Atjaunināt sīkdatņu piekrišanu

E-grāmata: High-Precision Methods in Eigenvalue Problems and Their Applications

, (Russian Academy of Sciences, Moscow, Russia)
  • Formāts: 260 pages
  • Izdošanas datums: 15-Oct-2004
  • Izdevniecība: CRC Press
  • Valoda: eng
  • ISBN-13: 9781134390212
Citas grāmatas par šo tēmu:
  • Formāts - EPUB+DRM
  • Cena: 62,60 €*
  • * ši ir gala cena, t.i., netiek piemērotas nekādas papildus atlaides
  • Ielikt grozā
  • Pievienot vēlmju sarakstam
  • Šī e-grāmata paredzēta tikai personīgai lietošanai. E-grāmatas nav iespējams atgriezt un nauda par iegādātajām e-grāmatām netiek atmaksāta.
  • Bibliotēkām
High-Precision Methods in Eigenvalue Problems and Their ApplicationsPalielināt
  • Formāts: 260 pages
  • Izdošanas datums: 15-Oct-2004
  • Izdevniecība: CRC Press
  • Valoda: eng
  • ISBN-13: 9781134390212
Citas grāmatas par šo tēmu:

DRM restrictions

  • Kopēšana (kopēt/ievietot):

    nav atļauts

  • Drukāšana:

    nav atļauts

  • Lietošana:

    Digitālo tiesību pārvaldība (Digital Rights Management (DRM))
    Izdevējs ir piegādājis šo grāmatu šifrētā veidā, kas nozīmē, ka jums ir jāinstalē bezmaksas programmatūra, lai to atbloķētu un lasītu. Lai lasītu šo e-grāmatu, jums ir jāizveido Adobe ID. Vairāk informācijas šeit. E-grāmatu var lasīt un lejupielādēt līdz 6 ierīcēm (vienam lietotājam ar vienu un to pašu Adobe ID).

    Nepieciešamā programmatūra
    Lai lasītu šo e-grāmatu mobilajā ierīcē (tālrunī vai planšetdatorā), jums būs jāinstalē šī bezmaksas lietotne: PocketBook Reader (iOS / Android)

    Lai lejupielādētu un lasītu šo e-grāmatu datorā vai Mac datorā, jums ir nepieciešamid Adobe Digital Editions (šī ir bezmaksas lietotne, kas īpaši izstrādāta e-grāmatām. Tā nav tas pats, kas Adobe Reader, kas, iespējams, jau ir jūsu datorā.)

    Jūs nevarat lasīt šo e-grāmatu, izmantojot Amazon Kindle.

Akulenko (Moscow Institute of Physics) and Nesterov (Bauman Moscow State Technical University) present a new numerical-analytical method, called the method of accelerated convergence, for solving boundary value problems for differential equations of the second or higher order, as well as for some systems of differential equations. Intended for graduate students and engineers, the text walks through the solution of the Sturm- Liouville problem for a system with slowly varying coefficients, treats generalized the Sturm-Liouville problem for higher vibration modes, and applies the new method to the investigation of elasticity, hydrodynamics, parametric instability, and the frequency and shape of free vibrations. Annotation ©2004 Book News, Inc., Portland, OR (booknews.com)

