Table Of ContentHIGH-PRECISION METHODS
IN EIGENVALUE PROBLEMS
AND THEIR APPLICATIONS
Differential and Integral Equations and Their Applications
A series edited by:
A.D. Polyanin
Institute for Problems in Mechanics, Moscow, Russia
Volume 1
Handbook of First Order Partial Differential Equations
A.D. Polyanin, V.F. Zaitsev and A. Moussiaux
Volume 2
Group-Theoretic Methods in Mechanics and Applied Mathematics
D.M. Klimov and V. Ph. Zhuravlev
Volume 3
Quantization Methods in the Theory of Differential Equations
V.E. Nazaikinskii, B.-W. Schulze and B. Yu. Sternin
Volume 4
Hypersingular Integral Equations and Their Applications
I.K. Lifanov, L.N. Poltavskii and G.M. Vainikko
Volume 5
Equations of Mathematical Diffraction Theory
Mezhlum A. Sumbatyan and Antonio Scalia
Volume 6
High-Precision Methods in Eigenvalue Problems and Their Applications
Leonid D. Akulenko and Sergei V. Nesterov
HIGH-PRECISION METHODS
IN EIGENVALUE PROBLEMS
AND THEIR APPLICATIONS
Leonid D. Akulenko
Sergei V. Nesterov
CHAPMAN & HALL/CRC
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Contents
Preface ............................................................... xiii
Authors ............................................................... xvii
BasicNotation.s.............................................................. xix
Chapter1. StatementofEigenvalueProblems.Basic MethodsofTheir Solution .... 1
1.1. StatementoftheSturm–LiouvilleProblem .................................. 1
1.1.1. Boundaryvalueproblemforeigenvaluesandeigenfunctions .............. 1
1.1.2. Variationalstatementoftheeigenvalueproblem ........................ 3
1.2. AnalyticalMethodsofSolvingthe Sturm–LiouvilleProblem .................... 5
1.2.1. Generalschemeofanalyticalsolution ................................ 5
1.2.2. Reductiontoa Fredholmintegralequationofthe secondkind ............. 9
1.2.3. Reductiontoa Volterraintegralequationofthe secondkind .............. 10
1.3. SolvingtheSturm–LiouvilleProblembytheMethodofRegularPerturbations ...... 12
1.3.1. Statementoftheperturbedproblem .................................. 12
1.3.2. Standardprocedureofasymptoticexpansions .......................... 12
1.3.3. Findingtheexpansioncoefficients ................................... 13
1.3.4. Justificationquestions ............................................. 14
1.4. NumericalMethodsforSolvingthe Sturm–LiouvilleProblem ................... 14
1.4.1. TheRayleigh–Ritzmethod ......................................... 15
1.4.2. Some generalfacts and remarkspertainingto other numericalmethodsin the
Sturm–Liouvilleproblem .......................................... 18
Chapter2. TheMethodofAcceleratedConvergencefortheSturm–LiouvilleProblem 21
2.1. Numerical-AnalyticalUpperandLowerBoundsforEigenvalues ................. 21
2.1.1. Theproblemofconstructingtwo-sidedestimates ....................... 21
2.1.2. Constructionandanalysisofcomparisonsystems ....................... 22
2.2. Criterion of Closeness between the First Eigenvalue and its Upper (Lower) Bound.
Introductionofa SmallParameter ......................................... 23
2.3. TheoryofPerturbations .................................................. 23
2.3.1. Constructionofanequivalentperturbedproblem ....................... 23
2.3.2. Approximatesolutionofthe perturbedproblem ........................ 24
2.3.3. Reductionofthecorrectiontermtodifferentialform .................... 25
2.4. DescriptionoftheMethodof AcceleratedConvergence ........................ 26
2.5. SomeApplicationsoftheAcceleratedConvergenceMethod .................... 27
2.5.1. Testmodelproblems .............................................. 27
2.5.2. A methodforthe calculationofweightednorms ........................ 28
vi CONTENTS
2.6. TheMethodofAcceleratedConvergenceforHigherEigenvalues ................ 29
2.6.1. Anexamplewith thecalculationoftwo eigenvalues ..................... 29
2.6.2. Somepropertiesoftheprocedureoffindingsubsequenteigenvalues ........ 30
2.7. Problemswith BoundaryConditionsofthe SecondKind ....................... 31
2.7.1. Constructionofacomparisonproblem ................................ 31
2.7.2. Approximatesolutionofthe problem ................................. 31
2.7.3. Testproblem .................................................... 31
2.8. Problemswith BoundaryConditionsofthe ThirdKind ........................ 32
2.8.1. Statementofthethirdboundaryvalueproblem ......................... 32
2.8.2. Constructionofacomparisonsystem ................................. 33
2.8.3. Solutionoftheperturbedproblem ................................... 34
2.8.4. Differentialrelationbetweeneigenvaluesandtheintervallength ........... 35
2.8.5. Themethodofacceleratedconvergence ............................... 35
2.8.6. Example ........................................................ 36
2.9. Problemswith PeriodicBoundaryConditions ................................ 37
2.9.1. Statementoftheperiodicboundaryvalueproblem ...................... 37
2.9.2. Mainpropertiesoftheperiodicproblem .............................. 37
2.9.3. Constructionofupperbounds ....................................... 38
2.9.4. Constructionofthecomparisonsystem ............................... 38
2.9.5. Introductionofa smallparameter .................................... 39
2.9.6. Approximatesolutionofthe perturbedproblem ........................ 40
2.9.7. Themethodofacceleratedconvergence ............................... 41
2.9.8. Examples ....................................................... 43
2.10. ProofofConvergenceof SuccessiveApproximations.ExistenceTheorem ......... 47
2.10.1.Transformationofthe perturbedboundaryvalueproblem ................. 47
2.10.2.Proofofconvergenceofsuccessiveapproximations ..................... 48
2.11. ProofofQuadraticConvergence ........................................... 50
2.12. TheMethodofHyperacceleratedConvergence ............................... 51
2.12.1.Third-orderrefinementprocedure .................................... 51
2.12.2.Anapplicationofthemethodofhyperacceleratedconvergence ............ 52
2.13. TakingintoAccountExplicitDependenceofBoundaryConditionsonEigenvalues .. 52
2.14. Exercises ............................................................. 53
Chapter3. ApproximateAnalyticalSolutionofPerturbedEigenvalueProblems .... 55
3.1. StatementandAnalysisofthePerturbedSturm–LiouvilleProblem ............... 55
3.1.1. Propertiesofthe perturbedspectrum ................................. 55
3.1.2. Theproblemofseculartermsandregularizationoftheproblem ........... 56
3.1.3. Separationofvariables ............................................ 57
3.2. ApproximateSolutionoftheBoundaryValueProblem ......................... 58
