High-Precision Methods In Eigenvalue Problems And Their Applications

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ISBN: 9780415309936 | 041530993X
Cover: Hardcover
Copyright: 10/15/2004

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This book presents a survey of analytical, asymptotic, numerical, and combined methods of solving eigenvalue problems. It considers the new method of accelerated convergence for solving problems of the Sturm-Liouville type as well as boundary-value problems with boundary conditions of the first, second, and third kind. The authors also present high-precision asymptotic methods for determining eigenvalues and eigenfunctions of higher oscillation modes and consider numerous eigenvalue problems that appear in oscillation theory, acoustics, elasticity, hydrodynamics, geophysics, quantum mechanics, structural mechanics, electrodynamics, and microelectronics.

Preface

xiii

Authors

xvii

Basic Notations

xix

Statement of Eigenvalue Problems. Basic Methods of Their Solution

(20)

Statement of the Sturm--Liouville Problem

(4)

Boundary value problem for eigenvalues and eigenfunctions

(2)

Variational statement of the eigenvalue problem

(2)

Analytical Methods of Solving the Sturm--Liouville Problem

(7)

General scheme of analytical solution

(4)

Reduction to a Fredholm integral equation of the second kind

(1)

Reduction to a Volterra integral equation of the second kind

(2)

Solving the Sturm--Liouville Problem by the Method of Regular Perturbations

(2)

Statement of the perturbed problem

(1)

Standard procedure of asymptotic expansions

(1)

Finding the expansion coefficients

(1)

Justification questions

(1)

Numerical Methods for Solving the Sturm--Liouville Problem

(7)

The Rayleigh--Ritz method

(3)

Some general facts and remarks pertaining to other numerical methods in the Sturm--Liouville problem

(3)

The Method of Accelerated Convergence for the Sturm--Liouville Problem

(34)

Numerical-Analytical Upper and Lower Bounds for Eigenvalues

(2)

The problem of constructing two-sided estimates

(1)

Construction and analysis of comparison systems

(1)

Criterion of Closeness between the First Eigenvalue and its Upper (Lower) Bound. Introduction of a Small Parameter

(1)

Theory of Perturbations

(3)

Construction of an equivalent perturbed problem

(1)

Approximate solution of the perturbed problem

(1)

Reduction of the correction term to differential form

(1)

Description of the Method of Accelerated Convergence

(1)

Some Applications of the Accelerated Convergence Method

(2)

Test model problems

(1)

A method for the calculation of weighted norms

(1)

The Method of Accelerated Convergence for Higher Eigenvalues

(2)

An example with the calculation of two eigenvalues

(1)

Some properties of the procedure of finding subsequent eigenvalues

(1)

Problems with Boundary Conditions of the Second Kind

(1)

Construction of a comparison problem

(1)

Approximate solution of the problem

(1)

Test problem

(1)

Problems with Boundary Conditions of the Third Kind

(5)

Statement of the third boundary value problem

(1)

Construction of a comparison system

(1)

Solution of the perturbed problem

(1)

Differential relation between eigenvalues and the interval length

(1)

The method of accelerated convergence

(1)

Example

(1)

Problems with Periodic Boundary Conditions

(10)

Statement of the periodic boundary value problem

(1)

Main properties of the periodic problem

(1)

Construction of upper bounds

(1)

Construction of the comparison system

(1)

Introduction of a small parameter

(1)

Approximate solution of the perturbed problem

(1)

The method of accelerated convergence

(2)

Examples

(4)

Proof of Convergence of Successive Approximations. Existence Theorem

(3)

Transformation of the perturbed boundary value problem

(1)

Proof of convergence of successive approximations

(2)

Proof of Quadratic Convergence

(1)

The Method of Hyperaccelerated Convergence

(1)

Third-order refinement procedure

(1)

An application of the method of hyperaccelerated convergence

(1)

Taking into Account Explicit Dependence of Boundary Conditions on Eigenvalues

(1)

Exercises

(2)

Approximate Analytical Solution of Perturbed Eigenvalue Problems

(12)

