REALIZATION ALGORITHM FOR DEFINING FRACTIONAL ORDER IN OSCILLATING SYSTEMS WITH LIQUID DAMPER

Year 2025, Volume: 15 Issue: 11, 2709 - 2717, 03.11.2025

Nazile Hajiyeva Irade Aliyeva Fikret Aliev

Abstract

In the paper the problem of defining the fractional order in oscillating systems with liquid damper. Firstly, the equation of the object is reduced to the Volterra integral equation of the second kind with respect to the second order derivative of the phase coordinate. Based on the statistical data the quadratic functional has been constructed. Using the method of successive approximations the obtained Volterra integral equation has been solved and its solution has the form of the Neumann series. By means of the least squares method, we ensure that the theoretical results coincide with the statistical data, and as a result, a more effective fractional order is determined. Then, an effective algorithm is proposed. Since some steps of this algorithm need explanation, the issue of the implementation of the algorithm is considered.

Keywords

fractional order differential equations , the least squares method , statistical data

References

• Agarwal G., Yadav L.K., Nisar K.S., Alqarni M.M., Mahmoud E.E., (2024), A hybrid method for the analytical solution of time fractional Whitham-Broer-Kaup equations, Appl. Comput. Math., 23(1), pp.3-17.
• Ahmad B., Alsaedi A., Ntouyas S.K., Alotaibi F.M., (2024), A coupled Hilfer-Hadamard fractional differential system with nonlocal fractional integral boundary conditions, TWMS J. Pure and Appl. Math., 15(1), pp.95-114.
• Aliev F.A., Aliev N.A., Mutallimov M.M., Namazov A.A., (2019), Identification method for determining the order of the fractional derivative of an oscillatory system, Proceedings of IAM, 8(1), pp.3-13. (in Russian)
• Aliev F.A., Aliev N.A., Rasulzade A.F., Hajiyeva N.S., (2023), Solution of the optimal program trajectory and control of the discretized equation of motion of sucker-rod pumping unit in a Newtonian fluid, TWMS J. App. and Eng. Math., 13(4), pp.1369-1382.
• Aliev F.A., Aliev N.A., Rasulzade A.F., Hajiyeva N.S., Alieva I.V., (2024), Development of discrete asymptotic algorithm for the optimal trajectory and control in oscillatory systems with liquid damper, SOCAR Proceedings, (2), pp.122-127.
• Aliev F.A., Aliev N.A., Velieva N.I., Gasimova K.G., (2021), A Method for the Discretization of Linear Systems of Ordinary Fractional Differential Equations with Constant Coefficients, Journal of Mathematical Sciences, 256, pp.567-575.
• Aliev F.A., Aliyev N.A, Hajiyeva N.S., Ismailov N.A., Magarramov I.A., Ramazanov A.B., Abdullayev V.C., (2021), Solution of an oscillatory system with fractional derivative including to equations of motion and to nonlocal boundary conditions, SOCAR Proceedings, (4), pp.115-121.
• Aliev F.A., Aliyev N.A, Hajiyeva N.S., Mahmudov N.I., (2021), Some mathematical problems and their solutions for the oscillating systems with liquid dampers: A review, Applied and Computational Mathematics, 20(3), pp.339-365.
• Aliev F.A., Aliyev N.A., Hajiyeva N.S., Safarova N.A., Aliyeva R., (2022), Asymptotic method for solution of oscillatory fractional derivative, Computational Methods for Differential Equations, 10(4), pp.1123-1130.
• Aliev F.A., Jamalbayov M.A., Valiyev N.A., Handajiyeva N.S., (2023), Computer model of pump--well--reservoir system based on the new concept of imitational modeling of dynamic systems, International Applied Mechanics, 59(3), pp.352-362.
• Aliev N.A., Hajiyeva N.S., Alieva I.V., Farajova Sh.A., (2024), Algorithm for defining the fractional order of an oscillatory system with liquid dampers, Proceedings of IAM, 13(2), pp.262-279. (in Russian)
• Aydinlik S., Kiris A., (2024), An efficient method for solving fractional integral and differential equations of Bratu type, TWMS J. App. and Eng. Math., 14\eqref{GrindEQ__1_}, pp.94-102.
• Benmerrous A., Chadli L.S., Moujahid A., Elomari M., Melliani S., (2024), Generalized solutions for fractional Schrodinger equation, TWMS J. App. and Eng. Math., 14(4), pp.1361-1374.
• Bonilla B., Rivero M., Trujillo J.J., (2007), On systems of linear fractional differential equations with constant coefficients, Appl. Math. Comput., \eqref{GrindEQ__187_}, pp.68-78.
