High-Precision Methods in Eigenvalue Problems and Their by Leonid D. Akulenko, Sergei V. Nesterov

By Leonid D. Akulenko, Sergei V. Nesterov

This publication provides a survey of analytical, asymptotic, numerical, and mixed equipment of fixing eigenvalue difficulties. It considers the hot approach to sped up convergence for fixing difficulties of the Sturm-Liouville sort in addition to boundary-value issues of boundary stipulations of the 1st, moment, and 3rd sort. The authors additionally current high-precision asymptotic equipment for picking eigenvalues and eigenfunctions of upper oscillation modes and examine a variety of eigenvalue difficulties that seem in oscillation concept, acoustics, elasticity, hydrodynamics, geophysics, quantum mechanics, structural mechanics, electrodynamics, and microelectronics.

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Difference methods are not very efficient for eigenvalue problems of Sturm–Liouville type. 4. NUMERICAL METHODS FOR SOLVING THE STURM–LIOUVILLE PROBLEM 19 The widely used finite-element method [18, 23, 30, 53, 60] is, in essence, a modification of the Rayleigh–Ritz method and that of finite-differences. According to the finite element method, one considers a partition of the interval and a polynomial (spline) approximation of the unknown function on the segments of that partition, the degree of the polynomial being not too high.

3) £ ¼ . 1 holds, since for £½ ¾ the solution of the Cauchy problem vanishes at the point ½ ¼ ½. Thus, we have a refined upper estimate, £ ½ Moreover, using ½, ½ ¾ ½ £ ¼ ½ we obtain another refined estimate from above, ½ £ ½ ¾ ¾ ½ The refinement formula in which the bound the relative error ½¼ . ¾¾¾ ££ ¾¾¾ ½ ££ ½ is used yields ½´½µ ¼¼ with Example 4. 4) and the system of test functions ½ ´Üµ Ü´½ ܵ ¾ ´Üµ ܾ ´½ ܵ ¿ Ü¿ ´½ ܵ The upper bounds obtained in this case are fairly rough, even though there are three test functions.

4), and it is implied that the integration is highly precise (“ideal”). © 2005 by Chapman & Hall/CRC 22 THE METHOD OF ACCELERATED CONVERGENCE FOR THE STURM–LIOUVILLE PROBLEM 2. Give a constructive method for obtaining a lower bound ¼ ½£ ½£ : £ ½ ½ 3. Develop a refinement procedure for eigenvalues and eigenfunctions. 4. Construct an algorithm for finding eigenvalues and eigenfunctions with arbitrary accuracy: both the absolute and the relative errors should be arbitrarily small. 5. Extend the method to subsequent Ò , ³Ò , Ò ¾.

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