By Leonid D. Akulenko, Sergei V. Nesterov
This e-book 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 type. The authors additionally current high-precision asymptotic equipment for picking eigenvalues and eigenfunctions of upper oscillation modes and view quite a few eigenvalue difficulties that seem in oscillation idea, 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
Example text
1) on the basis of the refined values ½´½µ , ½ , ½´¾µ , ¾ , etc. 4. Description of the Method of Accelerated Convergence The method or algorithm of accelerated convergence amounts to the implementation of the following operations: ½ Find an upper bound £ ½ (by the Rayleigh–Ritz or any other method). , find the function ν ´Ü £½ µ. , the point such that ν ´ £ ½µ ¼ Numerically, the point is found either by interpolation or by decreasing the step when integrating the Cauchy problem near the point Ü at which the function ν ´Ü £½ µ changes sign.
2) has already been constructed by some analytical or numerical procedure. 2) is a simple task. The solution ٠ν ´Ü £½ µ has the following property: ν ´ £ ½µ ¼ ¼ ½ where is the first null-point of the function ν . The inequality ½ follows from Sturm’s first oscillation theorem [24, 29, 33, 59] which claims that if the coefficient increases, the root of the solution of the Cauchy problem shifts to the left. The equality ½ is possible only for £ ½. ½ It would be natural to associate the bound ½£ with the root .
Approximate solution of the perturbed problem. 3), we obtain a sequence of boundary value problems. 5) is known and is given above. 3. 4) with ¾ , ¿ are taken into account. ´¼µ ´½µ ´½µ Since the function ͽ ´Ý µ is known, let us find the unknown quantities £½ and ͽ ´Ý µ. 3. Reduction of the correction term to differential form. The correction £½ has an intricate structure, but it can be simplified to a great extent, so that the integral in the numerator can be found in explicit form. 9) will frequently occur in the sequel.