Statement and Analysis of the Perturbed Sturm--Liouville Problem

(3)

Properties of the perturbed spectrum

(1)

The problem of secular terms and regularization of the problem

(1)

Separation of variables

(1)

Approximate Solution of the Boundary Value Problem

(3)

Construction of eigenfrequencies and phases of partial vibrations

(2)

Finding eigenfunctions and the construction of an orthonormal basis

(1)

Remarks

(1)

Approximation of Functions in Terms of the Approximate Basis

(3)

The problem of expansion in terms of an approximate basis

(2)

Uniform estimates

(1)

Applications to Initial Boundary Value Problems

(1)

Approximate solution

(1)

Error estimates

(1)

Exercises

(2)

Generalized Sturm--Liouville Problem

(12)

Statement of the Generalized Sturm--Liouville problem

(1)

Statement of the boundary value problem in differential form

(1)

Basic definitions

(1)

Some Sturm--Liouville Problems with Exact Solutions

(1)

Examples

(1)

Some basic general properties of solutions

(1)

Statement of an Auxiliary Variational Problem

(1)

Variational statement of the problem and its generalization

(1)

Derivation and analysis of the determining relation

(1)

Closeness Criterion and the Theory of Perturbations

(1)

Some properties of the solution of the comparison problem

(1)

Approximate solution of the perturbed problem

(1)

The Method of Accelerated Convergence for Generalized Sturm--Liouville Problems

(1)

Model Problems

(1)

Test example for an integrable equation

(1)

Numerical example; two-sided estimates

(1)

Generalized Parametric Vibrations

(3)

Statement of the generalized periodic problem

(1)

An example illustrating spectral properties

(1)

General properties of solutions of generalized periodic problems

(1)

An extended setting of the problem and the procedure of its approximate solution

(1)

Generalized Boundary Value Problems with Spectral Parameter in Boundary Conditions

(1)

Exercises

(2)

Asymptotics of Eigenvalues and Eigenfunctions of the Generalized Sturm--Liouville Problem for Higher Vibration Modes

(16)

General Notions Regarding the Asymptotic Behavior of Eigenvalues Corresponding to Higher Vibration Modes

(1)

Statement of the generalized problem

(1)

Classical results

(1)

Application of Asymptotic Methods of the Theory of Nonlinear Vibrations

(2)

``Amplitude--phase'' variables

(1)

Approximation of the phase

(1)

Determination of Eigenfrequencies and Vibration Phases

(3)

Introduction of intermediate parameters

(1)

Finding the original quantities

(1)

Procedure of successive approximations

(1)

Finding Amplitudes and Shapes of Free Vibrations

(2)

Approximate calculation of higher mode amplitudes

(1)

Finding eigenfunctions corresponding to higher modes

(1)

Other Types of Boundary Value Problems

(1)

Boundary conditions of the second kind

(1)

General boundary conditions of the third kind

(1)

Remarks about generalizations

(1)

Calculations for Some Specific Mechanical Systems

(5)

Longitudinal vibrations of an inhomogeneous rectilinear beam

(1)

Vibrations of an inhomogeneous string

(1)

Asymptotics of eigenvalues of the Hill problem

(1)

Spatial vibrations of a satellite

(2)

Exercises

(2)

Solutions of Fourth-Order Self-Conjugate Problems. Oscillation Properties

(14)

Statement of a Self-Conjugate Fourth-Order Boundary Value Problem

(4)

Statement of the problem in differential form. Some remarks

(1)

Statement of the problem in variational form

(1)

Introduction of natural physical variables

(1)

Scheme of solution

(2)

The Method of Sagittary Function. Sturm's Theorems

(3)

Construction of the characteristic equation and the sagittary function

(1)

Oscillation properties of the sagittary function

100

(2)

Computation Algorithms of the Shooting Method Based on the Sagittary Function

102

(2)

Algorithm of shooting with respect to the ordinate

102

(1)

Algorithm of shooting with respect to the abscissa

103

(1)

Examples

104

(5)

A model test example

105

(1)

Comparison with the results of S. Gould

106

(1)

