A Conformable Inverse Problem with Constant Delay
DOI:
https://doi.org/10.15377/2409-5761.2023.10.3Keywords:
Spectrum, Constant delay, Conformable derivative, Fractional sturm-liouville problemAbstract
This paper aims to express the solution of an inverse Sturm-Liouville problem with constant delay using a conformable derivative operator under mixed boundary conditions. For the problem, we stated and proved the specification of the spectrum. The asymptotics of the eigenvalues of the problem was obtained and the solutions were extended to the Regge-type boundary value problem. As such, a new result, as an extension of the classical Sturm-Liouville problem to the fractional phenomenon, has been achieved. The uniqueness theorem for the solution of the inverse problem is proved in different cases within the interval (0,π). The results in the classical case of this problem can be obtained at α=1.
2000 Mathematics Subject Classification. 34L20,34B24,34L30.
Downloads
References
Kilbas AA, Trujillo JJ, Srivastava H. Theory and applications of fractional differential equations. Elsevier 2006; 39: 52–7. https://doi.org/10.3182/20060719-3-PT-4902.00008 DOI: https://doi.org/10.3182/20060719-3-PT-4902.00008
Miller KS, Ross B. An introduction to the fractional calculus and fractional differential equations. John Wiley & Sons, New York, NY, USA; 1993.
Ortigueira MD, Tenreiro Machado JA. What is a fractional derivative? J Comput Phys. 2015; 293: 4-13. https://doi.org/10.1016/j.jcp.2014.07.019 DOI: https://doi.org/10.1016/j.jcp.2014.07.019
Khalil R, Al Horani M, Yousef A, Sababheh M. A new definition of fractional derivative. J Comput Appl Math. 2014; 264: 65-70. https://doi.org/10.1016/j.cam.2014.01.002 DOI: https://doi.org/10.1016/j.cam.2014.01.002
Abdeljawad T. On conformable fractional calculus. J Comput Appl Math. 2015; 279: 57-66. https://doi.org/10.1016/j.cam.2014.10.016 DOI: https://doi.org/10.1016/j.cam.2014.10.016
Atangana A, Baleanu D, Alsaedi A. New properties of conformable derivative. Open Mathematics. 2015; 13(1): 889-98. https://doi.org/10.1515/math-2015-0081 DOI: https://doi.org/10.1515/math-2015-0081
Abdelhakim AA, Machado JAT. A critical analysis of the conformable derivative. Nonlinear Dyn. 2019; 95: 3063-73. https://doi.org/10.1007/s11071-018-04741-5 DOI: https://doi.org/10.1007/s11071-018-04741-5
Baleanu D, Fernandez A. On fractional operators and their classifications. Mathematics. 2019; 7(9): 830. https://doi.org/10.3390/math7090830 DOI: https://doi.org/10.3390/math7090830
Tarasov VE. No nonlocality. No fractional derivative. Commun Nonlinear Sci Numer Simul. 2018; 62: 157-63. https://doi.org/10.1016/j.cnsns.2018.02.019 DOI: https://doi.org/10.1016/j.cnsns.2018.02.019
Allahverdiev BP, Tuna H, Yalçinkaya Y. Conformable fractional Sturm‐Liouville equation. Math Methods Appl Sci. 2019; 42: 3508-26. https://doi.org/10.1002/mma.5595 DOI: https://doi.org/10.1002/mma.5595
Ozarslan R, Bas E, Baleanu D. Representation of solutions for Sturm–Liouville eigenvalue problems with generalized fractional derivative. Chaos. 2020; 30(3): 033137. https://doi.org/10.1063/1.5131167 DOI: https://doi.org/10.1063/1.5131167
Sadabad MK, Akbarfam AJ, Shiri B. A numerical study of eigenvalues and eigenfunctions of fractional Sturm-Liouville problems via Laplace transform. Indian J Pure Appl Math. 2020; 51: 857-68. https://doi.org/10.1007/s13226-020-0436-2 DOI: https://doi.org/10.1007/s13226-020-0436-2
Rivero M, Trujillo J, Velasco M. A fractional approach to the Sturm-Liouville problem. Open Phys. 2013; 11: 1246–54. https://doi.org/10.2478/s11534-013-0216-2 DOI: https://doi.org/10.2478/s11534-013-0216-2
Sa’idu A, Koyunbakan H, Transmutation of conformable sturm-liouville operator with exactly solvable potential. Filomat. 2023; 37: 3383–90.