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
1(20)
Statement of the Sturm--Liouville Problem
1(4)
Boundary value problem for eigenvalues and eigenfunctions
1(2)
Variational statement of the eigenvalue problem
3(2)
Analytical Methods of Solving the Sturm--Liouville Problem
5(7)
General scheme of analytical solution
5(4)
Reduction to a Fredholm integral equation of the second kind
9(1)
Reduction to a Volterra integral equation of the second kind
10(2)
Solving the Sturm--Liouville Problem by the Method of Regular Perturbations
12(2)
Statement of the perturbed problem
12(1)
Standard procedure of asymptotic expansions
12(1)
Finding the expansion coefficients
13(1)
Justification questions
14(1)
Numerical Methods for Solving the Sturm--Liouville Problem
14(7)
The Rayleigh--Ritz method
15(3)
Some general facts and remarks pertaining to other numerical methods in the Sturm--Liouville problem
18(3)
The Method of Accelerated Convergence for the Sturm--Liouville Problem
21(34)
Numerical-Analytical Upper and Lower Bounds for Eigenvalues
21(2)
The problem of constructing two-sided estimates
21(1)
Construction and analysis of comparison systems
22(1)
Criterion of Closeness between the First Eigenvalue and its Upper (Lower) Bound. Introduction of a Small Parameter
23(1)
Theory of Perturbations
23(3)
Construction of an equivalent perturbed problem
23(1)
Approximate solution of the perturbed problem
24(1)
Reduction of the correction term to differential form
25(1)
Description of the Method of Accelerated Convergence
26(1)
Some Applications of the Accelerated Convergence Method
27(2)
Test model problems
27(1)
A method for the calculation of weighted norms
28(1)
The Method of Accelerated Convergence for Higher Eigenvalues
29(2)
An example with the calculation of two eigenvalues
29(1)
Some properties of the procedure of finding subsequent eigenvalues
30(1)
Problems with Boundary Conditions of the Second Kind
31(1)
Construction of a comparison problem
31(1)
Approximate solution of the problem
31(1)
Test problem
31(1)
Problems with Boundary Conditions of the Third Kind
32(5)
Statement of the third boundary value problem
32(1)
Construction of a comparison system
33(1)
Solution of the perturbed problem
34(1)
Differential relation between eigenvalues and the interval length
35(1)
The method of accelerated convergence
35(1)
Example
36(1)
Problems with Periodic Boundary Conditions
37(10)
Statement of the periodic boundary value problem
37(1)
Main properties of the periodic problem
37(1)
Construction of upper bounds
38(1)
Construction of the comparison system
38(1)
Introduction of a small parameter
39(1)
Approximate solution of the perturbed problem
40(1)
The method of accelerated convergence
41(2)
Examples
43(4)
Proof of Convergence of Successive Approximations. Existence Theorem
47(3)
Transformation of the perturbed boundary value problem
47(1)
Proof of convergence of successive approximations
48(2)
Proof of Quadratic Convergence
50(1)
The Method of Hyperaccelerated Convergence
51(1)
Third-order refinement procedure
51(1)
An application of the method of hyperaccelerated convergence
52(1)
Taking into Account Explicit Dependence of Boundary Conditions on Eigenvalues
52(1)
Exercises
53(2)
Approximate Analytical Solution of Perturbed Eigenvalue Problems
55(12)
Statement and Analysis of the Perturbed Sturm--Liouville Problem
55(3)
Properties of the perturbed spectrum
55(1)
The problem of secular terms and regularization of the problem
56(1)
Separation of variables
57(1)
Approximate Solution of the Boundary Value Problem
58(3)
Construction of eigenfrequencies and phases of partial vibrations
58(2)
Finding eigenfunctions and the construction of an orthonormal basis
60(1)
Remarks
61(1)
Approximation of Functions in Terms of the Approximate Basis
61(3)
The problem of expansion in terms of an approximate basis
61(2)
Uniform estimates
63(1)
Applications to Initial Boundary Value Problems
64(1)
Approximate solution
64(1)
Error estimates
64(1)
Exercises
65(2)
Generalized Sturm--Liouville Problem
67(12)
Statement of the Generalized Sturm--Liouville problem
67(1)
Statement of the boundary value problem in differential form
67(1)
Basic definitions
67(1)
Some Sturm--Liouville Problems with Exact Solutions
68(1)
Examples
68(1)
Some basic general properties of solutions
68(1)
Statement of an Auxiliary Variational Problem
69(1)
Variational statement of the problem and its generalization
69(1)
Derivation and analysis of the determining relation
69(1)
Closeness Criterion and the Theory of Perturbations
70(1)
Some properties of the solution of the comparison problem
70(1)
Approximate solution of the perturbed problem
71(1)
The Method of Accelerated Convergence for Generalized Sturm--Liouville Problems
71(1)
Model Problems
72(1)
Test example for an integrable equation
72(1)
Numerical example; two-sided estimates
73(1)
Generalized Parametric Vibrations
73(3)
Statement of the generalized periodic problem
73(1)
An example illustrating spectral properties
74(1)
General properties of solutions of generalized periodic problems
75(1)
An extended setting of the problem and the procedure of its approximate solution
75(1)
Generalized Boundary Value Problems with Spectral Parameter in Boundary Conditions
76(1)
Exercises
77(2)
Asymptotics of Eigenvalues and Eigenfunctions of the Generalized Sturm--Liouville Problem for Higher Vibration Modes
79(16)
General Notions Regarding the Asymptotic Behavior of Eigenvalues Corresponding to Higher Vibration Modes
79(1)
Statement of the generalized problem
79(1)
Classical results
80(1)
Application of Asymptotic Methods of the Theory of Nonlinear Vibrations
80(2)
``Amplitude--phase'' variables
80(1)
Approximation of the phase
81(1)
Determination of Eigenfrequencies and Vibration Phases
82(3)
Introduction of intermediate parameters
82(1)
Finding the original quantities
83(1)
Procedure of successive approximations
84(1)
Finding Amplitudes and Shapes of Free Vibrations
85(2)
Approximate calculation of higher mode amplitudes
85(1)
Finding eigenfunctions corresponding to higher modes
86(1)
Other Types of Boundary Value Problems
87(1)
Boundary conditions of the second kind
87(1)
General boundary conditions of the third kind
87(1)
Remarks about generalizations
88(1)
Calculations for Some Specific Mechanical Systems
88(5)
Longitudinal vibrations of an inhomogeneous rectilinear beam
88(1)
Vibrations of an inhomogeneous string
89(1)
Asymptotics of eigenvalues of the Hill problem
90(1)
Spatial vibrations of a satellite
91(2)
Exercises
93(2)
Solutions of Fourth-Order Self-Conjugate Problems. Oscillation Properties
95(14)
Statement of a Self-Conjugate Fourth-Order Boundary Value Problem
95(4)
Statement of the problem in differential form. Some remarks
95(1)
Statement of the problem in variational form
96(1)
Introduction of natural physical variables
97(1)
Scheme of solution
97(2)
The Method of Sagittary Function. Sturm's Theorems
99(3)
Construction of the characteristic equation and the sagittary function
99(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
Akulenko, Leonid D.; Nesterov, Sergei V.
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
Permanent link: https://www.kriso.lv/db/97811343902126e.html
Keywords: