High precision methods in eigenvalue problems and their applications

HIGH-PRECISION METHODS IN EIGENVALUE PROBLEMS AND THEIR APPLICATIONS......Page 3
Differential and Integral Equations and Their Applications Series......Page 2
Table of Contents......Page 5
Derivatives......Page 13
Preface......Page 14
Authors......Page 17
1.1.1. Boundary value problem for eigenvalues and eigenfunctions.......Page 18
1.1.2. Variational statement of the eigenvalue problem.......Page 20
1.2.1.General scheme of analytical solution.......Page 22
1.2.2. Reduction to a Fredholm integral equation of the second kind.......Page 26
1.2.3. Reduction to a Volterra integral equation of the second kind.......Page 27
1.3.2. Standard procedure of asymptotic expansions.......Page 29
1.3.3. Finding the expansion coefficients.......Page 30
1.4. Numerical Methods for Solving the Sturm–Liouville Problem......Page 31
1.4.1. The Rayleigh–Ritz method.......Page 32
1.4.2. Some general facts and remarks pertaining to other numerical methods in the Sturm–Liouville problem.......Page 35
2.1.1. The problem of constructing two-sided estimates.......Page 37
2.1.2. Construction and analysis of comparison systems.......Page 38
2.3.1. Construction of an equivalent perturbed problem.......Page 39
2.3.2. Approximate solution of the perturbed problem.......Page 40
2.3.3. Reduction of the correction term to differential form......Page 41
2.4. Description of the Method of Accelerated Convergence......Page 42
2.5.1. Test model problems.......Page 43
2.5.2. A method for the calculation of weighted norms.......Page 44
2.6.1. An example with the calculation of two eigenvalues.......Page 45
2.6.2. Some properties of the procedure of finding subsequent eigenvalues.......Page 46
2.7.3. Test problem.......Page 47
2.8.1. Statement of the third boundary value problem.......Page 48
2.8.2. Construction of a comparison system.......Page 49
2.8.3. Solution of the perturbed problem.......Page 50
2.8.5. The method of accelerated convergence.......Page 51
2.8.6. Example.......Page 52
2.9.2. Main properties of the periodic problem.......Page 53
2.9.4. Construction of the comparison system.......Page 54
2.9.5. Introduction of a small parameter.......Page 55
2.9.6. Approximate solution of the perturbed problem.......Page 56
2.9.7. The method of accelerated convergence.......Page 57
2.9.8. Examples.......Page 59
2.10.1. Transformation of the perturbed boundary value problem.......Page 63
2.10.2. Proof of convergence of successive approximations.......Page 64
2.11. Proof of Quadratic Convergence......Page 66
2.12.1. Third-order refinement procedure.......Page 67
2.13. Taking into Account Explicit Dependence of Boundary Conditions on Eigenvalues......Page 68
2.14. Exercises......Page 69
3.1.1. Properties of the perturbed spectrum.......Page 71
3.1.2. The problem of secular terms and regularization of the problem.......Page 72
3.1.3. Separation of variables.......Page 73
3.2.1. Construction of eigenfrequencies and phases of partial vibrations.......Page 74
3.2.2. Finding eigenfunctions and the construction of an orthonormal basis.......Page 76
3.3.1. The problem of expansion in terms of an approximate basis.......Page 77
3.3.2. Uniform estimates.......Page 79
3.4.2. Error estimates.......Page 80
3.5. Exercises......Page 81
4.1.2. Basic definitions.......Page 83
4.2.2. Some basic general properties of solutions.......Page 84
4.3.2. Derivation and analysis of the determining relation.......Page 85
4.4.1. Some properties of the solution of the comparison problem.......Page 86
4.5. The Method of Accelerated Convergence for Generalized Sturm–Liouville Problems......Page 87
4.6.1. Test example for an integrable equation.......Page 88
4.7.1. Statement of the generalized periodic problem.......Page 89
4.7.2. An example illustrating spectral properties.......Page 90
4.7.4. An extended setting of the problem and the procedure of its approximate solution.......Page 91
4.8. Generalized Boundary Value Problems with Spectral Parameter in Boundary Conditions......Page 92
4.9. Exercises......Page 93
5.1.1. Statement of the generalized problem.......Page 94
5.2.1. “Amplitude–phase” variables.......Page 95
5.2.2. Approximation of the phase.......Page 96
5.3.1. Introduction of intermediate parameters.......Page 97
5.3.2. Finding the original quantities.......Page 98
5.3.3. Procedure of successive approximations.......Page 99
5.4.1. Approximate calculation of higher mode amplitudes.......Page 100
5.4.2. Finding eigenfunctions corresponding to higher modes.......Page 101
5.5.2. General boundary conditions of the third kind.......Page 102
5.6.1. Longitudinal vibrations of an inhomogeneous rectilinear beam.......Page 103
5.6.2. Vibrations of an inhomogeneous string.......Page 104
5.6.3. Asymptotics of eigenvalues of the Hill problem.......Page 105
5.6.4. Spatial vibrations of a satellite.......Page 106
5.7. Exercises......Page 108
6.1.1. Statement of the problem in differential form. Some remarks.......Page 109
6.1.2. Statement of the problem in variational form.......Page 110
6.1.4. Scheme of solution.......Page 111
6.2.1. Construction of the characteristic equation and the sagittary function.......Page 113
6.2.2. Oscillation properties of the sagittary function.......Page 114
6.3.1. Algorithm of shooting with respect to the ordinate.......Page 116
6.3.2. Algorithm of shooting with respect to the abscissa.......Page 117
6.4. Examples......Page 118
6.4.1. A model test example.......Page 119