Wang WC. Some notes on conformable fractional Sturm–Liouville problems. Bound Value Probl. 2021; 2021: 1-8. https://doi.org/10.1186/s13661-021-01581-y DOI: https://doi.org/10.1186/s13661-021-01581-y
Al-Refai M, Abdeljawad T. Fundamental results of conformable Sturm-Liouville eigenvalue problems. Complexity. 2017; 2017: 1–7. https://doi.org/10.1155/2017/3720471 DOI: https://doi.org/10.1155/2017/3720471
Anderson D, Camrud E, Ulness D. On the nature of the conformable derivative and its applications to physics. J Fract Calc Appl. 2019; 10(2): 92-135.
Klimek M, Agrawal OP. Fractional sturm–liouville problem. Comput Math Appl. 2013; 66: 795-812. https://doi.org/10.1016/j.camwa.2012.12.011 DOI: https://doi.org/10.1016/j.camwa.2012.12.011
Koyunbakan H, Shah K, Abdeljawad T. Well-posedness of inverse sturm–liouville problem with fractional derivative. Qual Theory Dyn Syst. 2023; 22(1): Article 23. https://doi.org/10.1007/s12346-022-00727-2 DOI: https://doi.org/10.1007/s12346-022-00727-2
Liu T, Yu J, Zheng Y, Liu C, Yang Y, Qi Y. A nonlinear multigrid method for the parameter identification problem of partial differential equations with constraints. Mathematics. 2022; 10(16): 2938. https://doi.org/10.3390/math10162938 DOI: https://doi.org/10.3390/math10162938
Liu T, Xia K, Zheng Y, Yang Y, Qiu R, Qi Y, et al. A homotopy method for the constrained inverse problem in the multiphase porous media flow. Processes. 2022; 10(6): 1143. https://doi.org/10.3390/pr10061143 DOI: https://doi.org/10.3390/pr10061143
Liu T. Parameter estimation with the multigrid-homotopy method for a nonlinear diffusion equation. J Comput Appl Math. 2022; 413: 114393. https://doi.org/10.1016/j.cam.2022.114393 DOI: https://doi.org/10.1016/j.cam.2022.114393
Liu T. Porosity reconstruction based on Biot elastic model of porous media by homotopy perturbation method. Chaos Solitons Fractals. 2022; 158: 112007. https://doi.org/10.1016/j.chaos.2022.112007 DOI: https://doi.org/10.1016/j.chaos.2022.112007
Liu T. A wavelet multiscale-homotopy method for the parameter identification problem of partial differential equations. Comput Math Appl. 2016; 71: 1519-23. https://doi.org/10.1016/j.camwa.2016.02.036 DOI: https://doi.org/10.1016/j.camwa.2016.02.036
Liu T, Xia K, Zheng Y, Yang Y, Qiu R, Qi Y, et al. A homotopy method for the constrained inverse problem in the multiphase porous media flow. Processes. 2022; 10(6): 1143. https://doi.org/10.3390/pr10061143 DOI: https://doi.org/10.3390/pr10061143
Ambarzumian V. Über eine Frage der Eigenwerttheorie. Z. Physik 1929; 53: 690–5. https://doi.org/10.1007/BF01330827 DOI: https://doi.org/10.1007/BF01330827
Chern H-H, Law CK, Wang H-J. Extension of ambarzumyan’s theorem to general boundary conditions. J Math Anal Appl. 2001; 263: 333–42. https://doi.org/10.1006/jmaa.2001.7472 DOI: https://doi.org/10.1006/jmaa.2001.7472
Chuanfu Y, Xiaoping Y. Ambarzumyan’s theorem with eigenparameter in the boundary conditions. Acta Mathematica Scientia. 2011; 31: 1561–8. https://doi.org/10.1016/S0252-9602(11)60342-1 DOI: https://doi.org/10.1016/S0252-9602(11)60342-1
Amirov R, Ergun A, Durak S. Half-inverse problems for the quadratic pencil of the Sturm–Liouville equations with impulse. Numer Methods Partial Differ Equ. 2021; 37: 915–24. https://doi.org/10.1002/num.22559 DOI: https://doi.org/10.1002/num.22559
Buterin SA, Shieh CT. Incomplete inverse spectral and nodal problems for differential pencils. Results Math. 2012; 62: 167-79. https://doi.org/10.1007/s00025-011-0137-6 DOI: https://doi.org/10.1007/s00025-011-0137-6
Cheng YH, Law CK, Tsay J. Remarks on a new inverse nodal problem. J Math Anal Appl. 2000; 248: 145–55. https://doi.org/10.1006/jmaa.2000.6878 DOI: https://doi.org/10.1006/jmaa.2000.6878
Guliyev NJ. Inverse eigenvalue problems for sturm–liouville equations with spectral parameter linearly contained in one of the boundary conditions. Inverse Probl. 2005; 21: 1315-30. https://doi.org/10.1088/0266-5611/21/4/008 DOI: https://doi.org/10.1088/0266-5611/21/4/008
Gulsen T, Yilmaz E, Akbarpoor S. Numerical investigation of the inverse nodal problem by chebisyhev interpolation method. Therm Sci. 2018; 22: 123–36. https://doi.org/10.2298/TSCI170612278G DOI: https://doi.org/10.2298/TSCI170612278G
Mosazadeh S. A new approach to uniqueness for inverse sturm--liouville problems on finite intervals. Turkish J Math. 2017; 41: 1224–34. https://doi.org/10.3906/mat-1609-38 DOI: https://doi.org/10.3906/mat-1609-38
Sadovnichii VA, Sultanaev YaT, Akhtyamov AM. Solvability theorems for an inverse nonself-adjoint Sturm-Liouville problem with nonseparated boundary conditions. Differ Equ. 2015; 51(6): 717-25. https://doi.org/10.1134/S0012266115060026 DOI: https://doi.org/10.1134/S0012266115060026
Freiling G, Yurko VA. Inverse problems for Sturm-Liouville differential operators with a constant delay. Appl Math Lett. 2012; 25: 1999-2004. https://doi.org/10.1016/j.aml.2012.03.026. DOI: https://doi.org/10.1016/j.aml.2012.03.026
Freiling G, Yurko VA. Inverse Sturm-Liouville problems and their applications. Huntington: NOVA Science Publishers; 2001.
