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Khalil, Muayyad, Ibrahim, Najim. (1405). Comparative Accuracy Analysis of Spectral and Collocation Methods for the Fractional Bagley–Torvik Equation: A Systematic Numerical Study. فصلنامه علمی زیست فناوری گیاهان زراعی, (), -. doi: 10.30473/coam.2026.77807.1408
Muayyad Mahmood Khalil; Najim Abdullah Ibrahim. "Comparative Accuracy Analysis of Spectral and Collocation Methods for the Fractional Bagley–Torvik Equation: A Systematic Numerical Study". فصلنامه علمی زیست فناوری گیاهان زراعی, , , 1405, -. doi: 10.30473/coam.2026.77807.1408
Khalil, Muayyad, Ibrahim, Najim. (1405). 'Comparative Accuracy Analysis of Spectral and Collocation Methods for the Fractional Bagley–Torvik Equation: A Systematic Numerical Study', فصلنامه علمی زیست فناوری گیاهان زراعی, (), pp. -. doi: 10.30473/coam.2026.77807.1408
Khalil, Muayyad, Ibrahim, Najim. Comparative Accuracy Analysis of Spectral and Collocation Methods for the Fractional Bagley–Torvik Equation: A Systematic Numerical Study. فصلنامه علمی زیست فناوری گیاهان زراعی, 1405; (): -. doi: 10.30473/coam.2026.77807.1408

Comparative Accuracy Analysis of Spectral and Collocation Methods for the Fractional Bagley–Torvik Equation: A Systematic Numerical Study

Control and Optimization in Applied Mathematics
مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 07 خرداد 1405    اصل مقاله (8.49 M)
نوع مقاله: Research Article
شناسه دیجیتال (DOI): 10.30473/coam.2026.77807.1408
نویسندگان
Muayyad Mahmood Khalil* ؛ Najim Abdullah Ibrahim
Department of Mathematics, College of Education for Pure Sciences, Tikrit University, Tikrit, Iraq
چکیده
The Bagley--Torvik equation, which governs the motion of a rigid plate immersed in a Newtonian fluid with viscoelastic damping, represents one of the canonical benchmark problems in fractional calculus. This paper presents a systematic comparative numerical analysis of four established methods for solving this equation: the Fractional-Order Hybrid Jacobi Functions method (FOHJF), the Polynomial Least Squares Method (PLSM), the Vieta-Lucas Spectral Method (VLSM), and the Cubic Spline Collocation Method (CSCM). Three benchmark test problems with qualitatively distinct forcing functions---polynomial, oscillatory (cosine), and exponential---and varying initial conditions are used to evaluate absolute approximation errors at resolution levels N = 8, 16, 32, and 64. Detailed error tables and graphical convergence analyses are provided. The results consistently demonstrate that VLSM achieves the highest accuracy, with maximum absolute errors below 2.3×10⁻⁸ at N = 32, followed by FOHJF. PLSM and CSCM offer simpler implementation at the cost of reduced accuracy. Practical recommendations are provided for selecting a method based on the required precision level and the type of forcing function. The study identifies key limitations and directions for future work, including extension to nonlinear formulations, variable-order derivatives, and adaptive hybrid approaches.

تازه های تحقیق
  • Four numerical methods (VLSM, FOHJF, PLSM, CSCM) are systematically benchmarked for the Bagley–Torvik fractional equation.
  • Three benchmark test problems with polynomial, oscillatory, and exponential exact solutions are used for evaluation.
  • VLSM achieves the highest accuracy with maximum absolute errors below 2.3×10⁻⁸ at N = 32.
  • Convergence studies across resolutions N = 8, 16, 32, 64 confirm exponential convergence of spectral methods.
  • Practical guidelines for selecting methods are provided based on the required accuracy and the forcing function.
کلیدواژه‌ها
Bagley--Torvik equation؛ Caputo fractional derivative؛ Spectral methods؛ Collocation methods؛ Vieta-Lucas polynomials؛ Fractional-order Jacobi functions؛ Error analysis
مراجع

[1] Arqub, O.A., Rabah, A.B., Momani, S. (2023). “A spline construction scheme for numerically solving fractional Bagley-Torvik and Painlevé models correlating initial value problems concerning the Caputo-Fabrizio derivative approach”. International Journal of Modern Physics C, 34(9), Article 2350115. https://doi.org/10.1142/ S0129183123501152