• Celik B., Akdemir A.O., Set E., Aslan S., (2024), Ostrowski-Mercer type integral inequalities for differentiable convex functions via Atangana-Baleanu fractional integral operators conditions, TWMS J. Pure and Appl. Math., 15(1), pp.269-285.
• Çelik Bariş, Set Erhan, Akdemir Ahmet Ocak, Özdemir M. Emin, (2023), Novel generalizations for Grüss type inequalities pertaining to the constant proportional fractional integrals, Appl. Comput. Math., 22(2), pp.275-291.
• Fikhtengolts G.M., (1966), Course of Differential and Integral Calculus, Moscow: Nauka, 616p.
• Guliyev A. M., Jamalbayov M.A., (2017), The Prediction of the Development Indicators of Creeping Reservoirs of Light Oils, SOCAR Proceedings, (3), pp.51-57.
• Hajiyeva N.S., Aliev F.A., (2023), Algorithm for finding program trajectories and controls during oil production by the gas lift method in general case, Proceedings of IAM, 12(1), pp.76-84.
• Hamdy M. Ahmed, A.M. Sayed Ahmed, Maria Alessandra Ragusa, (2023), On some non-instantaneous impulsive differential equations with fractional Brownian motion and poisson jumps, TWMS J. Pure and Appl. Math., 14(1), pp.125-140.
• Jamalbayov M., Jamalbayli T., Hajiyeva N., Allahverdi J., Aliev F.A., (2022), Algorithm for Determining the Permeability and Compaction Properties of a Gas Condensate Reservoir based on a Binary Model, Journal of Applied and Computational Mechanics, 8(3), pp.1014-1022.
• Jamalbayov M., Valiyev N., (2024), The Discrete-Imitational Modeling of the Pump-Well-Reservior System with a Intermittent Sucker-Rod Pumping, Society of Petroleum Engineers - SPE Middle East Artificial Lift Conference and Exhibition, MEAL, Paper Number: SPE-221528-MS. https://doi.org/10.2118/221528-MS.
• Jamalbayov M.A., Valiyev N.A., (2024), The discrete-imitation modeling concept of the ``sucker-rod pump-well-reservoir'' system and the optimization of the pumping process, Petroleum Research, 9(4), pp.686-694.
• Kaczorek T., (2010), Positive linear systems with different fractional orders, Bull. Pol. Acad. Sci. Tech. Sci., 58, pp.453-458.
• Krushna B.M.B., (2024), Extremal points for a (n,p)-type Riemann-liouville fractional-order boundary value problems, TWMS J. App. and Eng. Math., 14\eqref{GrindEQ__1_}, pp.247-258.
• Mahdi N.K., Khudair A.R., (2024), Toward stability investigation of fractional dynamical systems on time scale, TWMS J. App. and Eng. Math., 14(4), pp.1495-1513.
• Miller K.S., Ross B., (1993), An Introduction to the Fractional Calculus and Fractional Differential Equations, New York:Wiley, 336p.
• Mirzajanzadeh A.Kh., (1997), Dynamic processes in oil and gas production, Moscow: Nauka, 246 p. (in Russian)
• Mittag-Leffler G., (1905), Sur la representation analytique d`une branche uniforme d'une function monogene: cinquieme note, Acta Mathematica, 29, pp.101-181.
• Monje C.A., Chen Y.Q, Vinagre B.M, Xue D., Feliu V., (2010), Fractional-Order Systems and Controls Fundamentals and Applications, London: Springer, 414p.
• Petrovski I.G., (1965), Lectures on the Theory of Integral Equations, Moscow: Science, 128 p.
• Petrovsky I.G., (1952), Lectures on the Theory of Ordinary Equations, Gostekhizdat, 232p. (in Russian).
• Sadek L., Yüzbaşi Ş., Alaoui H.T., (2024), Two numerical solutions for solving linear and nonlinear systems of differential equations, Appl. Comput. Math., 23(4), pp.421-436.
• Samarsky A.A., (1989), Numerical Methods, Moscow: Science, 432p.
• Samko S., Marichev O., Kilbas A., (1987), Fractional Integrals and Derivatives and Some of Their Applications, Science and Technica, Minsk.
• Temirbekov A.N., Temirbekova L.N., Zhumagulov B.T., (2023), Fictitious domain method with the idea of conjugate optimization for non-linear Navier-Stokes equations, Appl. Comput. Math., 22(2), pp.172-188.
• Thabet Abdeljawad, Kamal Shah, Mohammed S. Abdo, Fahd Jarad, (2023), An Analytical Study of Fractional Delay Impulsive Implicit Systems with Mittag-Leffler Law, Appl. Comput. Math., 22(1), pp.31-44.
• Xu C.J., Lin J., Zhao Y., Li P.L., Han L.Q., Qin Y.X., Peng X.Q., Shi S., (2023), Hopf bifurcation control of a fractional-order delayed turbidostat model via a novel extended hybrid controller, Appl. Comput. Math., 22(4), pp.495-519.