Advanced Engineering Mathematics with MATLAB (Dean G. Duffy) (z-library.sk, 1lib.sk, z-lib.sk)

Advanced Engineering Mathematics with MATLAB® In the five previous editions of Advanced Engineering Mathematics with MATLAB®, the author presented a text firmly grounded in mathematics that engineers and scientists must understand and know how to use. Tapping into decades of teaching at the US Navy Academy and the US Military Academy and serving for twenty-five years at (NASA) Goddard Space Flight, he combines teaching and practical experience that is rare among au- thors of advanced engineering mathematics books. This edition continues to refine a smaller, easier to read, and useful version of this classic textbook. While competing textbooks continue to grow, the book presents a slimmer, more practical option to align with the expectations of today’s students. The new edition of the author’s classic textbook continues on a path to creating the best possible learning re- source for instructors and students alike. Through extensive class testing over five previous editions, including the author’s current course at the US Naval Academy, the book has been steadily improved. The primary mission of this edition is to dramatically increase the quality and quantity of examples and prob- lems, especially in the chapters on differential equations and Laplace transforms. The chapters on differential equations, linear algebra, Fourier series, and Laplace transforms have seen the greatest changes. Of course, this edition continues to offer a wealth of examples and applications from scientific and engineering literature, a highlight of previous editions. MATLAB® remains central to the presentation and is employed to reinforce the concepts that are taught. Worked solutions are given in the back of the book. An Instructor’s Solutions Manual is also available. Dean G. Duffy has taught at the US Naval Academy and US Military Academy. He spent 25 years working on numerical weather prediction, oceanic wave modeling, and dynamical meteorology at NASA’s Goddard Space Flight Center. Prior to this, he was a numerical weather prediction officer in the US Air Force. He earned his Ph.D. in meteorology from MIT. Dr. Duffy has written several books on transform methods, engineering math- ematics, and mixed boundary value problems including Green’s Functions with Applications, Second Edition, published by CRC Press.

Advances in Applied Mathematics Series Editor: Daniel Zwillinger Quantum Computation Helmut Bez and Tony Croft Computational Mathematics An Introduction to Numerical Analysis and Scientific Computing with Python Dimitrios Mitsotakis Delay Ordinary and Partial Differential Equations Andrei D. Polyanin Vsevolod G. Sorkin Alexi I. Zhurov Clean Numerical Simulation Shijun Liao Multiplicative Partial Differential Equations Svetlin Georgiev and Khaled Zennir Engineering Statistics A Matrix-Vector Approach with MATLAB® Lester W. Schmerr Jr. General Quantum Numerical Analysis Svetlin Georgiev and Khaled Zennir An Introduction to Partial Differential Equations with MATLAB® Matthew P. Coleman and Vladislav Bukshtynov Handbook of Exact Solutions to Mathematical Equations Andrei D. Polyanin Introducing Game Theory and its Applications, Second Edition Elliott Mendelson and Dan Zwillinger Modeling Operations Research and Business Analytics William P. Fox and Robert E. Burks Decision Analysis through Modeling and Game Theory William P. Fox Advances in High-Order Predictive Modelling Dan Gabriel Cacuci Introduction to Financial Mathematics, Second Edition Kevin J. Hastings General Quantum Variational Calculus Svetlin G. Georgiev and Khaled Zennir Advanced Engineering Mathematics with MATLAB®, Sixth Edition Dean G. Duffy https://www.routledge.com/Advances-in-Applied-Mathematics/book-series/CRCADVAPPMTH?pd=publi shed,forthcoming&pg=1&pp=12&so=pub&view=list