[2] Atanackovic, T.M., Zorica, D. (2013). “On the Bagley-Torvik equation”. Journal of Applied Mechanics, 80(4), Article 041013. https://doi.org/10.1115/1.4007850

[3] Baleanu, D., Jajarmi, A., Sajjadi, S.S., Mozyrska, D. (2019). “A new fractional model and optimal control of a tumor-immune surveillance with non-singular derivative operator”. Chaos, 29, Article 083127. https://doi.org/10.1063/1.5096159

[4] Bansal, M.K., Jain, R. (2016). “Analytical solution of Bagley Torvik equation by generalize differential transform”. International Journal of Pure and Applied Mathematics, 110(2), 265–273. https://doi.org/10.12732/ijpam.v110i2.3

[5] Barary Darzinaghib, Z., Yazdani Cherati, A.B., Nemati, S. (2024). “An efficient numerical scheme for solving a general class of fractional differential equations via fractional-order hybrid Jacobi functions”. Communications in Nonlinear Science and Numerical Simulation, 128, Article 107599. https://doi.org/10.1016/j.cnsns.2023.107599

[6] Bhrawy, A., Zaky, M. (2016). “A fractional-order Jacobi Tau method for a class of timefractional PDEs with variable coefficients”. Mathematical Methods in the Applied Sciences, 39(7), 1765–1779. https://doi.org/10.1002/mma.3600

[7] Bota, C., Căruntu, B., Pașca, M.S., Țucu, D., Lăpădat, M. (2020). “Least squares differential quadrature method for the generalized Bagley–Torvik fractional differential equation”. Mathematical Problems in Engineering, 2020, Article 4806387. https://doi.org/10. 1155/2020/4806387

[8] Çenesiz, Y., Keskin, Y., Kurnaz, A. (2010). “The solution of the Bagley-Torvik equation with the generalized Taylor collocation method”. Journal of the Franklin Institute, 347(2), 452–466. https://doi.org/10.1016/j.jfranklin.2009.10.007

[9] Diethelm, K. (2010). The analysis of fractional Differential Equations: An application-oriented exposition using differential operators of Caputo type. Springer. https://doi. org/10.1007/978-3-642-14574-2

[10] Diethelm, K., Ford, N.J. (2002). “Numerical solution of the Bagley–Torvik equation”. BIT Numerical Mathematics, 42(3), 490–507. https://doi.org/10.1023/A: 1021973025166

[11] Doha, E.H., Bhrawy, A.H., Ezz-Eldien, S.S. (2011). “A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order”. Computers & Mathematics with Applications, 62(5), 2364–2373. https://doi.org/10. 1016/j.camwa.2011.07.024

[12] Ebrahimzadeh, A., Jajarmi, A., Yavuz, M. (2025). “Fractional optimal control of anthroponotic cutaneous leishmaniasis with behavioral and epidemiological extensions”. Mathematical and Computational Applications, 30(6), 122. https://doi.org/10. 3390/mca30060122

[13] Garrappa, R. (2015). “Numerical evaluation of two and three-parameter Mittag-Leffler functions”. SIAM Journal on Numerical Analysis, 53(3), 1350–1369. https://doi.org/ 10.1137/140971191

[14] Ghoreishi, F., Mokhtary, P. (2014). “Spectral collocation method for multi-order fractional differential equations”. International Journal of Computational Methods, 11(5), Article 1350072. https://doi.org/10.1142/S0219876213500722

[15] Gülsu, M., Öztürk, Y., Anapali, A. (2017). “Numerical solution of the fractional Bagley-Torvik equation arising in fluid mechanics”. International Journal of Computer Mathematics, 94(1), 173–184. https://doi.org/10.1080/00207160.2015.1099633

[16] Jajarmi, A. (2026). “Dynamics, stability, and optimal chaos control of a ψ-Caputo cancer-immune system”. International Journal of Dynamics and Control, 14, Article 33. https: //doi.org/10.1007/s40435-025-01974-2

[17] Jena, R.M., Chakraverty, S. (2019). “Analytical solution of Bagley–Torvik equations using Sumudu transformation method”. SN Applied Sciences, 1(3), Article 269. https: //doi.org/10.1007/s42452-019-0259-0

[18] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations. Elsevier. https://doi.org/10.1016/S0304-0208(06) 80001-0

