# American Institute of Mathematical Sciences

July  2021, 14(7): 2417-2434. doi: 10.3934/dcdss.2020171

## A Sturm-Liouville approach for continuous and discrete Mittag-Leffler kernel fractional operators

 1 Mechatronic Engineering Department, University of Turkish Aeronautical Association, 06790, Ankara, Turkey 2 Department of Mathematics and General Sciences, Prince Sultan University, P. O. Box 66833, 11586 Riyadh, Saudi Arabia 3 Department of Medical Research, China Medical University, Taichung 40402, Taiwan 4 Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan 5 Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE, 68588-0130, USA

Received  March 2019 Revised  April 2019 Published  July 2021 Early access  December 2019

In this work, we use integration by parts formulas derived for fractional operators with Mittag-Leffler kernels to formulate and investigate fractional Sturm-Liouville Problems ($FSLPs$). We analyze the self-adjointness, eigenvalue and eigenfunction properties of the associated Fractional Sturm-Liouville Operators ($FSLOs$). The discrete analogue of the obtained results is formulated and analyzed by following nabla analysis.

Citation: Raziye Mert, Thabet Abdeljawad, Allan Peterson. A Sturm-Liouville approach for continuous and discrete Mittag-Leffler kernel fractional operators. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2417-2434. doi: 10.3934/dcdss.2020171
##### References:

show all references

##### References:
 [1] Jean Daniel Djida, Juan J. Nieto, Iván Area. Parabolic problem with fractional time derivative with nonlocal and nonsingular Mittag-Leffler kernel. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 609-627. doi: 10.3934/dcdss.2020033 [2] Mehmet Yavuz, Necati Özdemir. Comparing the new fractional derivative operators involving exponential and Mittag-Leffler kernel. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 995-1006. doi: 10.3934/dcdss.2020058 [3] Ndolane Sene. Mittag-Leffler input stability of fractional differential equations and its applications. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 867-880. doi: 10.3934/dcdss.2020050 [4] Antonio Coronel-Escamilla, José Francisco Gómez-Aguilar. A novel predictor-corrector scheme for solving variable-order fractional delay differential equations involving operators with Mittag-Leffler kernel. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 561-574. doi: 10.3934/dcdss.2020031 [5] Moulay Rchid Sidi Ammi, Mostafa Tahiri, Delfim F. M. Torres. Necessary optimality conditions of a reaction-diffusion SIR model with ABC fractional derivatives. Discrete and Continuous Dynamical Systems - S, 2022, 15 (3) : 621-637. doi: 10.3934/dcdss.2021155 [6] Behzad Ghanbari, Devendra Kumar, Jagdev Singh. An efficient numerical method for fractional model of allelopathic stimulatory phytoplankton species with Mittag-Leffler law. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3577-3587. doi: 10.3934/dcdss.2020428 [7] Günter Leugering, Gisèle Mophou, Maryse Moutamal, Mahamadi Warma. Optimal control problems of parabolic fractional Sturm-Liouville equations in a star graph. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022015 [8] Russell Johnson, Luca Zampogni. On the inverse Sturm-Liouville problem. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 405-428. doi: 10.3934/dcds.2007.18.405 [9] Ricardo Almeida, M. Luísa Morgado. Optimality conditions involving the Mittag–Leffler tempered fractional derivative. Discrete and Continuous Dynamical Systems - S, 2022, 15 (3) : 519-534. doi: 10.3934/dcdss.2021149 [10] Chuan-Fu Yang, Natalia Pavlovna Bondarenko. A partial inverse problem for the Sturm-Liouville operator on the lasso-graph. Inverse Problems and Imaging, 2019, 13 (1) : 69-79. doi: 10.3934/ipi.2019004 [11] Muhammad Bilal Riaz, Syed Tauseef Saeed. Comprehensive analysis of integer-order, Caputo-Fabrizio (CF) and Atangana-Baleanu (ABC) fractional time derivative for MHD Oldroyd-B fluid with slip effect and time dependent boundary condition. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3719-3746. doi: 10.3934/dcdss.2020430 [12] Qiushuang Wang, Run Xu. A review of definitions of fractional differences and sums. Mathematical Foundations of Computing, 2022  doi: 10.3934/mfc.2022013 [13] N. A. Chernyavskaya, L. A. Shuster. Spaces admissible for the Sturm-Liouville equation. Communications on Pure and Applied Analysis, 2018, 17 (3) : 1023-1052. doi: 10.3934/cpaa.2018050 [14] Guglielmo Feltrin. Multiple positive solutions of a sturm-liouville boundary value problem with conflicting nonlinearities. Communications on Pure and Applied Analysis, 2017, 16 (3) : 1083-1102. doi: 10.3934/cpaa.2017052 [15] Chuan-Fu Yang, Natalia Pavlovna Bondarenko, Xiao-Chuan Xu. An inverse problem for the Sturm-Liouville pencil with arbitrary entire functions in the boundary condition. Inverse Problems and Imaging, 2020, 14 (1) : 153-169. doi: 10.3934/ipi.2019068 [16] Fahd Jarad, Thabet Abdeljawad. Generalized fractional derivatives and Laplace transform. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 709-722. doi: 10.3934/dcdss.2020039 [17] Peter Howard, Alim Sukhtayev. The Maslov and Morse indices for Sturm-Liouville systems on the half-line. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 983-1012. doi: 10.3934/dcds.2020068 [18] Fahd Jarad, Thabet Abdeljawad. Variational principles in the frame of certain generalized fractional derivatives. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 695-708. doi: 10.3934/dcdss.2020038 [19] Ebenezer Bonyah, Samuel Kwesi Asiedu. Analysis of a Lymphatic filariasis-schistosomiasis coinfection with public health dynamics: Model obtained through Mittag-Leffler function. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 519-537. doi: 10.3934/dcdss.2020029 [20] Daomin Cao, Guolin Qin. Liouville type theorems for fractional and higher-order fractional systems. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2269-2283. doi: 10.3934/dcds.2020361

2020 Impact Factor: 2.425