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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Extra resources for High-Precision Methods in Eigenvalue Problems and Their Applications (Differential and Integral Equations and Their Applications)
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 Üµ ¾ ´Üµ Ü¾ ´½ Üµ ¿ Ü¿ ´½ Üµ 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.