Nonlinear Ordinary Differential Equations: Analytical Approximation and Numerical Methods

320 Pages·2016·5.132 MB·English

by  Martin Hermann, Masoud Saravi (auth.)

Martin Hermann · Masoud Saravi
Nonlinear
Ordinary
Differential
Equations
Analytical Approximation and
Numerical Methods
Nonlinear Ordinary Differential Equations
Martin Hermann Masoud Saravi
(cid:129)
Nonlinear Ordinary
Differential Equations
Analytical Approximation and Numerical
Methods
123
Martin Hermann MasoudSaravi
Department ofNumerical Mathematics Department ofScience
Friedrich Schiller University Shomal University
Jena Amol
Germany Iran
ISBN978-81-322-2810-3 ISBN978-81-322-2812-7 (eBook)
DOI 10.1007/978-81-322-2812-7
LibraryofCongressControlNumber:2016937388
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Preface
Nonlinear ordinary differential equations (ODEs) play a fundamental role in sci-
entificmodeling,technicalandeconomicprocessesaswellasininnermathematical
questions.Examplesincludethecomplicatedcalculationofthetrajectoriesofspace
gliders and airplanes, forecast calculations, spread of AIDS, and the automation
of the parking maneuver of road vehicles on the basis of mechatronic systems.
However, the mathematical analysis of nonlinear ODEs is much more compli-
cated than the study of linear ones. The complexity of nonlinear problems can be
illustratedbythefollowingcomparison:iftheamountoflinearproblemsissetinto
relationtothesizeoftheearththentheamountofnonlinearproblemscorresponds
to the size of our universe. Therefore, the special type of the nonlinearity must be
taken into consideration if a nonlinear problem is theoretically and/or numerically
analyzed.
One of the greatest difficulties of nonlinear problems is that, in general, the
combination of known solutions is not again a solution. In linear problems, for
example, a family of linearly independent solutions can be used to construct the
general solution through the superposition principle.
Forsecond-andhigherordernonlinearODEs(moregenerallyspeaking,systems
of equations) the solutions can seldom be represented in closed form, though
implicit solutions and solutions involving non-elementary integrals are encoun-
tered.Itisthereforeimportantfortheapplicationstohaveanalyticalandnumerical
techniques at hand that can be used to compute approximate solutions.
The first part of this book presents some analytical approximation methods
which have been approved in practice. They are particularly suitable for nonlinear
lower order ODEs.
High-dimensional systems of nonlinear ODEs are very often found in the
applications. The solutions of these systems can only be approximated by appro-
priate numerical methods. Therefore, in the second part of this book, we deal with
numerical methods for the solution of systems of first-order nonlinear
(parameter-depending) ODEs, which are subjected to two-point boundary
vii
viii Preface
conditions.Itispreciselythisfocusonanalyticalaswellasnumericalmethodsthat
makes this book particularly attractive.
ThebookisintendedasaprimarytextforcoursesinthetheoryofODEsandthe
numericaltreatmentofODEsforgraduatestudents.Itisassumedthatthereaderhas
a basic knowledge of functional analysis, in particular analytical methods for
nonlinear operator equations, and numerical analysis. Moreover, the reader should
have substantial knowledge of mathematical techniques for solving initial and
boundaryvalueproblemsinlinearODEsastheyarerepresentedinourbook[63],A
First Course in Ordinary Differential Equations—Analytical and Numerical
Methods (Springer India, 2014). Physicists, chemists, biologists, computer scien-
tists, and engineers whose work involves the solution of nonlinear ODEs will also
find the book useful both as a reference and as a tool for self-studying.
InChaps.1–4,wehaveavoidedthetraditionaldefinition–theorem–proofformat.
Instead, the mathematical background is described by means of a variety of
problems, examples, and exercises ranging from the elementary to the challenging
problems. In Chap. 5, we deviate from this principle since the corresponding
mathematical proofs are constructive and can be used as the basis for the devel-
opment of appropriate numerical methods.
The book has been prepared within the scope of a German–Iranian research
project on mathematical methods for nonlinear ODEs, which was started in 2012.
We now outline the contents of the book. In Chap. 1, we present the basic
theoretical principles of nonlinear ODEs. The existence and uniqueness statements
for the solutions of the corresponding initial value problems (IVPs) are given. To
help the reader understand the structure and the methodology of nonlinear ODEs,
wepresentabriefreviewofelementarymethodsforfindinganalytical solutionsof
selected problems. Some worked examples focusing on the analytical solution of
ODEs from the applications are used to demonstrate mathematical techniques,
which may be familiar to the reader. In Chap. 2, the analytical approximation
methodsareintroduced.Specialattentionispaidtothevariationaliterationmethod
and the Adomian decomposition method. For smooth nonlinear problems both
methods generate series, which converge rapidly to an exact solution. These
approximation methods are applied to some nonlinear ODEs, which play an
important role in the pure and applied mathematics. In Chap. 3, another four
analyticalapproximationsmethodsareconsidered,namelytheperturbationmethod,
the energy balance method, the Hamiltonian approach, and the homotopy pertur-
bationmethod.Thesemethodshaveprovedtobepowerfultoolsforapproximating
solutions of strongly nonlinear ODEs. They are applied in many areas of applied
physics and engineering. The accuracy of the approximated solutions is demon-
strated by a series of examples which are based on strongly nonlinear ODEs.
The numerical part of the book begins with Chap. 4. Here, numerical methods
for nonlinear boundary value problems (BVPs) of systems offirst-order ODEs are
studied. Starting from the shooting methods for linear BVPs [63], we present the
simple shooting method, the method of complementary functions, the multiple
shootingmethod,andaversionofthestabilizedmarchmethodfornonlinearBVPs.
AMATLABcodeforthemultipleshootingmethodisprovidedsuchthatthereader
Preface ix
can perform his own numerical experiments. Chapter 5 is devoted to nonlinear
parametrized BVPs since in the applications the majority of nonlinear problems
dependonexternalparameters.ThefirststepinthetreatmentofsuchBVPsistheir
formulation as a nonlinear operator equation Tðy;λÞ¼0 in Banach spaces, and to
apply the Riesz–Schauder theory. Under the assumption that Tyðz0Þ is a Fredholm
operatorwithindexzero,wherez (cid:2)ðy ;λ Þisasolutionoftheoperatorequation,
0 0 0
the concept of isolated solutions and singular solutions (turning points and bifur-
cationpoints)isdiscussed.Appropriateextendedsystemsarepresentedtocompute
the singular points with standard techniques. In each case, the extended operator
equation is transformed back into an extended nonlinear BVP. Since the operator
equation is bad conditioned in the neighborhood of singular points, we use the
technique of transformed systems to determine solutions near to singularities. In
addition to the one-parameter problem, we study the operator equation under the
influenceofinitialimperfections,whichcanbedescribedbyasecond(perturbation)
parameter. Feasible path-following methods are required for the generation of
bifurcationdiagrams ofparametrized BVPs. We present some methods that follow
simple solution curves. The chapter is concluded with the description of two
real-life problems.
Atthispointitisourpleasuretothankallthosecolleagueswhohaveparticularly
helpeduswiththepreparationofthisbook.OurfirstthanksgotoDr.DieterKaiser,
without whose high motivation, deep MATLAB knowledge, and intensive com-
putational assistance this book could never have appeared. Moreover, our thanks
also go to Dr. Andreas Merker who read through earlier drafts or sections of the
manuscript and made important suggestions or improvements. We gratefully
acknowledgethefinancialsupportprovidedbytheErnstAbbeFoundationJenafor
our German–Iranian joint work.
It has been a pleasure working with the Springer India publication staff, in
particular with Shamim Ahmad and Ms. Shruti Raj.
Finally, the authors wish to emphasize that any helpful suggestion or comment
for improving this book will be warmly appreciated and can be sent by e-mail to:
[email protected] and [email protected].
