Analysis of Tridiagonal Recurrence Relations in Continuum Approximation

• Дата: 2001
• Автори: Bass, F.G., Gitterman, M.
• Формат: Стаття
• Мова: English
• Опубліковано: Радіоастрономічний інститут НАН України 2001
• Назва видання: Радиофизика и радиоастрономия
• Онлайн доступ: http://dspace.nbuv.gov.ua/handle/123456789/122226
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Цитувати: Analysis of Tridiagonal Recurrence Relations in Continuum Approximation / F.G. Bass, M. Gitterman // Радиофизика и радиоастрономия. — 2001. — Т. 6, № 1. — С. 71-78. — Бібліогр.: 11 назв. — англ.

Transition from difference to differential equation allows solving tridiagonal recurrence relations, which appear, among other things, in analysis of the rotation of an overdamped Brownian particle subjected to a periodic force. Replacement of the discrete integers in the Fourier series by continuum is justified for large numbers, i. e. for small angles. For the simplest case of the sinusoidal force, our solution, indeed, coincides with one obtained by expanding the sin in the original Fokker-Planck equation (The Ornstein-Uhlenbeck limit). However, for slightly more complicate potential the expansion for small angles does not transform the appropriate Fokker-Planck equation into the soluble. At the same time, the method suggested allows solving the problem for all periodic potentials which have finite number of terms in their Fourier series such as sinm(θ ) or cosm (θ). Even and odd functions require slightly different analysis, and are considered separately.