# Asymptotic method for solution of oscillatory fractional derivative

Document Type : Research Paper

Authors

1 Institute of Applied Mathematics, Baku State University, Baku, Azerbaijan.

2 Institute of Information Technologies of the National Academy of Sciences of Azerbaijan.

3 Nakhchivan State University, Nakhchivan, Azerbaijan.

Abstract

In the paper, an oscillatory system with liquid dampers is considered, when the mass of the head is large enough. By means of expedient transformations, the equation of motion with fractional derivatives is reduced to an equation of fractional order containing a small parameter. The corresponding nonlocal boundary value problem is solved and the zero and first approximations of solutions of the relative small parameter are constructed. The results are illustrated on the concrete example, where the solution differs from the analytical solution by 10−2 order.

Keywords

#### References

• [1]         R. S. Adıguzel, U¨ . Aksoy, E. Karapınar, and I˙. M. Erhan, On the solutions of fractional differential equations via geraghty type hybrid contractions, Appl. Comput. Math., 20(2) (2021), 313–333.
• [2]         F. A. Aliev, N. A. Aliev, and N. S. Hajiyeva, Some mathematical problems and their solutions for the oscil- lating systems with liquid dampers (Survey), 8th International Congress on Fundamental and Applied Sciences (ICFAS2021), Proceeding Book, (2021), 179–180.
• [3]         F. A. Aliev, N. A. Aliev, N. S. Hajiyeva, and N. I. Mahmudov, Some mathematical problems and their solutions for the oscillating systems with liquid dampers: A review, Appl. Comput. Math., 20(3) (2021), 339–365.
• [4]         F. A. Aliev, N. A. Aliev, M. M. Mutallimov, and A. A. Namazov, Algorithm for solving the identification problem for determining the fractional-order derivative of an oscillatory system, Appl. Comput. Math., 19(3) (2020), 415–422.
• [5]         F. A. Aliev, N. A. Aliev, N. A. Safarova, and Y. V. Mamedova, Solution of the problem of analytical construc- tion of optimal regulators for a fractional order oscillatory system in the general case, Journal of Applied and Computational Mechanics, 7(2) (2021), 970–976.
• [6]         F. A. Aliev, N. A. Aliev, N. A. Safarova, and N. I. Velieva, Algorithm for solving the Cauchy problem for stationary systems of fractional order linear ordinary differential equations, Comput. Methods Differ. Equ., 8(1) (2020), 212–221.
• [7]         E. Ashpazzadeh, M. Lakestani, and A. Fatholahzadeh, Spectral methods combined with operational matrices for fractional optimal control problems: A review, Appl. Comput. Math., 20(2) (2021), 209–235.
• [8]         S. Balaei, E. Eslami, and A. Borumand Saeid, Invertible square matrices over residuated lattices, TWMS J. Pure Appl. Math., 11(2) (2020), 173–188.
• [9]         R. E. Bellman, Introduction to matrix analysis, New York, 1970, 391 p.
• [10]       B. Bonilla, M. Rivero, and J. J. Trujillor, On systems of linear fractional differential equations with constant coefficients, Appl. Math. Comput., 187 (2007), 68–78.
• [11]       S. Khan, S. A. Wani, and M. Riyasat, Study of generalized Legendre-Appell polynomials via fractional operators, TWMS J. Pure Appl. Math., 11(2) (2020), 144–156.
• [12]       N. I. Mahmudov, I. T. Huseynov, N. A. Aliev, and F. A. Aliev, Analytical approach to a class of Bagley-Torvik equations, TWMS J. Pure Appl. Math., 11(2) (2020), 238–258.
• [13]       A. Kh. Mirzadjanzadeh, I. M. Akhmetov, A. M. Khasaev, and V. I. Gusev, Technology and technique of oil production, Moscow, Nedra, 1986.
• [14]       C. A. Monje, Y. Q. Chen, B. M. Vinagre, D. Xue, and V. Feliu, Fractional-order systems and controls. Funda- mentals and applications, Springer, London, 2010, 414 p.
• [15]       M. M. Mutallimov and F. A. Aliev, Methods for solving optimization problems during the operation of oil wells,, Saarbrucken (Deutscland), LAP LAMBERT, 2012, 164 p.
• [16]       A. A. Namazov, Computational algorithm for determining the order of fractional derivatives of oscillatory systems, Proceedings of IAM, 8(2) (2019), 202-210.
• [17]       Z. Odibat, Fractional power series solutions of fractional differential equations by using generalized taylor series, Appl. Comput. Math., 19(1) (2020), 47–58.
• [18]       A. Ozyapici and T. Karanfiller, New integral operator for solutions of differential equations, TWMS J. Pure Appl. Math., 11(2) (2020), 131–143.
• [19]       S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional integrals and derivatives: theory and applications, Gordon and Breach Science Publishers, 1993, 750 p.
• [20]       E. Set, A. O. Akdemir, and F. Ozata, Gruss type inequalities for fractional integral operator involving the extended generalized Mittag-Leffler function, Appl. Comput. Math., 19(3) (2020), 402–414.
• [21]       V. I. Shurov, Technology and technique of oil production, Moscow, Nedra(in Russian), 1983, 510 p.
• [22]       T. Tunc, M. Z. Sarikaya, and H. Yaldiz, Fractional Hermite Hadamards type inequality for the co-ordinated convex functions, 11(1) (2020), 3–29.

### History

• Receive Date: 17 July 2021
• Revise Date: 01 October 2021
• Accept Date: 11 October 2021
• First Publish Date: 19 November 2021