Advanced Engineering Mathematics with MATLAB® Sixth Edition Dean G. Duffy

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® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. Sixth edition published 2026 by CRC Press 2385 NW Executive Center Drive, Suite 320, Boca Raton FL 33431 and by CRC Press 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN CRC Press is an imprint of Taylor & Francis Group, LLC © 2026 Dean G. Duffy Fifth edition published by Chapman and Hall 2024 First edition published by CRC Press 1995 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, transmitted, 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, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact mpkbookspermissions@tandf.co.uk Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Names: Duffy, Dean G, author. Title: Advanced engineering mathematics with MATLAB / Dean G Duffy. Description: Sixth edition. | Boca Raton, FL : CRC Press, 2025. | Includes bibliographical references and index. Identifiers: LCCN 2024061338 (print) | LCCN 2024061339 (ebook) | ISBN 9781041018582 (hbk) | ISBN 9781041041986 (pbk) | ISBN 9781003627272 (ebook) Subjects: LCSH: Engineering mathematics--Data processing. | MATLAB. Classification: LCC TA345 .D84 2025 (print) | LCC TA345 (ebook) | DDC 510.285/536--dc23/eng/20250224 LC record available at https://lccn.loc.gov/2024061338 LC ebook record available at https://lccn.loc.gov/2024061339 ISBN: 978-1-041-01858-2 (hbk) ISBN: 978-1-041-04198-6 (pbk) ISBN: 978-1-003-62727-2 (ebk) DOI: 10.1201/9781003627272 Typeset in CMR10 font by KnowledgeWorks Global Ltd. Publisher’s note: This book has been prepared from camera-ready copy provided by the authors. Access the Support Material: www.routledge.com/9781041018582

Dedicated to the Brigade of Midshipmen and the Corps of Cadets v

Contents Acknowledgments xiii Author xv Introduction xvii List of Definitions xix −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 −1 −0.5 0 0.5 1 1.5 2 t x Chapter 1: First-Order Ordinary Differential Equations 1 1.1 Classification of Differential Equations 1 1.2 Separation of Variables 4 1.3 Homogeneous Equations 18 1.4 Exact Equations 19 1.5 Linear Equations 21 vii

viii Advanced Engineering Mathematics with MATLAB 1.6 Graphical Solutions 36 1.7 Numerical Methods 39 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 x v Chapter 2: Higher-Order Ordinary Differential Equations 53 2.1 Homogeneous Linear Equations with Constant Coefficients 57 2.2 Simple Harmonic Motion 67 2.3 Damped Harmonic Motion 71 2.4 Method of Undetermined Coefficients 78 2.5 Forced Harmonic Motion 87 2.6 Variation of Parameters 94 2.7 Euler-Cauchy Equation 99 2.8 Phase Diagrams 103 2.9 Numerical Methods 107   a11 a12 · · · a1n a21 a22 · · · a2n ... ... ... ... ... ... ... ... am1 am2 amn   · · · Chapter 3: Linear Algebra 117 3.1 Fundamentals 117 3.2 Determinants 125 3.3 Cramer’s Rule 129 3.4 Row Echelon Form and Gaussian Elimination 131 3.5 Eigenvalues and Eigenvectors 145 3.6 Systems of Linear Differential Equations 154 3.7 Matrix Exponential 160

Table of Contents ix x y z 3 n (1,0,0) (0,1,0) (0,0,1) C C 1 C 2 Chapter 4: Vector Calculus 167 4.1 Review 167 4.2 Divergence and Curl 174 4.3 Line Integrals 178 4.4 The Potential Function 183 4.5 Surface Integrals 184 4.6 Green’s Lemma 191 4.7 Stokes’ Theorem 195 4.8 Divergence Theorem 201 1 10 100 1000 10000 0.1 1.0 10.0 100.0 1000.0 10000.0 a m p li tu d e sp ec tr u m ( ft ) ti m es 1 0 0 0 0 Bay bridge and tunnel Chapter 5: Fourier Series 209 5.1 Fourier Series 210 5.2 Properties of Fourier Series 224 5.3 Half-Range Expansions 233 5.4 Fourier Series with Phase Angles 238 5.5 Complex Fourier Series 242 5.6 The Use of Fourier Series in the Solution of Ordinary Differential Equations 247 5.7 Finite Fourier Series 254 ω/ω 0 k 0.0 0.5 1.0 1.5 2.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 c /km = 0.012 c /km = 0.1 c /km = 1 2 2 | ) G (ω | Chapter 6: The Fourier Transform 271 6.1 Fourier Transforms 271

