https://oldena.lpnu.ua/handle/ntb/41464| Title: | Solving of differential equations systems in the presence of fractional derivatives using the orthogonal polynomials |
| Other Titles: | Застосування ортогональних многочленів для розв’язування систем диференціальних рівнянь за наявності похідних дробового порядку |
| Authors: | П’янило, Я. Браташ, О. П’янило, Г. Pyanylo, Ya. Bratash, O. Pyanylo, G. |
| Affiliation: | Центр математичного моделювання Інституту прикладних проблем механіки і математики ім. Я. С. Підстригача НАН України Centre of Mathematical Modelling of Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine |
| Bibliographic description (Ukraine): | Pyanylo Ya. Solving of differential equations systems in the presence of fractional derivatives using the orthogonal polynomials / Ya. Pyanylo, O. Bratash, G. Pyanylo // Mathematical Modeling and Computing. — Lviv : Lviv Politechnic Publishing House, 2017. — Vol 4. — No 1. — P. 87–95. |
| Bibliographic description (International): | Pyanylo Ya. Solving of differential equations systems in the presence of fractional derivatives using the orthogonal polynomials / Ya. Pyanylo, O. Bratash, G. Pyanylo // Mathematical Modeling and Computing. — Lviv : Lviv Politechnic Publishing House, 2017. — Vol 4. — No 1. — P. 87–95. |
| Is part of: | Mathematical Modeling and Computing, 1 (4), 2017 |
| Issue: | 1 |
| Volume: | 4 |
| Issue Date: | 15-Jun-2017 |
| Publisher: | Lviv Politechnic Publishing House |
| Place of the edition/event: | Lviv |
| UDC: | 519.6 539.3 |
| Keywords: | математична модель рух газу в трубопроводах спектральні методи ортогональні многочлени mathematical model gas motion in pipelines spectral methods orthogonal polynomials |
| Number of pages: | 9 |
| Page range: | 87-95 |
| Start page: | 87 |
| End page: | 95 |
| Abstract: | Побудовано математичну модель руху газу в трубопроводах для випадку, коли не-
усталений процес описано похiдною дробового порядку за часовою змiнною. Сфор-
мульовано крайову задачу. Рiшення задачi знаходять спектральним методом в бази-
сах многочленiв Чебишева–Лагерра за часовою змiнною та многочленiв Лежандра за
координатою. Знаходження рiшення в результатi зведено до системи алгебраїчних
рiвнянь. Проведено числовий експеримент. The mathematical model of the gas motion in the pipelines for the case where unstable process is described by the fractional time derivative is constructed in the paper. The boundary value problem is formulated. The solution of the problem is founded by the spectral method on Chebyshev-Laguerre polynomials bases with respect to the time variable and Legendre polynomials with respect to the coordinate variable. The finding of the solution eventually is reduced to the system of algebraic equations. The numerical experiment is conducted. |
| URI: | https://ena.lpnu.ua/handle/ntb/41464 |
| ISSN: | 2312-9794 |
| Copyright owner: | © 2017 Lviv Polytechnic National University CMM IAPMM NASU |
| References (Ukraine): | [1] KilbasA.A., SrivastavaH.M., Trujillo J. J. Theory and Applications of Fractional Differential Equations. In North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006). [2] Podlubny I. Fractional Differential Equations. Mathematics in Science and Engineering. Academic Press, New York (1999). [3] Sabatier J., AgrawalO.P., Tenreiro Machado J.A. (Eds.) Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht (2007). [4] Samko S.G., KilbasA.A., MarichevO. I. Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, Yverdon (1993). [5] VasilievV.V., Simak L.A. Fractional calculus and approximation methods in the modeling of dynamic systems. Kiev, Scientific publication of NAS of Ukraine (2008), (in Ukrainian). [6] AhmadB, SivasundaramS. Existence of solutions for impulsive integral boundary value problems of fractional order. Nonlinear Analysis: Hybrid Systems. 4, 134–141 (2010). [7] AhmadB, SivasundaramS. On four-point nonlocal boundary value problems of nonlinear integrodifferential equations of fractional order. Appl. Math. Comput. 217, 480–487 (2010). [8] DelboscoD., Rodino L. Existence and uniqueness for a nonlinear fractional differential equation. J. Math. Anal. Appl. 204, 609–625 (1996). [9] He J. Some applications of nonlinear fractional differential equations and their approximations. Bull. Sci. Technol. 15, 86–90 (1999). [10] Fylshtynsky L.A., MukomelT.V., KirochokT.A. One-dimensional initial-boundary problem for the fractional-differential equation of heat conductivity. Buletin of Zaporizhzhya National University. 1, 113–118 (2010), (in Ukrainian). [11] KhalimonO.O., BondarO. S. Approximation of a fractional derivative over space for generalized diffusion equations. Journal of Computational and Applied Mathematics. 