3.2.1. Constructionofeigenfrequenciesandphasesofpartialvibrations .......... 58
3.2.2. Findingeigenfunctionsandthe constructionofan orthonormalbasis ....... 60
3.2.3. Remarks ........................................................ 61
3.3. ApproximationofFunctionsin TermsoftheApproximateBasis ................. 61
3.3.1. Theproblemofexpansionintermsofanapproximatebasis .............. 61
3.3.2. Uniformestimates ................................................ 63
CONTENTS vii
3.4. ApplicationstoInitialBoundaryValueProblems ............................. 64
3.4.1. Approximatesolution ............................................. 64
3.4.2. Errorestimates ................................................... 64
3.5. Exercises ............................................................. 65
Chapter4. GeneralizedSturm–Liouville Problem .............................. 67
4.1. StatementoftheGeneralizedSturm–Liouvilleproblem ........................ 67
4.1.1. Statementoftheboundaryvalueproblemindifferentialform ............. 67
4.1.2. Basic definitions ................................................. 67
4.2. SomeSturm–LiouvilleProblemswithExactSolutions ......................... 68
4.2.1. Examples ....................................................... 68
4.2.2. Somebasicgeneralpropertiesofsolutions ............................ 68
4.3. StatementofanAuxiliaryVariationalProblem ............................... 69
4.3.1. Variationalstatementoftheproblemanditsgeneralization ............... 69
4.3.2. Derivationandanalysisofthedeterminingrelation ...................... 69
4.4. ClosenessCriterionandtheTheoryofPerturbations ........................... 70
4.4.1. Somepropertiesofthesolutionofthecomparisonproblem ............... 70
4.4.2. Approximatesolutionofthe perturbedproblem ........................ 71
4.5. TheMethodofAcceleratedConvergenceforGeneralizedSturm–LiouvilleProblems 71
4.6. ModelProblems ........................................................ 72
4.6.1. Testexampleforanintegrableequation ............................... 72
4.6.2. Numericalexample;two-sidedestimates .............................. 73
4.7. GeneralizedParametricVibrations ......................................... 73
4.7.1. Statementofthegeneralizedperiodicproblem ......................... 73
4.7.2. Anexampleillustratingspectralproperties ............................ 74
4.7.3. Generalpropertiesofsolutionsofgeneralizedperiodicproblems ........... 75
4.7.4. An extended setting of the problem and the procedure of its approximate
solution ........................................................ 75
4.8. Generalized Boundary Value Problems with Spectral Parameter in Boundary
Conditions ............................................................ 76
4.9. Exercises ............................................................. 77
Chapter5. Asymptotics of Eigenvalues and Eigenfunctions of the Generalized
Sturm–LiouvilleProblemforHigherVibrationModes ................ 79
5.1. GeneralNotionsRegardingtheAsymptoticBehaviorofEigenvaluesCorrespondingto
HigherVibrationModes ................................................. 79
5.1.1. Statementofthegeneralizedproblem ................................. 79
5.1.2. Classicalresults .................................................. 80
5.2. ApplicationofAsymptoticMethodsofthe TheoryofNonlinearVibrations ........ 80
5.2.1. “Amplitude–phase”variables ....................................... 80
5.2.2. Approximationofthephase ........................................ 81
5.3. DeterminationofEigenfrequenciesandVibrationPhases ....................... 82
5.3.1. Introductionofintermediateparameters ............................... 82
5.3.2. Findingtheoriginalquantities ...................................... 83
5.3.3. Procedureofsuccessive approximations .............................. 84
viii CONTENTS
5.4. FindingAmplitudesandShapesofFree Vibrations ............................ 85
5.4.1. Approximatecalculationofhighermodeamplitudes ..................... 85
5.4.2. Findingeigenfunctionscorrespondingtohighermodes ................... 86
5.5. OtherTypesofBoundaryValueProblems ................................... 87
5.5.1. Boundaryconditionsofthesecondkind .............................. 87
5.5.2. Generalboundaryconditionsofthethirdkind .......................... 87
5.5.3. Remarksaboutgeneralizations ...................................... 88
5.6. CalculationsforSomeSpecific MechanicalSystems ........................... 88
5.6.1. Longitudinalvibrationsofaninhomogeneousrectilinearbeam ............ 88
5.6.2. Vibrationsofaninhomogeneousstring ............................... 89
5.6.3. Asymptoticsofeigenvaluesofthe Hillproblem ........................ 90
5.6.4. Spatialvibrationsofa satellite ...................................... 91
5.7. Exercises ............................................................. 93
Chapter6. Solutions of Fourth-Order Self-Conjugate Problems. Oscillation
Properties ...................................................... 95
6.1. StatementofaSelf-ConjugateFourth-OrderBoundaryValueProblem ............ 95
6.1.1. Statementoftheproblemindifferentialform.Someremarks .............. 95
6.1.2. Statementoftheprobleminvariationalform ........................... 96
6.1.3. Introductionofnaturalphysicalvariables .............................. 97
6.1.4. Schemeofsolution ............................................... 97
