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Legendre-Galerkin spectral algorithm for fractional-order BVPs: Application to the Bagley-Torvik equation

S.M. Sayed, A.S. Mohamed, E.M. Abo-Eldahab, Y.H. Youssri

Article ID: 2733
Vol 2, Issue 1, 2024

DOI: https://doi.org/10.54517/mss.v2i1.2733
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Abstract

Herein, we provide an efficient spectral Galerkin algorithm, according to a special type of shifted Legendre basis for finding a semi-analytic solution to the Liouville-Caputo fractional boundary value problem. The algorithm’s main goal is to transform the fractional differential problem into a linear system with efficiently invertible, well-structured matrices. The convergence rates of the algorithm are carefully obtained as well as the error bound.

Keywords

Legendre polynomials; Galerkin method; fractional differential equations; convergence analysis

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References

1. He JH. Fractal calculus and its geometrical explanation. Results in Physics. 2018; 10: 272-276. doi: 10.1016/j.rinp.2018.06.011

2. He JH. Frontier of Modern Textile Engineering and Short Remarks on Some Topics in Physics. International Journal of Nonlinear Sciences and Numerical Simulation. 2010; 11(7). doi: 10.1515/ijnsns.2010.11.7.555

3. Abdelhakem M, Moussa H. Pseudo-spectral matrices as a numerical tool for dealing BVPs, based on Legendre polynomials’ derivatives. Alexandria Engineering Journal. 2023; 66: 301-313. doi: 10.1016/j.aej.2022.11.006

4. Abdelhakem M. Shifted Legendre fractional pseudo-spectral integration matrices for solving fractional Volterra Integro-differential equations and Abel’s integral equations. Fractals. 2023; 31(10). doi: 10.1142/s0218348x23401904

5. Abdelhakem M, Baleanu D, Agarwal P, et al. Approximating system of ordinary differential-algebraic equations via derivative of Legendre polynomials operational matrices. International Journal of Modern Physics C. 2022; 34(03). doi: 10.1142/s0129183123500365

6. Abd-Elhameed WM, Youssri YH. Fifth-kind orthonormal Chebyshev polynomial solutions for fractional differential equations. Computational and Applied Mathematics. 2017; 37(3): 2897-2921. doi: 10.1007/s40314-017-0488-z

7. Youssri YH, Abd‐Elhameed WM. Legendre‐Spectral Algorithms for Solving Some Fractional Differential Equations. Fractional Order Analysis. Published online August 10, 2020: 195-224. doi: 10.1002/9781119654223.ch8

8. Hafez RM, Youssri YH. Spectral Legendre-Chebyshev Treatment of 2D Linear and Nonlinear Mixed Volterra-Fredholm Integral Equation. Mathematical Sciences Letters. 2020; 9(2): 37-47.

9. Hafez RM, Youssri YH. Legendre-Collocation Spectral Solver for Variable-Order Fractional Functional Differential Equations. Comput. Methods Differential Equations. 2020; 8(1): 99-110.

10. Youssri YH, Abd-Elhameed WM. Numerical Spectral Legendre-Galerkin Algorithm for Solving Time Fractional Telegraph Equation. Romanian Journal of Physics. 2018; 63(107):1-16.

11. Torvik PJ, Bagley RL. On the Appearance of the Fractional Derivative in the Behavior of Real Materials. Journal of Applied Mechanics. 1984; 51(2): 294-298. doi: 10.1115/1.3167615

12. Zafar AA, Kudra G, Awrejcewicz J. An Investigation of Fractional Bagley–Torvik Equation. Entropy. 2019; 22(1): 28. doi: 10.3390/e22010028

13. Raja MAZ, Khan JA, Qureshi IM. Solution of Fractional Order System of Bagley‐Torvik Equation Using Evolutionary Computational Intelligence. Chou JH, ed. Mathematical Problems in Engineering. 2011; 2011(1). doi: 10.1155/2011/675075

14. Gülsu M, Öztürk Y, Anapali A. Numerical solution the fractional Bagley–Torvik equation arising in fluid mechanics. International Journal of Computer Mathematics. 2015; 94(1): 173-184. doi: 10.1080/00207160.2015.1099633

15. Pang D, Jiang W, Du J, et al. Analytical solution of the generalized Bagley–Torvik equation. Advances in Difference Equations. 2019; 2019(1). doi: 10.1186/s13662-019-2082-8

16. Ray SS, Bera RK. Analytical solution of the Bagley Torvik equation by Adomian decomposition method. Applied Mathematics and Computation. 2005; 168(1): 398-410. doi: 10.1016/j.amc.2004.09.006

17. Srivastava HM, Shah FA, Abass R. An Application of the Gegenbauer Wavelet Method for the Numerical Solution of the Fractional Bagley-Torvik Equation. Russian Journal of Mathematical Physics. 2019; 26(1): 77-93. doi: 10.1134/s1061920819010096

18. Sayed SM, Mohamed AS, El-Dahab EMA, et al. Alleviated Shifted Gegenbauer Spectral Method for Ordinary and Fractional Differential Equations. Contemporary Mathematics. Published online May 10, 2024: 4123-4149. doi: 10.37256/cm.5220244559

19. Izadi M, Yüzbaşı Ş, Cattani C. Approximating solutions to fractional-order Bagley-Torvik equation via generalized Bessel polynomial on large domains. Ricerche di Matematica. 2021; 72(1): 235-261. doi: 10.1007/s11587-021-00650-9

20. Izadi M, Negar MR. Local discontinuous Galerkin approximations to fractional Bagley‐Torvik equation. Mathematical Methods in the Applied Sciences. Published online January 28, 2020. doi: 10.1002/mma.6233

21. Srivastava HM, Adel W, Izadi M, et al. Solving Some Physics Problems Involving Fractional-Order Differential Equations with the Morgan-Voyce Polynomials. Fractal and Fractional. 2023; 7(4): 301. doi: 10.3390/fractalfract7040301

22. Podlubny I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press; 1998.

23. Abd-Elhameed WM, Youssri YH. Generalized Lucas polynomial sequence approach for fractional differential equations. Nonlinear Dynamics. 2017; 89(2): 1341-1355. doi: 10.1007/s11071-017-3519-9

24. Rainville ED. Special Functions. New York; 1960.

25. Shen J. Efficient Spectral-Galerkin Method I. Direct Solvers of Second- and Fourth-Order Equations Using Legendre Polynomials. SIAM Journal on Scientific Computing. 1994; 15(6): 1489-1505. doi: 10.1137/0915089

26. Ji T, Hou J, Yang C. Numerical solution of the Bagley–Torvik equation using shifted Chebyshev operational matrix. Advances in Difference Equations. 2020; 2020(1). doi: 10.1186/s13662-020-03110-0

27. Diethelm K, Ford NJ, Freed AD. Detailed Error Analysis for a Fractional Adams Method. Numerical Algorithms. 2004; 36: 31-52.

28. Yüzbaşı Ş. Numerical solution of the Bagley–Torvik equation by the Bessel collocation method. Mathematical Methods in the Applied Sciences. 2012; 36(3): 300-312. doi: 10.1002/mma.2588

29. Keskin Y, Karaoğlu O, Servi S. The Approximate Solution of High-Order Linear Fractional Differential Equations with Variable Coefficients in Terms of Generalized Taylor Polynomials. Mathematical and Computational Applications. 2011; 16(3): 617-629. doi: 10.3390/mca16030617

30. ur Rehman M, Khan RA. A numerical method for solving boundary value problems for fractional differential equations. Applied Mathematical Modelling. 2012; 36(3): 894-907. doi: 10.1016/j.apm.2011.07.045