Email this article Analysis of the difference scheme of wave equation equivalent with fractional differentiation operator
- Authors: Beybalaev V.D1, Yakubov A.Z2
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Affiliations:
- Dagestan State University
- Daghestan State Institute of National Economy
- Issue: Vol 18, No 1 (2014)
- Pages: 125-133
- Section: Articles
- URL: https://journal-vniispk.ru/1991-8615/article/view/20731
- DOI: https://doi.org/10.14498/vsgtu1209
- ID: 20731
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Abstract
Analysis of the difference scheme of boundary-value problem for the wave equation analogue is made. Explicit and implicit difference schemes for numerical solution of the first boundary-value problem for the wave equation analogue with Caputo fractional differentiation operator are investigated, and the stability criteria for these difference schemes are proved by the harmonic Fourier method. Estimates for eigenvalues of the operator of transition from one time layer to another are obtained. Computational experiment on the analysis of the given difference scheme has been performed for the example. The graphs of the numerical solution of the boundary-value problem for the wave equation with the operator of fractional differentiation having different values of parameters of fractional differentiation $\alpha$ and $\beta$ have been built. Change of the period of fluctuations under transition to a fractional derivative is established. On an example it is shown that parameters $\alpha$ and $\beta$ become managing directors.
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##article.viewOnOriginalSite##About the authors
Vetlugin D Beybalaev
Dagestan State University
Email: kaspij_03@mail.ru
(Cand. Phys. & Math. Sci.), Associate Professor, Dept. of Applied Mathematics 43a, M. Gadzhiev st., Makhachkala, 367025, Russian Federation
Amuchi Z Yakubov
Daghestan State Institute of National Economy
Email: yakubovaz@mail.ru
(Cand. Phys. & Math. Sci.), Associate Professor, Dept. of Computer Science & Computer Engineering 5. D. Ataev st., Makhachkala, 367008, Russian Federation
References
- Ю. И. Бабенко, Метод дробного дифференцирования в прикладных задачах теории тепломассообмена, СПб.: Профессионал, 2009. 584 с.
- K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, New York, Jon Wiley & Sons. Inc., 1993, xiii+366 pp.
- I. Podlubny, Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, vol. 198, San Diego, Academic Press, 1999, xxiv+340 pp.
- A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Amsterdam, Elsevier, 2006, xv+523 pp.
- В. В. Васильев, Л. А. Симак, Дробное исчисление и аппроксимационные методы в моделировании динамических систем, Киев: НАН Украины, 2008. 256 с.
- S. Das, Functional Fractional Calculus, Berlin, Heidelberg, Springer-Verlag, 2011, ix+612 pp. doi: 10.1007/978-3-642-20545-3.
- В. В. Учайкин, Метод дробных производных, Ульяновск: Артишок, 2008. 512 с.
- А. М. Нахушев, Уравнения математической биологии, М.: Высшая школа, 1995. 301 с.
- А. М. Нахушев, Дробное исчисление и его применение, М.: Физматлит, 2003. 271 с.
- А. М. Нахушев, Элементы дробного исчисления и их применение, Нальчик: КБНЦ РАН, 2003. 299 с.
- В. Д. Бейбалаев, “Одношаговые методы решения задачи Коши для обыкновенного дифференциального уравнения с производными дробного порядка” // Вестник Дагестанского государственного университета, 2011. No 6. С. 67-72.
- У. Г. Пирумов, Численные методы, М.: Дрофа, 2004. 224 с.
- Р. Р. Нигматуллин, “Дробный интеграл и его физическая интерпретация” // ТМФ, 1992. Т. 90, No 3. С. 354-368.
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