An efficient algorithm for computing the eigenvalues of conformable Sturm-Liouville problem

Mirzaei, Hanif; Emami, Mahmood; Ghanbari, Kazem; Shahriari, Mohammad

doi:10.22034/cmde.2023.57436.2403

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	An efficient algorithm for computing the eigenvalues of conformable Sturm-Liouville problem
Computational Methods for Differential Equations
مقاله 4، دوره 12، شماره 3، مهر 2024، صفحه 471-483 اصل مقاله (547.88 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22034/cmde.2023.57436.2403
نویسندگان
Hanif Mirzaei^* ¹؛ Mahmood Emami¹؛ Kazem Ghanbari¹؛ Mohammad Shahriari²
¹Faculty of Basic Sciences, Sahand University of Technology, Tabriz, Iran.
²Department of Mathematics, Faculty of Science, University of Maragheh, Maragheh, Iran.
چکیده
In this paper, Computing the eigenvalues of the Conformable Sturm-Liouville Problem (CSLP) of order $2 \alpha$, $\frac{1}{2}<\alpha \leq 1$, and dirichlet boundary conditions is considered. For this aim, CSLP is discretized to obtain a matrix eigenvalue problem (MEP) using finite element method with fractional shape functions. Then by a method based on the asymptotic form of the eigenvalues, we correct the eigenvalues of MEP to obtain efficient approximations for the eigenvalues of CSLP. Finally, some numerical examples to show the efficiency of the proposed method are given. Numerical results show that for the $n$th eigenvalue, the correction technique reduces the error order from $O(n^4h^2)$ to $O(n^2h^2)$.
کلیدواژه‌ها
Sturm-Liouville problem؛ Conformable derivative؛ Finite elements method؛ Correction idea

