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Boca Raton: Chapman & Hall/CRC, 2005. - 245 p.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.Contents :Derivatives

Preface

Authors

Boundary value problem for eigenvalues and eigenfunctions

Variational statement of the eigenvalue problem

General scheme of analytical solution

Reduction to a Fredholm integral equation of the second kind

Reduction to a Volterra integral equation of the second kind

Standard procedure of asymptotic expansions

Finding the expansion coefficients

Numerical Methods for Solving the Sturm–Liouville Problem

The Rayleigh–Ritz method

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

The problem of constructing two-sided estimates

Construction and analysis of comparison systems

Construction of an equivalent perturbed problem

Approximate solution of the perturbed problem

Reduction of the correction term to differential form

Description of the Method of Accelerated Convergence

Test model problems

A method for the calculation of weighted norms

An example with the calculation of two eigenvalues

Some properties of the procedure of finding subsequent eigenvalues

Test problem

Statement of the third boundary value problem

Construction of a comparison system

Solution of the perturbed problem

The method of accelerated convergence

Example

Main properties of the periodic problem

Construction of the comparison system

Introduction of a small parameter

Approximate solution of the perturbed problem

The method of accelerated convergence

Examples

Transformation of the perturbed boundary value problem

Proof of convergence of successive approximations

Proof of Quadratic Convergence

Third-order refinement procedure

Taking into Account Explicit Dependence of Boundary Conditions on Eigenvalues

Exercises

Properties of the perturbed spectrum

The problem of secular terms and regularization of the problem

Separation of variables

Construction of eigenfrequencies and phases of partial vibrations

Finding eigenfunctions and the construction of an orthonormal basis

The problem of expansion in terms of an approximate basis

Uniform estimates

Error estimates

Exercises

Basic definitions

Some basic general properties of solutions

Derivation and analysis of the determining relation

Some properties of the solution of the comparison problem

The Method of Accelerated Convergence for Generalized Sturm–Liouville Problems

Test example for an integrable equation

Statement of the generalized periodic problem

An example illustrating spectral properties

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

Generalized Boundary Value Problems with Spectral Parameter in Boundary Conditions

Exercises

Statement of the generalized problem

“Amplitude–phase” variables

Approximation of the phase

Introduction of intermediate parameters

Finding the original quantities

Procedure of successive approximations

Approximate calculation of higher mode amplitudes

Finding eigenfunctions corresponding to higher modes

General boundary conditions of the third kind

Longitudinal vibrations of an inhomogeneous rectilinear beam

Vibrations of an inhomogeneous string

Asymptotics of eigenvalues of the Hill problem

Spatial vibrations of a satellite

Exercises

Statement of the problem in differential form Some remarks

Statement of the problem in variational form

Scheme of solution

Construction of the characteristic equation and the sagittary function

Oscillation properties of the sagittary function

Algorithm of shooting with respect to the ordinate

Algorithm of shooting with respect to the abscissa

Examples

A model test example

Parametric synthesis for conical beams

Differential and variational statements of the problem

Construction of upper bounds

Relation between the upper bound and the length of the interval

Introduction of a small parameter

An approximate solution of the perturbed problem

Algorithm of the accelerated convergence method

Other Types of Boundary Conditions

General remarks about calculations

Test examples with analytically integrable equations

Problem of transverse vibrations of an inhomogeneous beam occurring in applications

