Ordinary Differential Equations for Engineers: Problems with MATLAB Solutions

791 Pages·2019·49.09 MB·English

by  Ali Ümit Keskin

Ali Ümit Keskin
Ordinary
Differential
Equations for
Engineers
Problems with MATLAB Solutions
Ordinary Differential Equations for Engineers
Ü
Ali mit Keskin
Ordinary Differential
Equations for Engineers
Problems with MATLAB Solutions
123
Ali ÜmitKeskin
Department ofBiomedical Engineering
Yeditepe University
Istanbul,Turkey
ISBN978-3-319-95242-0 ISBN978-3-319-95243-7 (eBook)
https://doi.org/10.1007/978-3-319-95243-7
LibraryofCongressControlNumber:2018947489
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Preface
The theory of ODEs is a well-established field, and there are quite a number of
excellent books on ODEs. The aim of this book (which is written from the view-
pointoftheappliedmathematicsinengineering)istoteachsomeofitsfundamental
ideas,results,andapplicationsindifferentengineeringfields.However,thestyleof
this book is centered on the learning procedure that is based upon improving
problem-solving techniques using a modern software tool. In most of these
(over 600all-solvedproblemsthatareincludedinthisbook),eachrelevantconcept
is introduced within the problem statement and solutions are illustrated computa-
tionally with the help of short software scripts in MATLAB®. This kind of
approach to learn and apply ODEs in particular cases makes everything practical
andeasytograsp,demystifying existing theoretical complexitiesofthesubjectvia
the numerical experiments.
As the advent of computers has changed many things in the world, they also
influenced perceptions of many engineers. Because modern computers crunch
numbersofbigdata,carryoutsymbolicmanipulations,andputtheresultsofthese
computations into graphical form easily, many of the earlier difficult and
hard-to-solve problems of ODEs are best approached with computational
techniques.
Most of the problems on the applications of ODEs concentrate on various
engineering projects that are aimed to engage students in the understanding and
application of ODEs using MATLAB®. These may illustrate direct numerical
applications of the explicit formulas or more complicated methods using iterative
algorithms or symbolic MATLAB® solutions.
EngineersareratherinterestedinthesolutionofanODEdescribingamodeland
the physical interpretations of the solution. Therefore, many problems and their
solutions presented here are designed to provide the engineers gain best possible
understanding.
The originator of each of the main ideas was cited as a historical footnote.
Almostallofthechaptersinthebookincludesufficientamountofusefulreferences
that have been cited in the related problems of the book. The readers who wish to
delvedeeperintoaspecifictopiccanthenfollowthese(morethan200)references.
v
vi Preface
There are no problems left to students as homework assignments or “do it by
yourself” studies in the book. An outstanding feature of the book is the large
numberandvarietyoftheall-solvedproblemsthatareincludedinit.Someofthese
problems can be found relatively simple, while others are more challenging and
used for research projects. All solutions to the problems and scripts introduced in
the book have been tested using MATLAB®.
Mathematical models lead to ODEs, and they are part of research in many
different fields; once a particular solution method of a model equation is well
understood,itcanbeusedinanyotherfieldofapplicationinwhichanODEarises.
ThisbookpresentsaquantitativetreatmentofODEsthatarisefromthesemodelsin
various areas of engineering.
This book was evolved from the courses taught at Yeditepe University,
Biomedical Engineering Department. The book endeavors to prepare the reader to
solve realistic problems, answer the needs in the field, and it is expected to be
helpfulforundergraduatestudentsaswellastograduatesandexperts.Itisassumed
that the reader is comfortable with fundamental mathematical principles and basic
MATLAB® use.
Acknowledgements I would like to thank Prof. A.Okay Çelebi of Yeditepe
University,HeadofMathematicsDepartment,forhisfruitfuldiscussions,andProf.
Lawrence F. Shampine (Mathematics Department, Southern Methodist University,
Dallas, TX) for his correspondence and material supply. I also thank our research
assistants and graduate students in Biomedical Engineering Department; Sibel
Ozbal, Kubra Ozturk, Ibrahim Kapici, Ilayda Hasdemir, Hayrettin Can Sudor, and
AhmetYetkinwhoallhaveofferedvaluableopinionsandsuggestions,workedwith
the problems and proofreading.
Finally, I would like to express my special thanks to my wife Naciye, for her
endless patience, encouragement, and support.
Istanbul, Turkey Ali Ümit Keskin
Disclaimer
The software presented in this book is provided “as is” and for academic purpose.
Any express or implied warranties, including, but not limited to, the implied
warranties of merchantability and fitness for a particular purpose are disclaimed.
Innoeventshallthecopyrightownerorpublisherbeliableforanydirect,indirect,
incidental,special,exemplary,orconsequentialdamages(including,butnotlimited
to, procurement of substitute goods or services; loss of use, data, or profits; or
business interruption) however, caused and on any theory of liability, whether in
contract, strict liability, or tort (including negligence or otherwise) arising in any
way out of the use of this software, even if advised of the possibility of such
damage.
MATLAB® is registered trademark of The MathWorks, Inc. Used with
permission.
For MATLAB® product information, please contact:
The Mathworks, Inc.
3 Apple Hill Drive
Natwick, MA, 01760-2098 USA
Tel: [phone]
Fax: [phone]
E-mail: [e-mail]
Web: https://www.mathworks.com
vii
Contents