Parametric synthesis for conical beams

106

(3)

The Method of Accelerated Convergence for Eigenvalue Problems for Fourth-Order Equations

109

(12)

Two-Sided Estimates for Lower Mode Eigenvalues

109

(4)

Differential and variational statements of the problem

109

(1)

Construction of upper bounds

110

(1)

Relation between the upper bound and the length of the interval

111

(2)

Construction of lower bounds and two-sided estimates

113

(1)

Closeness Criterion and Perturbation Theory

113

(2)

Introduction of a small parameter

113

(1)

An approximate solution of the perturbed problem

114

(1)

The Method of Accelerated Convergence for Fourth-Order Boundary Value Problems

115

(1)

A differential relation between the eigenvalue and the length of the interval

115

(1)

Algorithm of the accelerated convergence method

115

(1)

Other Types of Boundary Conditions

116

(1)

Procedure of Continuation in a Parameter

117

(1)

Model Problems

117

(4)

General remarks about calculations

117

(1)

Test examples with analytically integrable equations

118

(1)

Problem of transverse vibrations of an inhomogeneous beam occurring in applications

119

(2)

Perturbation Method in Eigenvalue Problems for Fourth-Order Equations

121

(12)

Reduction of the Original Problem to the Standard Perturbed Boundary Value Problem

121

(3)

Statement of the initial boundary value problem; preliminary remarks

121

(2)

Reduction to perturbed boundary value problems

123

(1)

Some features of the standard procedure of the perturbation method

123

(1)

Regularization of the Perturbation Method

124

(4)

Transformation of the independent variable

124

(1)

Regular procedure of the perturbation method

125

(1)

Justification of the perturbation method

126

(2)

Motion Control Problem

128

(1)

Finding the Eigenvalues and the Eigenfunctions in the First Approximation

128

(3)

Exercises

131

(2)

Sturm--Liouville Problems for Vector-Valued Functions

133

(8)

Setting of the Problem. Preliminary Remarks

133

(1)

Statement of the problem in differential form

133

(1)

Variational statement of the problem

133

(1)

Closeness Criterion and Perturbation Theory

134

(2)

Construction of the comparison problem; analysis of its properties

134

(1)

Introduction of a small parameter

135

(1)

Approximate solution of the problem

135

(1)

The Method of Accelerated Convergence for the Sturm--Liouville Problem for Vector-Valued Functions

136

(2)

Properties of the first approximation of the solution

136

(1)

Algorithm of accelerated convergence for vector problems

137

(1)

Model Problems

138

(1)

A system of Euler type

138

(1)

A system with periodic coefficients

139

(1)

Exercises

139

(2)

Vibrations and Stability of Elastic Systems

141

(20)

Plane Vibrations of a Rotating Heavy Thread and Their Stability

141

(11)

Statement of the initial boundary value problem. Its solution by the Fourier method

141

(4)

Free vibrations of a rotating heavy homogeneous string subjected to tension

145

(4)

Vibrations of an inhomogeneous thread

149

(3)

Parametric Synthesis in the Problem of Instability of an Inhomogeneous Beam

152

(5)

Setting of the problem of longitudinal bending of an elastic beam

152

(2)

Calculation of the critical force for some rigidity distributions

154

(3)

The Problem of Lateral Buckling of a Long Beam with Narrow Cross-Section

157

(1)

Statement of the Prandtl problem

157

(1)

A numerical-analytical solution

157

(1)

Longitudinal Vibrations of an Inhomogeneous Beam with Transverse Inertia

158

(2)

Approaches of Rayleigh and Love

158

(2)

Experimental determination of Poisson's ratio on the basis of measurements of longitudinal frequencies by the resonance method

160

(1)

Exercises

160

(1)

Surface and Internal Waves in Heavy Ideal Fluid

161

(18)

Free Vibrations of the Surface of a Rotating Spherical Layer of Heavy Fluid

161

(9)

Preliminary remarks and statement of the problem

161

(2)

Solving the eigenvalue problem

163

(3)

Calculation results and their analysis

166

(4)