Statement of the initial boundary value problem; preliminary remarks

Some features of the standard procedure of the perturbation method

Transformation of the independent variable

Regular procedure of the perturbationmethod

Justification of the perturbation method

Finding Eigenvalues and Eigenfunctions in the First Approximation

Exercises

Variational statement of the problem

Construction of the comparison problem; analysis of its properties

Approximate solution of the problem

Properties of the first approximation of the solution

Algorithm of accelerated convergence for vector problems

A system of Euler type

Exercises

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

Free vibrations of a rotating heavy homogeneous string subjected to tension

Vibrations of an inhomogeneous thread

Setting of the problem of longitudinal bending of an elastic beam

Calculation of the critical force for some rigidity distributions

A numerical-analytical solution

Approaches of Rayleigh and Love

Exercises

Preliminary remarks and statement of the problem

Solving the eigenvalue problem

Calculation results and their analysis

Statement of the problem and some mathematical aspects of its solution

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

Calculations for some specific stratified fluids

Exercises

Setting of the problem

Perturbation method

Numerical-analytical analysis

Vibrations of crankshafts

Setting of the problem

Results of numerical-analytical investigation

Exercises

Statement of the initial boundary value problem

Separation of variables

Structural properties of eigenvalues and eigenfunctions

Introduction of small parameters

A parallel scheme of the algorithm of accelerated convergence

Iterative refinement procedure

Perturbation of the surface density function

The presence of elastic environment

Inhomogeneity with respect to one coordinate

Symmetric inhomogeneity

Multi-coordinate approximation

Exercises

Preliminary remarks

Statement of the boundary value problem

A scheme for the construction of the generating solution

Approximation of the density function

Brief description of the algorithm

Software

Calculation Results and Conclusions

Calculation results for the symmetrical cross

Calculation results for the shifted cross

Calculation results for the nonsymmetric cross

Conclusions

Preliminary remarks regarding the present state of the investigations

Setting of the problem

Variational approach and the construction of highly precise estimates

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

Asymptotic expansions of eigenvalues for large eccentricity values

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

Conclusions

Setting of the problem

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

Estimates for the second vibration modes

Estimates of eigenfrequencies for higher vibration modes

Exercises

References

Preface

Authors

Boundary value problem for eigenvalues and eigenfunctions

Variational statement of the eigenvalue problem

General scheme of analytical solution

Reduction to a Fredholm integral equation of the second kind

Reduction to a Volterra integral equation of the second kind

Standard procedure of asymptotic expansions

Finding the expansion coefficients

Numerical Methods for Solving the Sturm–Liouville Problem

The Rayleigh–Ritz method

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

The problem of constructing two-sided estimates

Construction and analysis of comparison systems

Construction of an equivalent perturbed problem

Approximate solution of the perturbed problem

Reduction of the correction term to differential form

Description of the Method of Accelerated Convergence

Test model problems

A method for the calculation of weighted norms

An example with the calculation of two eigenvalues

Some properties of the procedure of finding subsequent eigenvalues

Test problem

Statement of the third boundary value problem

Construction of a comparison system

Solution of the perturbed problem

The method of accelerated convergence

Example

Main properties of the periodic problem

Construction of the comparison system

Introduction of a small parameter

Approximate solution of the perturbed problem

The method of accelerated convergence

Examples

Transformation of the perturbed boundary value problem

Proof of convergence of successive approximations

Proof of Quadratic Convergence

Third-order refinement procedure

Taking into Account Explicit Dependence of Boundary Conditions on Eigenvalues

Exercises

Properties of the perturbed spectrum

The problem of secular terms and regularization of the problem

Separation of variables

Construction of eigenfrequencies and phases of partial vibrations

Finding eigenfunctions and the construction of an orthonormal basis

The problem of expansion in terms of an approximate basis

Uniform estimates

Error estimates

Exercises

Basic definitions

Some basic general properties of solutions

Derivation and analysis of the determining relation

Some properties of the solution of the comparison problem

The Method of Accelerated Convergence for Generalized Sturm–Liouville Problems

Test example for an integrable equation

Statement of the generalized periodic problem

An example illustrating spectral properties

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

Generalized Boundary Value Problems with Spectral Parameter in Boundary Conditions

Exercises

Statement of the generalized problem

“Amplitude–phase” variables

Approximation of the phase

Introduction of intermediate parameters

Finding the original quantities

Procedure of successive approximations

Approximate calculation of higher mode amplitudes

Finding eigenfunctions corresponding to higher modes

General boundary conditions of the third kind

Longitudinal vibrations of an inhomogeneous rectilinear beam

Vibrations of an inhomogeneous string

Asymptotics of eigenvalues of the Hill problem

Spatial vibrations of a satellite

Exercises

Statement of the problem in differential form Some remarks

Statement of the problem in variational form

Scheme of solution

Construction of the characteristic equation and the sagittary function

Oscillation properties of the sagittary function

Algorithm of shooting with respect to the ordinate

Algorithm of shooting with respect to the abscissa

Examples

A model test example

Parametric synthesis for conical beams

Differential and variational statements of the problem

Construction of upper bounds

Relation between the upper bound and the length of the interval

Introduction of a small parameter

An approximate solution of the perturbed problem

Algorithm of the accelerated convergence method

Other Types of Boundary Conditions

General remarks about calculations

Test examples with analytically integrable equations

Problem of transverse vibrations of an inhomogeneous beam occurring in applications

Statement of the initial boundary value problem; preliminary remarks

Some features of the standard procedure of the perturbation method

Transformation of the independent variable

Regular procedure of the perturbationmethod

Justification of the perturbation method

Finding Eigenvalues and Eigenfunctions in the First Approximation

Exercises

Variational statement of the problem

Construction of the comparison problem; analysis of its properties

Approximate solution of the problem

Properties of the first approximation of the solution

Algorithm of accelerated convergence for vector problems

A system of Euler type

Exercises

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

Free vibrations of a rotating heavy homogeneous string subjected to tension

Vibrations of an inhomogeneous thread

Setting of the problem of longitudinal bending of an elastic beam

Calculation of the critical force for some rigidity distributions

A numerical-analytical solution

Approaches of Rayleigh and Love

Exercises

Preliminary remarks and statement of the problem

Solving the eigenvalue problem

Calculation results and their analysis

Statement of the problem and some mathematical aspects of its solution

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

Calculations for some specific stratified fluids

Exercises

Setting of the problem

Perturbation method

Numerical-analytical analysis

Vibrations of crankshafts

Setting of the problem

Results of numerical-analytical investigation

Exercises

Statement of the initial boundary value problem

Separation of variables

Structural properties of eigenvalues and eigenfunctions

Introduction of small parameters

A parallel scheme of the algorithm of accelerated convergence

Iterative refinement procedure

Perturbation of the surface density function

The presence of elastic environment

Inhomogeneity with respect to one coordinate

Symmetric inhomogeneity

Multi-coordinate approximation

Exercises

Preliminary remarks

Statement of the boundary value problem

A scheme for the construction of the generating solution

Approximation of the density function

Brief description of the algorithm

Software

Calculation Results and Conclusions

Calculation results for the symmetrical cross

Calculation results for the shifted cross

Calculation results for the nonsymmetric cross

Conclusions

Preliminary remarks regarding the present state of the investigations

Setting of the problem

Variational approach and the construction of highly precise estimates

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

Asymptotic expansions of eigenvalues for large eccentricity values

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

Conclusions

Setting of the problem

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

Estimates for the second vibration modes

Estimates of eigenfrequencies for higher vibration modes

Exercises

References

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