1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Definitions and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 First Order Ordinary Differential Equations . . . . . . . . . . . . . . . . . 9
2.1 Fundamentals of First Order ODEs . . . . . . . . . . . . . . . . . . . . 9
2.2 Direction Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.3 Systems of First Order ODEs . . . . . . . . . . . . . . . . . . . . . . . . 54
2.4 Applications of First Order ODEs . . . . . . . . . . . . . . . . . . . . . 61
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3 Second and Higher Order Ordinary Differential Equations . . . . . . 91
3.1 Linearity and the Wronskian . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.2 Linear Second-Order ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.3 Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.4 Cauchy–Euler ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.5 Method of Undetermined Coefficients. . . . . . . . . . . . . . . . . . . 129
3.6 Variation of Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
4 Series Solutions of Second-Order Ordinary Differential
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.1 Power Series and Convergence . . . . . . . . . . . . . . . . . . . . . . . 156
4.2 Taylor Series and Polynomials. . . . . . . . . . . . . . . . . . . . . . . . 166
4.3 Ordinary and Singular Points. . . . . . . . . . . . . . . . . . . . . . . . . 180
4.4 Series Solutions Near an Ordinary Point. . . . . . . . . . . . . . . . . 192
4.5 Series Solutions Near a Singular Point; the Method
of Frobenius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
ix
x Contents
5 Special Differential Equations, Functions, and Polynomials . . . . . . 283
5.1 Gamma Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
5.2 Bessel Equation, Bessel Functions, and Polynomials. . . . . . . . 286
5.3 Chebyshev Equation, Chebyshev Functions,
and Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
5.4 Legendre Equation, Legendre Functions and Polynomials . . . . 335
5.5 Laguerre Equation and Polynomials . . . . . . . . . . . . . . . . . . . . 370
5.6 Hermite Equation and Polynomials . . . . . . . . . . . . . . . . . . . . 378
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
6 Laplace Transform Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
6.1 Laplace Transform and Its Properties . . . . . . . . . . . . . . . . . . . 385
6.2 Inverse Laplace Transforms, Initial and Final Values. . . . . . . . 409
6.3 Solutions of Linear ODEs Using Laplace Transforms . . . . . . . 425
6.4 Applications of Laplace Transform Methods. . . . . . . . . . . . . . 451
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
7 Systems of First-Order Linear Equations . . . . . . . . . . . . . . . . . . . . 465
7.1 Review of Matrices, Linear Independence, Eigenvalues,
Eigenvectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466
7.2 Order Reduction of Second- and Higher Order ODEs
in Matrix Form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486
7.3 Homogeneous Systems with Constant Coefficients . . . . . . . . . 495
7.4 The Matrix Exponential Function. . . . . . . . . . . . . . . . . . . . . . 512
7.5 The Jordan Form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522
7.6 Matrix Methods and Solutions of Nonhomogeneous ODEs
in MATLAB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533
8 Numerical Solutions of Differential Equations. . . . . . . . . . . . . . . . . 535
8.1 Euler Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536
8.2 Second-Order Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548
8.3 Numerical Solution of Second-Order ODEs (Backward
Euler Method). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559
8.4 Fourth and Higher Order Numerical Methods. . . . . . . . . . . . . 573
8.5 Variable Step Size Methods. . . . . . . . . . . . . . . . . . . . . . . . . . 590
8.6 Multistep Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598
8.7 Runge–Kutta–Nystrom (RKN) Method. . . . . . . . . . . . . . . . . . 610
8.8 Stiff ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613
8.9 Numerical Solution of Implicit ODEs. . . . . . . . . . . . . . . . . . . 616
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629
Contents xi
9 Nonlinear Ordinary Differential Equations. . . . . . . . . . . . . . . . . . . 631
9.1 Phase Plane Analysis of Linear Systems. . . . . . . . . . . . . . . . . 632
9.2 Autonomous Equations and Stability . . . . . . . . . . . . . . . . . . . 657
9.3 Almost (Locally) Linear Systems. . . . . . . . . . . . . . . . . . . . . . 671
9.4 Limit Cycles, Competing Species, Chaos . . . . . . . . . . . . . . . . 685
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696
10 More Applications of ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697
10.1 Buoyancy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697
10.2 Mass–Spring Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 700
10.3 Numerical Solutions: Flame Propagation,
Logistic Growth, Vertical Projectile . . . . . . . . . . . . . . . . . . . . 705
10.4 Belousov–Zhabotinsky Oscillating Chemical Reactions . . . . . . 710
10.5 Electric Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713
10.6 Hodgin–Huxley and Fitzhugh–Nagumo Spiking Neuron
Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735
10.7 Mixing Tank and Chemical Reactions in a Batch
Reactor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 751
10.8 Modeling Quadrotor Dynamics . . . . . . . . . . . . . . . . . . . . . . . 756
10.9 Pendulum Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761
10.10 Satellite Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772
Appendix: Mathematical Formulas and Tables ... .... .... ..... .... 775
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 781