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

Yıl 2023, Cilt: 18 Sayı: 1, 23 - 31, 29.03.2023
https://doi.org/10.55525/tjst.1155648

Öz

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.

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Kaynakça

  • 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.
Yıl 2023, Cilt: 18 Sayı: 1, 23 - 31, 29.03.2023
https://doi.org/10.55525/tjst.1155648

Öz

Proje Numarası

NONE

Kaynakça

  • 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.
Toplam 25 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Mühendislik
Bölüm TJST
Yazarlar

Falade Kazeem Iyanda 0000-0001-7572-5688

Abd'gafar Tiamiyu 0000-0003-1641-7196

Adesina Adio 0000-0002-7330-4852

Huzaifa Muhammad Tahir 0000-0001-8392-2542

Umar Muhammad Abubakar 0000-0003-3935-4829

Sahura Badamasi 0000-0002-2016-3988

Proje Numarası NONE
Yayımlanma Tarihi 29 Mart 2023
Gönderilme Tarihi 4 Ağustos 2022
Yayımlandığı Sayı Yıl 2023 Cilt: 18 Sayı: 1

Kaynak Göster

APA Kazeem Iyanda, F., Tiamiyu, A., Adio, A., Tahir, H. M., vd. (2023). Computational Relationship of The Surface Area and Stiffness of the Spring Constant on Fractional Bagley-Torvik Equation. Turkish Journal of Science and Technology, 18(1), 23-31. https://doi.org/10.55525/tjst.1155648
AMA Kazeem Iyanda F, Tiamiyu A, Adio A, Tahir HM, Abubakar UM, Badamasi S. Computational Relationship of The Surface Area and Stiffness of the Spring Constant on Fractional Bagley-Torvik Equation. TJST. Mart 2023;18(1):23-31. doi:10.55525/tjst.1155648
Chicago Kazeem Iyanda, Falade, Abd’gafar Tiamiyu, Adesina Adio, Huzaifa Muhammad Tahir, Umar Muhammad Abubakar, ve Sahura Badamasi. “Computational Relationship of The Surface Area and Stiffness of the Spring Constant on Fractional Bagley-Torvik Equation”. Turkish Journal of Science and Technology 18, sy. 1 (Mart 2023): 23-31. https://doi.org/10.55525/tjst.1155648.
EndNote Kazeem Iyanda F, Tiamiyu A, Adio A, Tahir HM, Abubakar UM, Badamasi S (01 Mart 2023) Computational Relationship of The Surface Area and Stiffness of the Spring Constant on Fractional Bagley-Torvik Equation. Turkish Journal of Science and Technology 18 1 23–31.
IEEE F. Kazeem Iyanda, A. Tiamiyu, A. Adio, H. M. Tahir, U. M. Abubakar, ve S. Badamasi, “Computational Relationship of The Surface Area and Stiffness of the Spring Constant on Fractional Bagley-Torvik Equation”, TJST, c. 18, sy. 1, ss. 23–31, 2023, doi: 10.55525/tjst.1155648.
ISNAD Kazeem Iyanda, Falade vd. “Computational Relationship of The Surface Area and Stiffness of the Spring Constant on Fractional Bagley-Torvik Equation”. Turkish Journal of Science and Technology 18/1 (Mart 2023), 23-31. https://doi.org/10.55525/tjst.1155648.
JAMA Kazeem Iyanda F, Tiamiyu A, Adio A, Tahir HM, Abubakar UM, Badamasi S. Computational Relationship of The Surface Area and Stiffness of the Spring Constant on Fractional Bagley-Torvik Equation. TJST. 2023;18:23–31.
MLA Kazeem Iyanda, Falade vd. “Computational Relationship of The Surface Area and Stiffness of the Spring Constant on Fractional Bagley-Torvik Equation”. Turkish Journal of Science and Technology, c. 18, sy. 1, 2023, ss. 23-31, doi:10.55525/tjst.1155648.
Vancouver Kazeem Iyanda F, Tiamiyu A, Adio A, Tahir HM, Abubakar UM, Badamasi S. Computational Relationship of The Surface Area and Stiffness of the Spring Constant on Fractional Bagley-Torvik Equation. TJST. 2023;18(1):23-31.