6.2. TheMethodofSagittaryFunction.Sturm’sTheorems ......................... 99
6.2.1. Constructionofthecharacteristicequationandthesagittaryfunction ....... 99
6.2.2. Oscillationpropertiesofthesagittaryfunction ......................... 100
6.3. ComputationAlgorithmsofthe ShootingMethodBasedonthe SagittaryFunction .. 102
6.3.1. Algorithmofshootingwith respectto theordinate ...................... 102
6.3.2. Algorithmofshootingwith respectto theabscissa ...................... 103
6.4. Examples ............................................................. 104
6.4.1. A modeltestexample ............................................. 105
6.4.2. Comparisonwith theresultsofS. Gould .............................. 106
6.4.3. Parametricsynthesisforconicalbeams ............................... 106
Chapter7. The Method of Accelerated Convergence for Eigenvalue Problems for
Fourth-OrderEquations .......................................... 109
7.1. Two-SidedEstimatesforLowerModeEigenvalues ........................... 109
7.1.1. Differentialandvariationalstatementsoftheproblem ................... 109
7.1.2. Constructionofupperbounds ....................................... 110
7.1.3. Relationbetweenthe upperboundandthelengthoftheinterval ........... 111
7.1.4. Constructionoflowerboundsandtwo-sidedestimates ................... 113
7.2. ClosenessCriterionandPerturbationTheory ................................. 113
7.2.1. Introductionofa smallparameter .................................... 113
7.2.2. Anapproximatesolutionofthe perturbedproblem ...................... 114
7.3. TheMethodofAcceleratedConvergenceforFourth-OrderBoundaryValueProblems 115
7.3.1. A differentialrelationbetweentheeigenvalueandthelengthofthe interval .. 115
7.3.2. Algorithmoftheacceleratedconvergencemethod ...................... 115
7.4. OtherTypesofBoundaryConditions ....................................... 116
CONTENTS ix
7.5. ProcedureofContinuationina Parameter ................................... 117
7.6. ModelProblems ........................................................ 117
7.6.1. Generalremarksaboutcalculations .................................. 117
7.6.2. Testexampleswith analyticallyintegrableequations .................... 118
7.6.3. Problem of transverse vibrations of an inhomogeneous beam occurring in
applications ..................................................... 119
Chapter8. PerturbationMethodinEigenvalueProblemsforFourth-OrderEquations 121
8.1. ReductionoftheOriginalProblemtotheStandardPerturbedBoundaryValueProblem 121
8.1.1. Statementoftheinitialboundaryvalueproblem;preliminaryremarks ...... 121
8.1.2. Reductiontoperturbedboundaryvalueproblems ....................... 123
8.1.3. Somefeaturesofthe standardprocedureoftheperturbationmethod ........ 123
8.2. Regularizationofthe PerturbationMethod ................................... 124
8.2.1. Transformationofthe independentvariable ............................ 124
8.2.2. Regularprocedureofthe perturbationmethod .......................... 125
8.2.3. Justificationoftheperturbationmethod ............................... 126
8.3. MotionControlProblem ................................................. 128
8.4. FindingtheEigenvaluesandtheEigenfunctionsinthe FirstApproximation ........ 128
8.5. Exercises ............................................................. 131
Chapter9. Sturm–LiouvilleProblemsforVector-ValuedFunctions ............... 133
9.1. SettingoftheProblem.PreliminaryRemarks ................................ 133
9.1.1. Statementoftheproblemindifferentialform .......................... 133
9.1.2. Variationalstatementoftheproblem ................................. 133
9.2. ClosenessCriterionandPerturbationTheory ................................. 134
9.2.1. Constructionofthecomparisonproblem;analysisofits properties ......... 134
9.2.2. Introductionofa smallparameter .................................... 135
9.2.3. Approximatesolutionofthe problem ................................. 135
9.3. The Method of Accelerated Convergence for the Sturm–Liouville Problem for
Vector-ValuedFunctions ................................................. 136
9.3.1. Propertiesofthe firstapproximationofthesolution ..................... 136
9.3.2. Algorithmofacceleratedconvergenceforvectorproblems ............... 137
9.4. ModelProblems ........................................................ 138
9.4.1. A systemofEulertype ............................................ 138
9.4.2. A systemwith periodiccoefficients .................................. 139
9.5. Exercises ............................................................. 139
Chapter10. VibrationsandStabilityofElastic Systems .......................... 141
10.1. PlaneVibrationsofa RotatingHeavyThreadandTheirStability ................ 141
10.1.1.Statement of the initial boundary value problem. Its solution by the Fourier
method ......................................................... 141
10.1.2.Freevibrationsofa rotatingheavyhomogeneousstringsubjectedtotension . 145
10.1.3.Vibrationsofaninhomogeneousthread ............................... 149
10.2. ParametricSynthesisin theProblemof Instabilityofan InhomogeneousBeam ..... 152
10.2.1.Settingoftheproblemoflongitudinalbendingofanelastic beam .......... 152
10.2.2.Calculationofthe criticalforceforsome rigiditydistributions ............. 154
