Falade KAZEEM IYANDA, Abd'gafar TİAMİYU, Adesina ADİO, Huzaifa Muhammad TAHİR, Umar Muhammad ABUBAKAR, Sahura BADAMASİ

Computational Relationship of The Surface Area and Stiffness of the Spring Constant on Fractional Bagley-Torvik Equation

In this paper, we formulate an efficient algorithm based on a new iterative method for the numerical solution of the Bagley-Torvik equation. The fractional differential equation arises in many areas of applied mathematics including viscoelasticity problems and applied mechanics of the oscillation process. We construct the fractional derivatives via the Caputo-type fractional operator to formulate a three-step algorithm using the MAPLE 18 software package. We further investigate the relationships between the surface area and stiffness of the spring constants of the Bagley-Torvik equation on three case problems and numerical results are presented to demonstrate the efficiency of the proposed algorithm.

Keywords:

Fractional Bagley-Torvik equation, new iterative method, Caputo derivative, Riemann-Liouville fractional integral operator, MAPLE 18 software package.,

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Agrawal OP, Kumar P. Comparison of five schemes for fractional differential equations, J. Sabatier et al. (eds.), Advances in Fractional Calculus: Theoretical Developments and App in Phy and Eng 2007; 43-60.
Chen W, Ye L, Sun H. Fractional diffusion equation by the Kansa method, Comp Math App 2010;59:1614-1620.
Jiang CX, Carletta JE, Hartley TT. Implementation of fractional order operators on field programmable gate arrays, J. Sabatier et al. (eds.), Advances in Fractional Calculus:Theoretical Developments and App in Phy and Eng 2007; 333-346.
Kilbas AA, Srivastava HM, Trujillo JJ. Theory of application of fractional differential equations, first ed., Belarus, 2006.
Lakshmikantham V, Vatsala AS. Basic theory of fractional differential equations, Noon Anal. 2008;(69): 2677-2682.
Miller KS, Ross B. An Introduction to the fractional calculus and differential equations, John Wiley, New York, 1993.
Momani S, Noor MN. Numerical methods for fourth-order fractional integrodifferential equations, App Math and Comp 2006 (182);754-760.
Torvik PJ, Bagley RL. On the appearance of the fractional derivative in the behavior of real materials. J Appl Mech. 1984; (51): 294-298.
Azhar AZ, Grzegorz K, Jan A. An investigation of fractional Bagley–Torvik equation, MDPI Entropy 2020; (22): 1-13.
Witkowski K, Kudra G, Wasilewski G, Awrejcewicz J. Modelling and experimental validation of 1-degree-of-freedom impacting oscillator. Journal of System Control Engineering 2019; (233): 418–430.
Bagley RL, Torvik PJ. A theoretical basis for the application of fractional calculus to viscoelasticity, J Rheol 1983; (27): 201-210.
Torvik PJ, Bagley RL. Fractional calculus in the transient analysis of viscoelastically damped structures. AIAA J 1985(23): 918-925.
Diethelm K, Ford NJ, Numerical solution of the Bagley-Torvik equation, BIT Numerical Mathematics, 2002; (42): 490-507.
Çenesiz Y, Keskin Y, Kurnaz A. The solution of the Bagley-Torvik equation with the generalized Taylor collocation method, J Frankl Inst 2010:(347), 452-466.
Zahra WK. Van Daele M. Discrete spline methods for solving two-point fractional Bagley-Torvik equation, Appl Math Comput 2017; (296):42-56.
Mekkaoui T, Hammouch Z. Approximate analytical solutions to the Bagley-Torvik equation by the fractional iteration method, Ann Univ Craiova Math Comput Sci Ser 2012; 39 (2): 251-256.
Mohammadi F. Numerical solution of Bagley-Torvik equation using Chebyshev wavelet operational matrix of fractional derivative, Int J Adv in Appl Math and Mech 2014; 2(1): 83-91.
Ray S. S., Bera R. K. Analytical solution of the Bagley-Torvik equation by Adomian decomposition method, Appl Math Comput 2005; 168 (1): 398-410.
Vijay S, Sushil K. Numerical scheme for solving two-point fractional Bagley-Torvik equation using Chebyshev collocation method, WSEAS Transactions on Systems 2018; (17):166-177.
Uzbas S. Y. Numerical solution of the Bagley-Torvik equation by the Bessel collocation method, Mathematical Methods in the Applied Sciences 2013; 36:300-312.
Tianfu J, Jianhua H and Changqing Y. Numerical solution of the Bagley–Torvik equation using shifted Chebyshev operational matrix, Advances in Difference Equations 2020; 3-14.
Hossein F, Juan J. N. An investigation of fractional Bagley-Torvik equation, DE GRUYTER Open Math. 2019, 17: 499–512.
Daftardar-Gejji. V and Jafari H. An iterative method for solving nonlinear functional equations, Journal of Mathematical Analysis and Applications 2006; 316(2): 753–763.
Falade KI, Tiamiyu AT. Numerical solution of partial differential equations with fractional variable coefficients using new iterative method (NIM), IJ Mathematical Sciences and Computing, 2020; 3: 12-21.
Hemeda AA. New iterative method: an application for solving fractional physical differential equations, Hindawi Publishing Corporation Abstract and Applied Analysis 2013; 1-10.