Martin Hermann
Masoud Saravi
Contents
1 A Brief Review of Elementary Analytical Methods for Solving
Nonlinear ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Analytical Solution of First-Order Nonlinear ODEs . . . . . . . . . . . 1
1.3 High-Degree First-Order ODEs. . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Analytical Solution of Nonlinear ODEs by Reducing
the Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5 Transformations of Nonlinear ODEs . . . . . . . . . . . . . . . . . . . . . 21
1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2 Analytical Approximation Methods . . . . . . . . . . . . . . . . . . . . . . . . 33
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 The Variational Iteration Method. . . . . . . . . . . . . . . . . . . . . . . . 33
2.3 Application of the Variational Iteration Method. . . . . . . . . . . . . . 38
2.4 The Adomian Decomposition Method . . . . . . . . . . . . . . . . . . . . 44
2.5 Application of the Adomian Decomposition Method . . . . . . . . . . 48
2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3 Further Analytical Approximation Methods and Some
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.1 Perturbation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.1.1 Theoretical Background. . . . . . . . . . . . . . . . . . . . . . . . . 61
3.1.2 Application of the Perturbation Method . . . . . . . . . . . . . . 63
3.2 Energy Balance Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2.1 Theoretical Background. . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2.2 Application of the Energy Balance Method . . . . . . . . . . . 74
3.3 Hamiltonian Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.3.1 Theoretical Background. . . . . . . . . . . . . . . . . . . . . . . . . 87
3.3.2 Application of the Hamiltonian Approach. . . . . . . . . . . . . 88
xi
xii Contents
3.4 Homotopy Analysis Method. . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.4.1 Theoretical Background. . . . . . . . . . . . . . . . . . . . . . . . . 92
3.4.2 Application of the Homotopy Analysis Method. . . . . . . . . 107
3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4 Nonlinear Two-Point Boundary Value Problems. . . . . . . . . . . . . . . 121
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.2 Simple Shooting Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.3 Method of Complementary Functions. . . . . . . . . . . . . . . . . . . . . 131
4.4 Multiple Shooting Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.5 Nonlinear Stabilized March Method. . . . . . . . . . . . . . . . . . . . . . 146
4.6 MATLAB PROGRAMS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5 Numerical Treatment of Parametrized Two-Point Boundary
Value Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.2 Two-Point BVPs and Operator Equations. . . . . . . . . . . . . . . . . . 169
5.3 Analytical and Numerical Treatment of Limit Points . . . . . . . . . . 172
5.3.1 Simple Solution Curves. . . . . . . . . . . . . . . . . . . . . . . . . 172
5.3.2 Extension Techniques for Simple Turning Points . . . . . . . 179
5.3.3 An Extension Technique for Double Turning Points . . . . . 186
5.3.4 Determination of Solutions in the Neighborhood
of a Simple Turning Point . . . . . . . . . . . . . . . . . . . . . . . 190
5.4 Analytical and Numerical Treatment of Primary Simple
Bifurcation Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
5.4.1 Bifurcation Points, Primary and Secondary Bifurcation
Phenomena. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
5.4.2 Analysis of Primary Simple Bifurcation Points . . . . . . . . . 195
5.4.3 An Extension Technique for Primary Simple Bifurcation
Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
5.4.4 Determination of Solutions in the Neighborhood
of a Primary Simple Bifurcation Point. . . . . . . . . . . . . . . 204
5.5 Analytical and Numerical Treatment of Secondary Simple
Bifurcation Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
5.5.1 Analysis of Secondary Simple Bifurcation Points . . . . . . . 232
5.5.2 Extension Techniques for Secondary Simple
Bifurcation Points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
5.5.3 Determination of Solutions in the Neighborhood
of a Secondary Simple Bifurcation Point . . . . . . . . . . . . . 244
5.6 Perturbed Bifurcation Problems. . . . . . . . . . . . . . . . . . . . . . . . . 247
5.6.1 Nondegenerate Initial Imperfections. . . . . . . . . . . . . . . . . 247
5.6.2 Nonisolated Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 252
5.6.3 Solution Curves Through Nonisolated Solutions . . . . . . . . 267