[19] Krishnasamy, V.S., Razzaghi, M. (2016). “The numerical solution of the Bagley-Torvik equation with fractional Taylor methods”. Journal of Computational and Nonlinear Dynamics, 11(5), Article 051010. https://doi.org/10.1115/1.4032390

[20] Mainardi, F. (2010). Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press. https://doi.org/10.1142/ P614

[21] Mekkaoui, T., Hammouch, Z. (2012). “Approximate analytical solutions to the Bagley–Torvik equation by the fractional iteration method”. Annals of the University of Craiova, Mathematics and Computer Science Series, 39(2), 251–256. https://doi.org/10. 52846/ami.v39i2.476

[22] Momani, S., Odibat, Z. (2007). “Numerical comparison of methods for solving linear differential equations of fractional order”. Chaos, Solitons & Fractals, 31(5), 1248–1255. https://doi.org/10.1016/j.chaos.2005.10.068

[23] Noor, Z.A., Talib, I., Abdeljawad, T., Alqudah, M.A. (2022). “Numerical study of Caputo fractional-order differential equations by developing new operational matrices of Vieta– Lucas polynomials”. Fractal and Fractional, 6(2), 79. https://doi.org/10.3390/ fractalfract6020079

[24] Podlubny, I. (1999). Fractional Differential Equations. Academic Press. https://doi. org/10.1016/s0076-5392(99)x8001-5

[25] Raja, M.A.Z., Samar, R., Manzar, M.A., Shah, S.M. (2017). “Design of unsupervised fractional neural network model optimized with interior point algorithm for solving Bagley– Torvik equation”. Mathematics and Computers in Simulation, 132, 139–158. https: //doi.org/10.1016/j.matcom.2016.08.002

[26] Rawashdeh, E.A. (2006). “Numerical solution of fractional integro-differential equations by collocation method”. Applied Mathematics and Computation, 176(1), 1–6. https:// doi.org/10.1016/j.amc.2005.09.059

[27] Ray, S.S., Bera, R.K. (2005). “Analytical solution of the Bagley-Torvik equation by Adomian decomposition method”. Applied Mathematics and Computation, 168(1), 398–410. https://doi.org/10.1016/j.amc.2004.09.006

[28] Saadatmandi, A., Dehghan, M. (2010). “A new operational matrix for solving fractional-order differential equations”. Computers & Mathematics with Applications, 59(3), 1326– 1336. https://doi.org/10.1016/j.camwa.2009.07.006

[29] Salati, S., Matinfar, M., Jafari, H. (2023). “A numerical approach for solving Bagley–Torvik and fractional oscillation equations”. Advanced Mathematical Models & Applications, 8(2), 241–252.

[30] Srivastava, H.M., Jena, R.M., Chakraverty, S., Jena, S.K. (2020). “Dynamic response analysis of fractionally-damped generalized Bagley-Torvik equation subject to external loads”. Russian Journal of Mathematical Physics, 27(2), 254–268. https://doi.org/ 10.1134/S1061920820020120

[31] Torvik, P.J., and Bagley, R.L. (1984). “On the appearance of the fractional derivative in the behavior of real materials”. Journal of Applied Mechanics, 51(2), 294–298. https: //doi.org/10.1115/1.3167615

[32] Verma, A., Kumar, M. (2021). “Numerical solution of Bagley–Torvik equations using Legendre artificial neural network method”. Evolving Systems, 14(4), 2027–2037. https://doi.org/10.1007/s12065-020-00481-x

[33] Yüzbaşı, Ş. (2013). “Numerical solution of the Bagley-Torvik equation by the Bessel collocation method”. Mathematical Methods in the Applied Sciences, 36(3), 312–320. https://doi.org/10.1002/mma.2588

[34] Yüzbaşı, Ş., Yıldırım, G. (2022). “Numerical solutions of the Bagley–Torvik equation by using generalized functions with fractional powers of Laguerre polynomials”. International Journal of Nonlinear Sciences and Numerical Simulation, 24(3), 1003–1021. https://doi.org/10.1515/ijnsns-2021-0120

[35] Zolfaghari, M., Ghaderi, R., Sheikhol-Eslami, A., Ranjbar, A., Hosseinnia, S.H., Momani, S., Sadati, J. (2009). “Application of the enhanced homotopy perturbation method to solve the fractional-order Bagley–Torvik differential equation”. Physica Scripta, T136, Article 014032. https://doi.org/10.1088/0031-8949/2009/T136/014032

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