x Advanced Engineering Mathematics with MATLAB 6.2 Fourier Transforms Containing the Delta Function 285 6.3 Properties of Fourier Transforms 287 6.4 Inversion of Fourier Transforms 298 6.5 Convolution 302 6.6 The Solution of Ordinary Differential Equations by Fourier Transforms 306 6.7 The Solution of Laplace’s Equation on the Upper Half-Plane 308 6.8 The Solution of the Heat Equation 310 1 0.0 0.5 1.0 1.5 2.0 time (seconds) −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 ω (t ) Chapter 7: The Laplace Transform 317 7.1 Definition and Elementary Properties 317 7.2 The Heaviside Step and Dirac Delta Functions 322 7.3 Some Useful Theorems 329 7.4 The Laplace Transform of a Periodic Function 338 7.5 Inversion by Partial Fractions: Heaviside’s Expansion Theorem 340 7.6 Convolution 347 7.7 Solution of Linear Differential Equations with Constant Coefficients 352 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 4 −1.5 −1 −0.5 0 0.5 1 1.5 R TIME S O L U T IO N Chapter 8: The Wave Equation 379 8.1 The Vibrating String 380 8.2 Initial Conditions: Cauchy Problem 383 8.3 Separation of Variables 383 8.4 D’Alembert’s Formula 399 8.5 Numerical Solution of the Wave Equation 407

Table of Contents xi 0.5 1 1.5 2 S O L U T IO N 0 0.5 1 0 2 4 6 0 DIS TANCE TIME Chapter 9: The Heat Equation 421 9.1 Derivation of the Heat Equation 421 9.2 Initial and Boundary Conditions 423 9.3 Separation of Variables 424 9.4 The Superposition Integral 442 9.5 Numerical Solution of the Heat Equation 446 −1 0 1 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 1.5 ZX u (R ,θ ) Chapter 10: Laplace’s Equation 457 10.1 Derivation of Laplace’s Equation 457 10.2 Boundary Conditions 459 10.3 Separation of Variables 460 10.4 Poisson’s Equation on a Rectangle 469 10.5 Numerical Solution of Laplace’s Equation 474 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 x −1.0 0.0 1.0 2.0 3.0 4.0 y y=tan( x) y=x π Chapter 11: The Sturm-Liouville Problem 483 11.1 Eigenvalues and Eigenfunctions 484 11.2 Orthogonality of Eigenfunctions 497 11.3 Expansion in Series of Eigenfunctions 501 11.4 Finite Element Method 527

xii Advanced Engineering Mathematics with MATLAB 0.0 2.0 4.0 6.0 8.0 x -0.5 0.0 0.5 1.0 J 0 J (x) J (x) J (x) (x) 1 2 3 Chapter 12: Special Functions 535 12.1 Legendre sPolynomials 537 12.2 Bessel Functions 561 12.A Appendix A: Derivation of the Laplacian in Polar Coordinates 611 12.B Appendix B: Derivation of the Laplacian in Spherical Polar Coordinates 612 Answers to the Odd-Numbered Problems 615 Index 635 ’

Acknowledgments I would like to thank the many midshipmen and cadets who have taken engineering mathematics from me. They have been willing or unwilling guinea pigs in testing out many of the ideas and problems in this book. Special thanks go to Dr. Mike Marcozzi for his many useful suggestions for improving this book. Most of the plots and calculations were done using MATLAB R©. MATLAB is a registered trademark of The MathWorks Inc. 24 Prime Park Way Natick, MA 01760-1500 Phone: (508) 647-7000 Email: info@mathworks.com www.mathworks.com xiii

Author Dean G. Duffy received his bachelor of science in geophysics from Case Institute of Technology (Cleveland, Ohio) and his doctorate of science in meteorology from the Mas- sachusetts Institute of Technology (Cambridge, Massachusetts). He served in the United States Air Force from September 1975 to December 1979 as a numerical weather prediction officer. After his military service, he began a twenty-five-year (1980 to 2005) association with NASA at the Goddard Space Flight Center (Greenbelt, Maryland) where he focused on numerical weather prediction, oceanic wave modeling, and dynamic meteorology. He also wrote papers in the areas of Laplace transforms, antenna theory, railroad tracks, and heat conduction. In addition to his NASA duties, he taught engineering mathematics, differential equations, and calculus at the United States Naval Academy (Annapolis, Maryland) and the United States Military Academy (West Point, New York). Drawing from his teaching experience, he has written several books on transform methods, engineering mathematics, Green’s functions, and mixed-boundary-value problems. xv

