In this paper, the eigenvalues and corresponding eigenfunctions of a fractional order Sturm-Liouville problem (FSLP) are approximated by using the fractional differential transform method (FDTM), which is a generalization of the differential transform method (DTM). FDTM reduces the proposed fourth-order FSLP to a system of algebraic equations. The resulting coefficient matrix defines a characteristic polynomial which its roots correspond to the eigenvalues of FSLP. The obtained numerical results which are compared with the results of other papers confirm the efficiency of the method.
[1] S. Abbasbandy and A. Shirzadi, Homotopy analysis method for multiple solutions of the fractional sturm-liouville problems, Numer. Algorithms, 54(4) (2010), 521–532.
[2] R. Agarwal, M. Benchohra, and S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math., 109(3) (2010), 1033.
[3] A. J. Akbarfam and A. Mingarelli, Duality for an indefnite inverse sturm-liouville problem, J. Math. Anal. Appl., 312(2) (2005), 435–463.
[4] P. Antunes and R. Ferreira, An augmented-rbf method for solving fractional sturm-liouville eigenvalue problems, SIAM J. Sci. Comput., 37(1) (2017), A515–A535.
[5] Q. Al-Mdallal, M. Al-Refai, M. Syam, and M. Sirhin, Theoretical and computational perspectives on the eigenvalues of fourth order fractional sturm-liouville problem, International Journal of Computer Mathematics, (2017), 1–18.
[6] Q. Al-Mdallal, On the numerical solution of fractional sturm-liouville problems, Int. J. Comput. Math., 87(12) (2010), 2837–2845.
[7] Q. Al-Mdallal, An effcient method for solving fractional sturm-liouville problems, Chaos Solitons Fractals, 40(1) (2009), 183–189.
[8] M. Al-Refai, Basic results on nonlinear eigenvalue problems of fractional order, Electron. J. Differential Equations, 2012(191) (2012), 1–12.
[9] A. Arikoglu and I. Ozkol, Solution of fractional differential equation by using differential transform method, Chaos Solitons Fract. 34 (2007), 1473–1481.
[10] A. M. F. V. Atkinson, Multiparameter Eigenvalue Problems: Sturm-Liouville Theory, CRC Press, 2010.
[11] B. S. Attili and D. Lesnic, An efcient method for computing eigen elements of Sturm-Liouville fourth-order boundary value problems, Appl. Math. Comput. 182 (2006), 1247–1254.
[12] M. Bahmanpour, M. Tavassoli Kajani, and M. Maleki, Solving Fredholm integral equations of the first kind using Mu¨ntz wavelets, Applied Numerical Mathematics, 143 (2019), 159–171.
[13] M. Bahmanpour, M. Tavassoli Kajani, and M. Maleki, A Mu¨ntz wavelets collocation method for solving fractional differential equations, Computational and Applied Mathematics, 37(4) (2018), 5514–5526.
[14] E. Bas and F. Metin, Spectral analysis for fractional hydrogen atom equation, Adv. Pure Appl. Math. 5(13) (2015), 767.
[15] J. Biazar, M. Dehghan, and T. Houlari, An efficient method to approximate eigenvalues and eigenfunctions of high order Sturm-Liouville problems, Computational Methods for Differential Equations, 8(2) (2020), 389-400.
[16] J. Biazar and M. Eslami, Analiytic solution of Telegraph equation by differential transform method, Phys. Lett. A, 374 (2010), 2904–2906.
[17] A. Cetinkaya and O. Kiymaz, The solution of the time-fractional diffusion equation by the generalize differential transform method, Math. Comput. Model., 57 (2013), 2349–2354.
[18] B. Chanane, Acurate solutions of fourth order sturm-liouville problems, Journal of computational and Applied Mathematics, 234 (2010), 3064–3071.
[19] C. K. Chen and S. H. Ho, Solving partial differential equations by two-dimensional differential transform method, Appl. Math. Comput., 106 (1999), 171–179.
[20] M. Dehghan and A. Mingarelli, Fractional sturm-liouville eigenvalue problems, i, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM, 114(2) (2020), 1–15.
[21] G. Fix and J. Roof, Least squares fnite-element solution of a fractional order two-point boundary value problem, Comput. Math. Appl., 48(7-8) (2004), 1017–1033.
[22] M. Garg, P. Manohar, and S.L. Kalla, Generalized differential transform method to space- time fractional telegraph equation, International Journal of Differential equations, 2011 (2011), 1–9.
[23] M. Ghasemi, E. Babolian, and M. Tavassoli Kajani, Numerical solution of linear Fredholm integral equations using sine–cosine wavelets, International Journal of Computer Mathematics, 84(7) (2007), 979–987.
[24] E. F. D. Goufo and S. Kumar, Shallow water wave models with and without singular kernel:existenc, uniqueness, and similarities, Math. Probl. Eng., 2017 (2017), 1–9.
[25] R. M. Hani, Existence and uniqueness of the solution for fractional sturm-liouville boundary value problem, Coll. Basic Educ. Res. J., 11(2) (2011), 698–710.
[26] B. Jin and W. Rundell, An inverse sturm-liouville problem with a fractional derivative, J. Comput. Phys., 231(14) (2012), 4954–4966.
