and numerical experimentation based on mathematical models. It is now widely acknowledged that computational modeling and computer simulations serve as a cost-effective alternative, bridging the gap between theory and practice as well as complementing the traditional theoretical and experimental approaches to problem solving. Mathematical modeling is essentially an abstract art of formulating the mathematical models from their corresponding real-world problems. The mastery of this art requires practice and experience, and it is not easy to teach such skills as the style of mathematical modeling largely depends on each person’s own insight, abstraction, type of problems, and experience of dealing with similar problems. Even for the same physical process, different models could be obtained, depending on the emphasis of some part of the process, say, based on your interest in certain quantities in a particular problem, while the same quantities could be viewed as unimportant in other processes and other problems.

28.2

MODEL FORMULATION

Mathematical modeling often starts with the analysis of the physical process and attempts to make an abstract physical model by idealization and approximations. From this idealized physical model, we can use the various first principles such as the conservation of mass, momentum, energy and Newton’s laws to translate into mathematical equations. However, such transformation from practice to theory can rarely be achieved in a single step, thus an iterative loop between theory and practice is needed, as pointed out by the famous statistician George Box. As an example, let us now look at the example of the diffusion process of sugar in a glass of water. We know that the diffusion of sugar will occur if there is any spatial difference in the sugar concentration. The physical process is complicated

Mathematical Modeling Chapter | 28

327

FIGURE 28.2 Representative element volume (REV).

and many factors could affect the distribution of sugar concentration in water, including the temperature, stirring, mass of sugar, type of sugar, how you add the sugar, even geometry of the container and others. We can idealize the process by assuming that the temperature is constant (so as to neglect the effect of heat transfer)...