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
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