Internal Waves in Essentially Inhomogeneous Fluids

170

(7)

Statement of the problem and some mathematical aspects of its solution

170

(2)

A version of the perturbation method for approximate solution of the Sturm--Liouville Problem

172

(3)

Calculations for some specific stratified fluids

175

(2)

Exercises

177

(2)

Parametric Vibrations of One-Dimensional Systems

179

(8)

Parametric Vibrations of Systems of Hill's Type

179

(5)

Setting of the problem

179

(1)

Perturbation method

180

(1)

Numerical-analytical analysis

181

(1)

Vibrations of crankshafts

182

(2)

Stability of Plane Vibrations and Rotations of a Satellite on a Circular Orbit

184

(2)

Setting of the problem

184

(1)

Results of numerical-analytical investigation

185

(1)

Exercises

186

(1)

Vibrations of a Distributed Inhomogeneous System in a Rectangular Domain

187

(14)

Vibrations of an Inhomogeneous Membrane

187

(2)

Statement of the initial boundary value problem

187

(1)

Separation of variables

188

(1)

Scheme of the Construction of a Solution of the Membrane Eigenvalue Problem

189

(2)

Separation of spatial variables

189

(1)

Structural properties of eigenvalues and eigenfunctions

189

(2)

Method of Accelerated Convergence

191

(3)

Two-parameter scheme of solution

191

(1)

Introduction of small parameters

191

(1)

A parallel scheme of the algorithm of accelerated convergence

192

(1)

Numerical-graphical solution of the problem

193

(1)

Iterative refinement procedure

193

(1)

Some Generalizations

194

(2)

Perturbation of the surface density function

194

(1)

Nonuniform membrane tension

195

(1)

The presence of elastic environment

195

(1)

Taking into account perturbations of general form

196

(1)

Examples

196

(4)

Inhomogeneity with respect to one coordinate

196

(1)

Symmetric inhomogeneity

197

(1)

Multicoordinate approximation

198

(2)

Exercises

200

(1)

Free Vibrations of a Rectangular Membrane with Sharply Varying Surface Density

201

(14)

Statement of the Problem of Free Vibrations of an Inhomogeneous Rectangular Membrane

201

(2)

Preliminary remarks

201

(1)

Statement of the boundary value problem

202

(1)

Construction of the Generating Solution

203

(1)

Separation of the variables in the unperturbed problem

203

(1)

A scheme for the construction of the generating solution

203

(1)

Membrane Model with Sharply Changing Surface Density

204

(4)

Approximation of the density function

204

(1)

Brief description of the algorithm

205

(2)

Software

207

(1)

Calculation Results and Conclusions

208

(7)

Calculation results for the symmetrical cross

209

(1)

Calculation results of the shifted cross

210

(1)

Calculation results for the nonsymmetric cross

211

(1)

Conclusions

212

(3)

Free Vibrations of Elastic Systems in Elliptic Domains

215

(18)

Free Vibrations of a Homogeneous Elliptic Membrane

215

(12)

Preliminary remarks regarding the present state of the investigations

215

(1)

Setting of the problem

216

(1)

Variational approach and the construction of highly precise estimates

217

(5)

Construction of approximate analytical expressions for eigenvalues of elliptic membranes with small eccentricity

222

(1)

Asymptotic expansions of eigenvalues for large eccentricity values

223

(2)

Finding eigenfrequencies and vibration shapes of an elliptic membrane by the method of accelerated convergence

225

(1)

Conclusions

226

(1)

Free Vibrations of an Elliptic Plate with Clamped Edge

227

(5)

Preliminary remarks

227

(1)

Setting of the problem

227

(1)

Estimates for the frequency of the lowest vibration mode with the help of an elliptically symmetrical test function

228

(2)

Estimates for the second vibration modes

230

(1)

Estimates of eigenfrequencies for higher vibration modes

231

(1)

Conclusions

232

(1)

Exercises

232

(1)

References

233

(4)

Index

237

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High-Precision Methods In Eigenvalue Problems And Their Applications

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High-Precision Methods In Eigenvalue Problems And Their Applications

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