مراجع
[1] T. Abdeljawad, On conformable fractional calculus, J Comput Appl Math., 279 (2015), 57-66. [2] B. P. Allahverdiev and H. Tuna, and Yal¸cınkaya, Spectral expansion for singular conformable Sturm-Liouville problem, Math. Commun., 25 (2020), 237-252. [3] D. R. Anderson and D. J. Ulness, Properties of the Katugampola fractional derivative with potential application in quantum mechanics, J Math Phys., 56 (2015), 063502. [4] A. L. Andrew and J. W. Paine, Correction of finite element estimates for Sturm-Liouville eigenvalues, Numer. Math., 50 (1986), 205-215. [5] A. L. Andrew and J. W. Paine, Correction of Numerov’s eigenvalue estimates, Numer. Math., 47 (1985), 289-300. [6] A. Atangana, D. Baleanu, and A. Alsaedi, New properties of conformable derivative, Open Math., 13 (2015), 889-898. [7] D. Avci, B. Iskender Eroglu, and N. Ozdemir, The Dirichlet problem of a conformable advection diffusion equation, Thermal Sci., 21 (2017), 9-18. [8] N. Benkhettou, S. Hassani, and D. F. M. Torres, A conformable fractional calculus on arbitrary time scales, J King Saud Univ Sci., 28 (2016), 93-98. [9] Y. Cenesiz, A. Kurt, and E. Nane, Stochastic solutions of conformable fractional Cauchy problems, Statist Probab Lett. 124 (2017), 126-131. [10] M. Ciesielski, M. Klimek, and T. Blaszczyk, The fractional Sturm-Liouville problem-Numerical approximation and application in fractional diffusion, Journal of computational and applied mathematics., 317 (2017), 573-588. [11] W. S. Chung, Fractional Newton mechanics with conformable fractional derivative, J Comput Appl Math. 290 (2015), 150-158. [12] M. Dehghan and A. B. Mingarelli, Fractional Sturm-Liouville eigenvalue problems, I. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matem´aticas, 114(46) (2020). [13] M. H. Derakhshan and A. Ansari, Numerical approximation to Prabhakar fractional Sturm-Liouville problem, Computational and Applied Mathematics, 38(71) (2019), 1-20. [14] A. Ebaid, B. Masaedeh, and E. El-Zahar, A new fractional model for the falling body problem, Chin Phys Lett. 34 (2017), 020201-1. [15] G. M. L. Gladwell, Inverse problem in vibration, Kluwer academic publishers, New York, 2004. [16] T. Gulshen, E. Yilmaz, and H. Kemaloglu, Conformable fractional Sturm-Liouville equation and some existence results on time scales, Turkish Journal of Mathematics, 42 (2018), 1348-1360. [17] H. Hochstadt, Asymptotic estimates for the Sturm-Liouville spectrum, Communications on pure and applied mathematics, XIV (1961), 740-764. [18] O. Iyiola, O. Tasbozan, and A. Kurt, On the analytical solutions of the system of conformable time-fractional Robertson equations with 1-D diffusion, Chaos Solitons Fractals., 94 (2017), 1-7. [19] Z. Kavooci, K. Ghanbari, and H. Mirzaei, New form of Laguerre Fractional Differential Equation and Applications, Turk J Math, 46 (2022), 2998-3010. [20] R. Khalil, M. Al Horani, and A. Yousef, new definition of fractional derivative, J Comput Appl Math. 264 (2014), 65-70. [21] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer-Verlag, New York, 1996. [22] M. Klimek and O. P. Agrawal, Fractional Sturm-Liouville problem, Comput Math Appl., 66 (2013), 795-812. [23] M. Klimek, M. Ciesielski, and T. Blaszczyk, Exact and numerical solutions of the fractional Sturm-Liouville problem, Fractional calculus and applied analysis, 21(1) (2018), 45-71. [24] M. J. Lazo and D. F. M. Torres, Variational calculuswith conformable fractional derivatives, IEEE/CAA J Automat Sinica., 4 (2017), 340-352. [25] V. Ledoux, M. V. Daele, and G. V. Berghe, Effcient numerical solution of the 1D schr¨odinger problem using magnus integrators, IMA journal of numerical analysis, 30 (2010), 751-776. [26] V. Ledoux, M. V. Daele, and G. V. Berghe, Matslise: A matlab package for the numerical solution of SturmLiouville and schr¨odinger equations, ACM transactions on mathematical software, 31 (2005), 532-554. [27] A. B. Makhlouf, O. Naifar, and M. A. Hammami, FTS and FTB of conformable fractional order linear systems, Math Probab Eng., 5 (2018), 2572986. [28] H. Mirzaei, Computing the eigenvalues of fourth order Sturm-Liouville problems with Lie Group method, J Numer Anal Optim. 7(1) (2017), 1-12. [29] H. Mirzaei, K. Ghanbari, and M. Emami, Direct and inverse problems of string equation by Numerov’s method, Iranian Journal of Science, 47 (2023), 871-884. [30] H. Mortazaasl and A. Jodayree Akbarfam, Trace formula and inverse nodal problem for a conformable fractional Sturm-Liouville problem, Inverse Probl Sci Eng., 28(4) (2020), 524-555. [31] E. R. Nwaeze and D. F. M. Torres, Chain rules and inequalities for the BHT fractional calculus on arbitrary timescales, Arab J Math., 6 (2017), 13-20. [32] A. S. Ozkan and I. Adalar, Inverse problems for a conformable fractional Sturm-Liouville operator, (2019) arXiv: 1908.03457. [33] A. P´alfalvi, Efcient solution of a vibration equation involving fractional derivatives, Int J Nonlin Mech., 45 (2010), 169-175. [34] J. W. Paine, F. R. Hoog, and R. S. Anderssen, On the correction of finite difference eigenvalue approximations for Sturm-Liouville problems, Computing, 26 (1981), 123-139. [35] M. Shahriari and H. Mirzaei, Inverse Sturm-Liouville problem with conformable derivative and transmission conditions, Hacet. J. Math. Stat. 52 (2023), 753 -767. [36] M. Shahriari, B. Nemati, B. Mohammadalipour, and S. Saeidian, Pseudospectral method for solving the fractional one-dimensional Dirac operator using Chebyshev cardinal functions, Physica Scripta, 98 (2023), 055205. [37] G. Teschl, Mathematical Methods in Quantum Mechanics, With Applications to Schr¨odinger Operators, Graduate Studies in Mathematics, Amer. Math. Soc., Rhode Island, 2009. [38] D. Zhao and M. Luo, General conformable fractional derivative and its physical interpretation, Calcolo. 54 (2017), 903-917. [39] H. W. Zhou, S. Yang, and S. Q. Zhang, Conformable derivative approach to anomalous diffusion, Phys A., 491 (2018), 1001-1013. [40] M. Zayernouri and G. E. Karniadakis, Fractional Sturm-Liouville eigen-problems: Theory and numerical approximation, J Comput Phys., 252 (2013), 495-517.
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An efficient algorithm for computing the eigenvalues of conformable Sturm-Liouville problem