[27] A. Kılı¸cman, I. Hashim, M. Tavassoli Kajani, and M. Maleki, On the rational second kind Chebyshev pseudospectral method for the solution of the Thomas–Fermi equation over an infinite interval, Journal of Computational and Applied Mathematics, 257 (2014), 79–85.
[28] M. Klimek and M. Blasik, Regular sturm-liouville problem with riemann-liouville derivatives of order in (1, 2): discrete spectrum, solutions and applications, in: Advances in Modelling and Control of Non-Integer-Order Systems, (2015), 25–36.
[29] W. Luo, T. Huang, and G. Wu, X. Gu, Quadratic spline collocation method for the time fractional subdiffusion equation, Appl. Math. Comput., 276 (2016), 252–265.
[30] A. Ma, E. Sayed, and F. Gaafar, Existence and uniqueness of solution for sturm-liouville fractional differential equation with multi-point boundary condition via caputo derivative, Adv. Difference Equ., 2019(1) (2019), 1–17.
[31] M. Maleki and M. Tavassoli Kajani, Numerical approximations for Volterra’s population growth model with frac- tional order via a multi-domain pseudospectral method, Applied Mathematical Modelling, 39(15) (2015), 4300– 4308.
[32] S. Mohammadian, Y. Mahmoudi, and F. D. Saei, Numerical solution of fractional multi-delay differential equa- tions, Int. J. Appl. Comput. Math., 8(71) (2022).
[33] S. Mohammadian, Y. Mahmoudi, and F. D. Saei, Analytical approximation of time-fractional telegraph equation with Riesz space-fractional derivative, Differential Equations & Applications, 12(3) (2020), 243–258.
[34] S. Mohammadian, Y. Mahmoudi, and F. D. Saei, Solution of fractional telegraph equation with Riesz space- fractional derivative, AIMS Mathematics, 4(6)(2019), 1664–1683.
[35] Z. Odibat, S. Momani, and V. S. Erturk, Generalized differential transform method: Application to differential equations of fractional order, Appl. Math. Comput., 197 (2008), 467–477.
[36] Z. Odibat and S. Momani, A generalized differential transform method for linear partial differential equations of fractional order, Appl. Math. Lett., 21 (2008), 194–199.
[37] I. Podlubny, Fractional differential equations, San Diego, Academic Press, (1999).
[38] M. Rasouli Gandomani and M. Tavassoli Kajani, Numerical solution of a fractional order model of HIV infection of CD4+ T cells using Mu¨ntz-Legendre polynomials, International Journal Bioautomation, 20(2) (2016), 193.
[39] S. S. Ray, Numerical solution and solitary wave solutions of fractional KDV equations using modified fractional reduced differential transform method, Computational Mathematics and Mathematical Physics, 53 (2013), 1870– 1881.
[40] Y. Rossikhin and M. Shitikova, Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results, Appl. Mech. Rev., 63(1) (2010), 010801.
[41] M. K. Sadabad and A. J. Akbarfam, An efficient numerical method for estimating eigenvalues and eigenfunctions of fractional sturm-liouville problems, Mathematics and Computers in Simulation 185 (2021), 547–569.
[42] V. K. Sirvastava, M. K. Awasthi, and R. K. Chaurasia, Reduced differential transform method to solve two and three dimensional second order hyperbolic telegraph equations, Journal of King Saud University: Engineering Sciences, 29 (2017), 166–171.
[43] B. Soltanalizadeh, Differential transform method for solving one-space-dimensional telegraph equation, Comput. Appl. Math. 30 (2011), 639–653.
[44] A. H. S. Taher and A. Malek, An efficient algorithm for solving high-order Sturm-Liouville problems using vari- ational iteration method, Fixed Point Theory, 14 (2013), 193–210.
[45] S. Zhang, Existence of solution for a boundary value problem of fractional order, Acta Math. Sci., 26(2) (2006), 220–228.
[46] J. K. Zhou, Differential Transformation and Its Application for Electrical Circuits, Wuhan, China: Huazhong University Press, 1986.
[47] L. Zou, Z. Wang, and Z. Zong, Generalized differential transform method to differential-difference equation, Phys. Lett. A, 373 (2009), 4142–4151.
Aghazadeh, A., & Mahmoudi, Y. (2023). On approximating eigenvalues and eigenfunctions of fractional order Sturm-Liouville problems. Computational Methods for Differential Equations, 11(4), 811-821. doi: 10.22034/cmde.2022.52790.2221
MLA
Arezu Aghazadeh; Yaghoub Mahmoudi. "On approximating eigenvalues and eigenfunctions of fractional order Sturm-Liouville problems". Computational Methods for Differential Equations, 11, 4, 2023, 811-821. doi: 10.22034/cmde.2022.52790.2221
HARVARD
Aghazadeh, A., Mahmoudi, Y. (2023). 'On approximating eigenvalues and eigenfunctions of fractional order Sturm-Liouville problems', Computational Methods for Differential Equations, 11(4), pp. 811-821. doi: 10.22034/cmde.2022.52790.2221
VANCOUVER
Aghazadeh, A., Mahmoudi, Y. On approximating eigenvalues and eigenfunctions of fractional order Sturm-Liouville problems. Computational Methods for Differential Equations, 2023; 11(4): 811-821. doi: 10.22034/cmde.2022.52790.2221
)
