[1] Abbasbandy, S. and Shirzadi, A. Homotopy analysis method for multiplesolutions of the fractional Sturm–Liouville problems, Numer. Algorithms 54 (4) (2010) 521–532.
[2] Agarwal, R., Benchohra, M. and Hamani, S. A survey on existence re-sults for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math. 109(3) (2010) 973–1033.
[3] Akbarfam, A.J. and Mingarelli, A. Duality for an indefinite inverse Sturm–Liouville problem, J. Math. Anal. Appl. 312(2) (2005) 435–463.
[4] Al-Mdallal, Q. An eﬀicient method for solving fractional Sturm–Liouville problems, Chaos Solitons Fractals 40(1) (2009) 183–189.
[5] Al-Mdallal, Q. On the numerical solution of fractional Sturm–Liouville problems, Int. J. Comput. Math. 87(12) (2010) 837–2845.
[6] Al-Refai, M. Basic results on nonlinear eigenvalue problems of fractional order, Electron. J. Differential Equations 2012, No. 191, 12 pp.
[7] Antunes, P. and Ferreira, R.A. An augmented-RBF method for solving fractional Sturm-Liouville eigenvalue problems, SIAM J. Sci. Comput. 37(1) (2017) A515–A535.
[8] Asl, M. and Javidi, M. An improved pc scheme for nonlinear fractional differential equations: Error and stability analysis, J. Comput. Appl. Math. 324 (2017) 101–117.
[9] Atkinson, F.V. and Mingarelli, A.B. Multiparameter eigenvalue prob-lems: Sturm-Liouville theory, CRC Press, 2010.
[10] Bas, E. and Metin, F. Spectral analysis for fractional hydrogen atom equation, Adv. Pure Appl. Math. 5(13) (2015) 767.
[11] Blaszczyk, T. and Ciesielski, M. Numerical solution of fractional Sturm–Liouville equation in integral form, Fract. Calc. Appl. Anal. 17(2) (2014) 307–320.
[12] Dehghan, M. and Mingarelli, A. Fractional Sturm–Liouville eigenvalue problems, I. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RAC-SAM 114(2) (2020) Paper No. 46, 15 pp.
[13] El-Sayed, A. and Gaafar, F. M. Existence and uniqueness of solution for Sturm–Liouville fractional differential equation with multi-point bound-ary condition via Caputo derivative, Adv. Difference Equ. 2019(1) (2019) 1–17.
[14] Fix, G. and Roof, J. Least squares finite-element solution of a fractional order two-point boundary value problem, Comput. Math. Appl. 48(7-8) (2004) 1017–1033.
[15] Hani, R.M. Existence and uniqueness of the solution for fractional sturm–liouville boundary value problem, Coll. Basic Educ. Res. J. 11(2) (2011) 698–710.
[16] Isah, A. and Phang, C. Operational matrix based on Genocchi polyno-mials for solution of delay differential equations, Ain Shams Eng. J. 9(4) (2018) 2123–2128.
[17] Isah, A., Phang, C. and Phang, P. Collocation method based on Genocchi operational matrix for solving generalized fractional pantograph equa-tions, Int. J. Differ. Equ. 2017, Art. ID 2097317, 10 pp.
[18] Jin, B., Lazarov, R., Pasciak, J. and Rundell, W. A finite element method for the fractional Sturm–Liouville problem, arXiv preprint arXiv:1307.5114 (2013).
[19] Jin, B. and Rundell, W. An inverse Sturm–Liouville problem with a fractional derivative, J. Comput. Phys. 231(14) (2012) 4954–4966.
[20] Kexue, L. and Jigen, P. Laplace transform and fractional differential equations, Appl. Math. Lett. 24(12) (2011) 2019–2023.
[21] Klimek, M. and Blasik, M. 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, Springer, Cham.
[22] Luchko, Y. Initial–boundary-value problems for the one-dimensional time-fractional diffusion equation, Fract. Calc. Appl. Anal. 15(1) (2012) 141–160.
[23] Luo, W., Huang, T., Wu, G. and Gu, X. Quadratic spline collocation method for the time fractional subdiffusion equation, Appl. Math. Com-put. 276 (2016) 252–265.
[24] Luo, W., Li, C., Huang, T., Gu, X. and Wu, G. A high-order accu-rate numerical scheme for the Caputo derivative with applications to
fractional diffusion problems, Numer. Funct. Anal. Optim. 39(5) (2018) 600–622.
[25] Metzler, R. and Klafter, J. Boundary value problems for fractional dif-fusion equations, Physica A 278 (1-2) (2000) 107–125.
[26] Metzler, R. and Nonnenmacher, T. Space-and time-fractional diffusion and wave equations, fractional Fokker–Planck equations, and physical motivation, Coll. Basic Educ. Res. J. 284 (1-2) (2002) 67–90.
[27] Mirzaei, H. Computing the eigenvalues of fourth order Sturm–Liouville problems with lie group method, Iranian Journal of Numerical Analysis and Optimization 7(1) (2017) 1–12.
[28] Phang, C., Ismail, N.F., Isah, A., Loh, J.R. A new eﬀicient numerical scheme for solving fractional optimal control problems via a Genocchi operational matrix of integration, J. Vib. Control 24(14) (2018) 3036–3048.
[29] Podlubny, I. Fractional differential equations, Academic Press, San Diego, Calif, USA, 1999.
[30] Rossikhin, Y. and Shitikova, M. Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results, Appl. Mech. Rev. 63 (1) (2010) 010801.
[31] Sadabad, M. K. and Akbarfam, A.J. An eﬀicient numerical method for estimating eigenvalues and eigenfunctions of fractional Sturm–Liouville
problems, Math. Comput. Simulation 185 (2021) 547–569.
[32] Tohidi, E., Bhrawy, A. and Erfani, K. A collocation method based on
Bernoulli operational matrix for numerical solution of generalized pan-
tograph equation, Appl. Math. Model. 37(6) (2013) 4283–4294.
[33] Zayernouri, M. and Karniadakis, G. Fractional Sturm–Liouville eigen-
problems: theory and numerical approximation, J. Comput. Phys. (2013)
495–517.
[34] Zhang, S. Existence of solution for a boundary value problem of fractional
order, Acta Math. Sci. 26(2) (2006) 220–228.
Send comment about this article