Akulenko L.D., Nesterov S.V. High precision methods in eigenvalue problems and their applications

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.DerivativesAuthors
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