Solving of differential equations systems in the presence of fractional derivatives using the orthogonal polynomials

dc.citation.epage95
dc.citation.issue1
dc.citation.spage87
dc.citation.volume4
dc.contributor.affiliationЦентр математичного моделювання Інституту прикладних проблем механіки і математики ім. Я. С. Підстригача НАН України
dc.contributor.affiliationCentre of Mathematical Modelling of Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine
dc.contributor.authorП’янило, Я.
dc.contributor.authorБраташ, О.
dc.contributor.authorП’янило, Г.
dc.contributor.authorPyanylo, Ya.
dc.contributor.authorBratash, O.
dc.contributor.authorPyanylo, G.
dc.coverage.placenameLviv
dc.date.accessioned2018-06-05T14:12:24Z
dc.date.available2018-06-05T14:12:24Z
dc.date.created2017-06-15
dc.date.issued2017-06-15
dc.description.abstractПобудовано 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.
dc.description.abstractThe 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.
dc.format.extent87-95
dc.format.pages9
dc.identifier.citationPyanylo 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.
dc.identifier.citationenPyanylo 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.
dc.identifier.issn2312-9794
dc.identifier.urihttps://ena.lpnu.ua/handle/ntb/41464
dc.language.isoen
dc.publisherLviv Politechnic Publishing House
dc.relation.ispartofMathematical Modeling and Computing, 1 (4), 2017
dc.relation.references[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).
dc.relation.references[2] Podlubny I. Fractional Differential Equations. Mathematics in Science and Engineering. Academic Press, New York (1999).
dc.relation.references[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).
dc.relation.references[4] Samko S.G., KilbasA.A., MarichevO. I. Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, Yverdon (1993).
dc.relation.references[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).
dc.relation.references[6] AhmadB, SivasundaramS. Existence of solutions for impulsive integral boundary value problems of fractional order. Nonlinear Analysis: Hybrid Systems. 4, 134–141 (2010).
dc.relation.references[7] AhmadB, SivasundaramS. On four-point nonlocal boundary value problems of nonlinear integrodifferential equations of fractional order. Appl. Math. Comput. 217, 480–487 (2010).
dc.relation.references[8] DelboscoD., Rodino L. Existence and uniqueness for a nonlinear fractional differential equation. J. Math. Anal. Appl. 204, 609–625 (1996).
dc.relation.references[9] He J. Some applications of nonlinear fractional differential equations and their approximations. Bull. Sci. Technol. 15, 86–90 (1999).
dc.relation.references[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).
dc.relation.references[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).
dc.relation.references[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).
dc.relation.references[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).
dc.relation.references[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).
dc.relation.references[15] PskhuA.V. Equations of partial fractional. Research Institute of Mathematics and Automatization of Kabardino-Balkar Scientific Center. Moscow, Science (2005), (in Russian).
dc.relation.references[16] Samko S.G., KilbasA.A., MarichevO. I. Integrals and fractional derivatives and some of their applications. Minsk, Science and Technology (1987), (in Russian).
dc.relation.references[17] PyanyloYa.D. Projection-iterative methods of solving of direct and inverse problems of transport. Lviv, Spline (2011), (in Ukrainian).
dc.relation.references[18] DitkinV.A., PrudnikovA.P. Operational calculus. Moscow, High school (1975), (in Russian).
dc.relation.referencesen[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).
dc.relation.referencesen[2] Podlubny I. Fractional Differential Equations. Mathematics in Science and Engineering. Academic Press, New York (1999).
dc.relation.referencesen[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).
dc.relation.referencesen[4] Samko S.G., KilbasA.A., MarichevO. I. Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, Yverdon (1993).
dc.relation.referencesen[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).
dc.relation.referencesen[6] AhmadB, SivasundaramS. Existence of solutions for impulsive integral boundary value problems of fractional order. Nonlinear Analysis: Hybrid Systems. 4, 134–141 (2010).
dc.relation.referencesen[7] AhmadB, SivasundaramS. On four-point nonlocal boundary value problems of nonlinear integrodifferential equations of fractional order. Appl. Math. Comput. 217, 480–487 (2010).
dc.relation.referencesen[8] DelboscoD., Rodino L. Existence and uniqueness for a nonlinear fractional differential equation. J. Math. Anal. Appl. 204, 609–625 (1996).
dc.relation.referencesen[9] He J. Some applications of nonlinear fractional differential equations and their approximations. Bull. Sci. Technol. 15, 86–90 (1999).
dc.relation.referencesen[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).
dc.relation.referencesen[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).
dc.relation.referencesen[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).
dc.relation.referencesen[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).
dc.relation.referencesen[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).
dc.relation.referencesen[15] PskhuA.V. Equations of partial fractional. Research Institute of Mathematics and Automatization of Kabardino-Balkar Scientific Center. Moscow, Science (2005), (in Russian).
dc.relation.referencesen[16] Samko S.G., KilbasA.A., MarichevO. I. Integrals and fractional derivatives and some of their applications. Minsk, Science and Technology (1987), (in Russian).
dc.relation.referencesen[17] PyanyloYa.D. Projection-iterative methods of solving of direct and inverse problems of transport. Lviv, Spline (2011), (in Ukrainian).
dc.relation.referencesen[18] DitkinV.A., PrudnikovA.P. Operational calculus. Moscow, High school (1975), (in Russian).
dc.rights.holder© 2017 Lviv Polytechnic National University CMM IAPMM NASU
dc.subjectматематична модель
dc.subjectрух газу в трубопроводах
dc.subjectспектральні методи
dc.subjectортогональні многочлени
dc.subjectmathematical model
dc.subjectgas motion in pipelines
dc.subjectspectral methods
dc.subjectorthogonal polynomials
dc.subject.udc519.6
dc.subject.udc539.3
dc.titleSolving of differential equations systems in the presence of fractional derivatives using the orthogonal polynomials
dc.title.alternativeЗастосування ортогональних многочленів для розв’язування систем диференціальних рівнянь за наявності похідних дробового порядку
dc.typeArticle

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