1(119), 95–101 (2015), (in Ukrainian). [12] LopuhN.B., PyanyloYa.D. Numerical model of gas filtration in porous media using fractional time derivatives. Mathematical methods in Chemistry and Biology. 2, n.1, 98–104 (2014), (in Ukrainian). [13] PyanyloYa., VasyunykM., Vasyunyk I. Investigation of the spectral method of solving of fractional time derivatives in Laguerre polynomials basis. Physical-Mathematical Modeling and Informational Technologies. 18, 173–179 (2013), (in Ukrainian). [14] PyanyloYa. Use of fractional derivatives for analysis of nonstationary gas motion in pipelines in the presense of compressor stations and outlets. Physical-Mathematical Modeling and Informational Technologies. 16, 122–132 (2012), (in Ukrainian). [15] PskhuA.V. Equations of partial fractional. Research Institute of Mathematics and Automatization of Kabardino-Balkar Scientific Center. Moscow, Science (2005), (in Russian). [16] Samko S.G., KilbasA.A., MarichevO. I. Integrals and fractional derivatives and some of their applications. Minsk, Science and Technology (1987), (in Russian). [17] PyanyloYa.D. Projection-iterative methods of solving of direct and inverse problems of transport. Lviv, Spline (2011), (in Ukrainian). [18] DitkinV.A., PrudnikovA.P. Operational calculus. Moscow, High school (1975), (in Russian). |
| References (International): | [1] KilbasA.A., SrivastavaH.M., Trujillo J. J. Theory and Applications of Fractional Differential Equations. In North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006). [2] Podlubny I. Fractional Differential Equations. Mathematics in Science and Engineering. Academic Press, New York (1999). [3] Sabatier J., AgrawalO.P., Tenreiro Machado J.A. (Eds.) Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht (2007). [4] Samko S.G., KilbasA.A., MarichevO. I. Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, Yverdon (1993). [5] VasilievV.V., Simak L.A. Fractional calculus and approximation methods in the modeling of dynamic systems. Kiev, Scientific publication of NAS of Ukraine (2008), (in Ukrainian). [6] AhmadB, SivasundaramS. Existence of solutions for impulsive integral boundary value problems of fractional order. Nonlinear Analysis: Hybrid Systems. 4, 134–141 (2010). [7] AhmadB, SivasundaramS. On four-point nonlocal boundary value problems of nonlinear integrodifferential equations of fractional order. Appl. Math. Comput. 217, 480–487 (2010). [8] DelboscoD., Rodino L. Existence and uniqueness for a nonlinear fractional differential equation. J. Math. Anal. Appl. 204, 609–625 (1996). [9] He J. Some applications of nonlinear fractional differential equations and their approximations. Bull. Sci. Technol. 15, 86–90 (1999). [10] Fylshtynsky L.A., MukomelT.V., KirochokT.A. One-dimensional initial-boundary problem for the fractional-differential equation of heat conductivity. Buletin of Zaporizhzhya National University. 1, 113–118 (2010), (in Ukrainian). [11] KhalimonO.O., BondarO. S. Approximation of a fractional derivative over space for generalized diffusion equations. Journal of Computational and Applied Mathematics. 1(119), 95–101 (2015), (in Ukrainian). [12] LopuhN.B., PyanyloYa.D. Numerical model of gas filtration in porous media using fractional time derivatives. Mathematical methods in Chemistry and Biology. 2, n.1, 98–104 (2014), (in Ukrainian). [13] PyanyloYa., VasyunykM., Vasyunyk I. Investigation of the spectral method of solving of fractional time derivatives in Laguerre polynomials basis. Physical-Mathematical Modeling and Informational Technologies. 18, 173–179 (2013), (in Ukrainian). [14] PyanyloYa. Use of fractional derivatives for analysis of nonstationary gas motion in pipelines in the presense of compressor stations and outlets. Physical-Mathematical Modeling and Informational Technologies. 16, 122–132 (2012), (in Ukrainian). [15] PskhuA.V. Equations of partial fractional. Research Institute of Mathematics and Automatization of Kabardino-Balkar Scientific Center. Moscow, Science (2005), (in Russian). [16] Samko S.G., KilbasA.A., MarichevO. I. Integrals and fractional derivatives and some of their applications. Minsk, Science and Technology (1987), (in Russian). [17] PyanyloYa.D. Projection-iterative methods of solving of direct and inverse problems of transport. Lviv, Spline (2011), (in Ukrainian). [18] DitkinV.A., PrudnikovA.P. Operational calculus. Moscow, High school (1975), (in Russian). |
| Content type: | Article |
| Appears in Collections: | Mathematical Modeling And Computing. – 2017. – Vol. 4, No. 1 |
| File | Description | Size | Format | |
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| 2017v4n1_Pyanylo_Ya-Solving_of_differential_87-95.pdf | 857.36 kB | Adobe PDF | View/Open | |
| 2017v4n1_Pyanylo_Ya-Solving_of_differential_87-95__COVER.png | 424.83 kB | image/png | View/Open |
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