6.4.3. Parametric synthesis for conical beams.......Page 120
7.1.1. Differential and variational statements of the problem.......Page 123
7.1.2. Construction of upper bounds.......Page 124
7.1.3. Relation between the upper bound and the length of the interval.......Page 125
7.2.1. Introduction of a small parameter.......Page 127
7.2.2. An approximate solution of the perturbed problem.......Page 128
7.3.2. Algorithm of the accelerated convergence method.......Page 129
7.4. Other Types of Boundary Conditions......Page 130
7.6.1. General remarks about calculations.......Page 131
7.6.2. Test examples with analytically integrable equations.......Page 132
7.6.3. Problem of transverse vibrations of an inhomogeneous beam occurring in applications.......Page 133
8.1.1. Statement of the initial boundary value problem; preliminary remarks.......Page 135
8.1.3. Some features of the standard procedure of the perturbation method.......Page 137
8.2.1. Transformation of the independent variable.......Page 138
8.2.2. Regular procedure of the perturbationmethod.......Page 139
8.2.3. Justification of the perturbation method.......Page 140
8.4. Finding Eigenvalues and Eigenfunctions in the First Approximation......Page 142
8.5. Exercises......Page 145
9.1.2. Variational statement of the problem.......Page 146
9.2.1. Construction of the comparison problem; analysis of its properties.......Page 147
9.2.3. Approximate solution of the problem.......Page 148
9.3.1. Properties of the first approximation of the solution.......Page 149
9.3.2. Algorithm of accelerated convergence for vector problems.......Page 150
9.4.1. A system of Euler type.......Page 151
9.5. Exercises......Page 152
10.1.1. Statement of the initial boundary value problem. Its solution by the Fourier method,......Page 153
10.1.2. Free vibrations of a rotating heavy homogeneous string subjected to tension.......Page 157
10.1.3. Vibrations of an inhomogeneous thread.......Page 161
10.2.1. Setting of the problem of longitudinal bending of an elastic beam.......Page 164
10.2.2. Calculation of the critical force for some rigidity distributions.......Page 166
10.3.2. A numerical-analytical solution.......Page 169
10.4.1. Approaches of Rayleigh and Love.......Page 170
10.5. Exercises......Page 172
11.1.1. Preliminary remarks and statement of the problem.......Page 173
11.1.2. Solving the eigenvalue problem.......Page 175
11.1.3. Calculation results and their analysis.......Page 178
11.2.1. Statement of the problem and some mathematical aspects of its solution.......Page 182
11.2.2. A version of the perturbation method for approximate solution of the Sturm– Liouville Problem.......Page 184
11.2.3. Calculations for some specific stratified fluids.......Page 187
11.3. Exercises......Page 189
12.1.1. Setting of the problem.......Page 190
12.1.2. Perturbation method.......Page 191
12.1.3. Numerical-analytical analysis.......Page 192
12.1.4. Vibrations of crankshafts.......Page 193
12.2.1. Setting of the problem.......Page 195
12.2.2. Results of numerical-analytical investigation.......Page 196
12.3. Exercises......Page 197
13.1.1. Statement of the initial boundary value problem.......Page 198
13.1.2. Separation of variables.......Page 199
13.2.2. Structural properties of eigenvalues and eigenfunctions.......Page 200
13.3.2. Introduction of small parameters.......Page 202
13.3.3. A parallel scheme of the algorithm of accelerated convergence.......Page 203
13.3.5. Iterative refinement procedure.......Page 204
13.4.1. Perturbation of the surface density function.......Page 205
13.4.3. The presence of elastic environment.......Page 206
13.5.1. Inhomogeneity with respect to one coordinate.......Page 207
13.5.2. Symmetric inhomogeneity.......Page 208
13.5.3. Multi-coordinate approximation.......Page 209
13.6. Exercises......Page 211
14.1.1. Preliminary remarks.......Page 212
14.1.2. Statement of the boundary value problem.......Page 213
14.2.2. A scheme for the construction of the generating solution.......Page 214
14.3.1. Approximation of the density function.......Page 215
14.3.2. Brief description of the algorithm.......Page 216
14.3.3. Software.......Page 218
14.4. Calculation Results and Conclusions......Page 219
14.4.1. Calculation results for the symmetrical cross.......Page 220
14.4.2. Calculation results for the shifted cross.......Page 221
14.4.3. Calculation results for the nonsymmetric cross.......Page 222
14.4.4. Conclusions.......Page 223
15.1.1. Preliminary remarks regarding the present state of the investigations.......Page 225
15.1.2. Setting of the problem.......Page 226
15.1.3. Variational approach and the construction of highly precise estimates.......Page 227
15.1.4. Construction of approximate analytical expressions for eigenvalues of elliptic membranes with small eccentricity.......Page 232
15.1.5. Asymptotic expansions of eigenvalues for large eccentricity values.......Page 233
15.1.6. Finding eigenfrequencies and vibration shapes of an elliptic membrane by the method of accelerated convergence.......Page 235
15.1.7. Conclusions.......Page 236
15.2.2. Setting of the problem.......Page 237
15.2.3. Estimates for the frequency of the lowest vibration mode with the help of an elliptically symmetrical test function.......Page 238
15.2.4. Estimates for the second vibration modes.......Page 240
15.2.5. Estimates of eigenfrequencies for higher vibration modes.......Page 241
15.3. Exercises......Page 242
References......Page 243