Introduction In January 2023 I returned to the classroom to teach two sections of differential equa- tions to sophomores (youngsters) at the United States Naval Academy. After two decades away from teaching, I was pleased that my 5th edition contained all of the topics that we currently teach our midshipmen. However, the number of examples and homework prob- lems was inadequate and the primarily mission of this edition is to dramatically increase the quality and quantity of examples and problems, especially in the area of differential equations and Laplace transforms, and eliminate typos. The chapters on differential equa- tions, linear algebra, Fourier series, and Laplace transforms have seen the greatest changes. The chapters on wave, heat and Laplace’s equations are slightly modified. The chapters on vector calculus, the Fourier transform, the Sturm-Liouville problem and special functions are unchanged. The book begins with first- and higher-order ordinary differential equations, Chapters 1 and 2, respectively. After some introductory remarks, Chapter 1 devotes itself to present- ing general methods for solving first-order ordinary differential equations. These methods include separation of variables, employing the properties of homogeneous, linear, and exact differential equations, and finding and using integrating factors. The reason most students study ordinary differential equations is for their use in ele- mentary physics, chemistry, and engineering courses. Because these differential equations contain constant coefficients, we focus on how to solve them in Chapter 2, along with a detailed analysis of the simple, damped, and forced harmonic oscillator. Furthermore, we include the commonly employed techniques of undetermined coefficients and variation of parameters for finding particular solutions. Finally, the special equation of Euler and Cauchy is included because of its use in solving partial differential equations in spherical coordinates. The next few chapters are given in arbitrary order because they require no additional knowledge beyond calculus and ordinary differential equations. Chapter 3 presents linear algebra as a method for solving systems of linear equations and includes such topics as matrices, determinants, Cramer’s rule, and the solution of systems of ordinary differential xvii

xviii Advanced Engineering Mathematics with MATLAB equations via the classic eigenvalue problem. Vector calculus is presented in Chapter 4 and focuses on the gradient operator as it applies to line integrals, surface integrals, the divergence theorem, and Stokes’ theorem. We now turn to a trio of chapters built upon the concept of Fourier series. Fourier series allow us to reconstruct any arbitrary, well-behaved function defined over the interval (−L,L) in terms of cosines and sines. We then use this concept to solve ordinary differential equations, Section 5.6. Finally we show how these ideas may be applied when the function is given as data points, Section 5.7. Fourier series are expanded to functions that are defined over the interval (−∞,∞) in Chapter 6. We then use Fourier integrals to find the particular solution for ordinary differential equations in Section 6.6. Having presented the general aspects of differential equations, we introduce in Chapter 7 the technique of Laplace transforms. Derived from Fourier transforms when the function is defined over the interval [0,∞), this method allows us to solve initial-value problems where the forcing function turns “on” and “off.” Engineers use it extensively in their disciplines. The reason why Fourier series are so important is their application in solving partial differential equations using separation of variables. We explore this topic using the pro- totypical wave equation (Section 8.3), heat equation (Section 9.3) and Laplace’s equation (Section 10.3). We also investigate numerical solutions for each of these equations in Sec- tions 8.5, 9.5 and 10.5, respectively. Having taught the method of separation of variables using Fourier series, we expand the student’s knowledge by generalizing Fourier series to eigenfunction expansions. Sections 11.1–11.3 explain how any piece-wise continuous func- tion can be re-expressed in an eigenfunction expansion using eigenfunctions from the classic Sturm-Liouville problem. Then separation of variables is retaught from this generalized viewpoint. We conclude the book by focusing on Bessel functions (Section 12.2) and Legendre polynomials (Section 12.1). These eigenfunctions appear in the solution of the wave, heat, and Laplace’s equations in cylindrical and spherical coordinates, respectively. MATLAB is still employed to reinforce the concepts that are taught. Of course, this book still continues my principle of including a wealth of examples from the scientific and engineering literature. Worked solutions are given in the back of the book.

List of Definitions Function Definition δ(t− a) = { ∞, t = a, 0, t 6= a, ∫ ∞ −∞ δ(t− a) dt = 1 erf(x) = 2√ π ∫ x 0 e−y2 dy Γ(x) gamma function H(t− a) = { 1, t > a, 0, t < a. ℑ(z) imaginary part of the complex variable z In(x) modified Bessel function of the first kind and order n Jn(x) Bessel function of the first kind and order n Kn(x) modified Bessel function of the second kind and order n Pn(x) Legendre polynomial of order n ℜ(z) real part of the complex variable z sgn(t− a) = { −1, t < a, 1, t > a. Yn(x) Bessel function of the second kind and order n xix