Koyunbakan H, Mosazadeh S. Inverse nodal problem for discontinuous sturm–liouville operator by new Prüfer Substitutions. Math Sci. 2021; 15: 387–94. https://doi.org/10.1007/s40096-021-00383-8 DOI: https://doi.org/10.1007/s40096-021-00383-8
Adalar İ, Ozkan AS. Inverse problems for a Fractional Sturm-Liouville operators. J Inverse Ill Posed Probl. 2020; 28: 775-82. https://doi.org/10.1515/jiip-2019-0058 DOI: https://doi.org/10.1515/jiip-2019-0058
Çakmak Y. Inverse nodal problem for a conformable fractional diffusion operator. Inverse Probl Sci Eng. 2021; 29: 1308–22. https://doi.org/10.1080/17415977.2020.1847103 DOI: https://doi.org/10.1080/17415977.2020.1847103
Mortazaasl H, Jodayree Akbarfam A. Trace formula and inverse nodal problem for a conformable fractional sturm-liouville problem. Inverse Probl Sci Eng. 2020; 28: 524-55. https://doi.org/10.1080/17415977.2019.1615909 DOI: https://doi.org/10.1080/17415977.2019.1615909
Sa’idu A, Koyunbakan H. Inverse fractional Sturm‐Liouville problem with eigenparameter in the boundary conditions. Math Methods Appl Sci. 2022; 45: 1–10. https://doi.org/10.1002/mma.8433 DOI: https://doi.org/10.1002/mma.8433
Buterin SA, Yurko VA. An inverse spectral problem for Sturm–Liouville operators with a large constant delay. Anal Math Phys. 2019; 9: 17–27. https://doi.org/10.1007/s13324-017-0176-6 DOI: https://doi.org/10.1007/s13324-017-0176-6
Bondarenko N, Yurko V. An inverse problem for sturm–iiouville differential operators with deviating argument. Appl Math Lett. 2018; 83: 140–4. https://doi.org/10.1016/j.aml.2018.03.025 DOI: https://doi.org/10.1016/j.aml.2018.03.025
Buterin S, Vasilev S. An inverse sturm--liouville-type problem with constant delay and non-zero initial function. ArXiv Preprint 2023.
Myshkis AD. Linear differential equations with retarded argument. Moscow: Nauka; 1972.
Norkin S. Second order differential equations with a delay argument. Moscow: Nauka; 1965.
Yang CF. Inverse nodal problems for the Sturm–Liouville operator with a constant delay. J Differ Equ. 2014; 257: 1288–306. https://doi.org/10.1016/j.jde.2014.05.011 DOI: https://doi.org/10.1016/j.jde.2014.05.011
Yang CF. Trace and inverse problem of a discontinuous Sturm–Liouville operator with retarded argument. J Math Anal Appl. 2012; 395: 30–41. https://doi.org/10.1016/j.jmaa.2012.04.078 DOI: https://doi.org/10.1016/j.jmaa.2012.04.078
Bayramov A, Öztürk Uslu S. Computation of eigenvalues and eigenfunctions of a discontinuous boundary value problem with retarded argument. Appl Math Comput. 2007; 191: 592–600. https://doi.org/10.1016/j.amc.2007.02.118 DOI: https://doi.org/10.1016/j.amc.2007.02.118
Conway JB. Functions of One Complex Variable, second ed., vol. I, Springer-Verlag, New York, 1995. DOI: https://doi.org/10.1007/978-1-4612-0817-4
Downloads
Published
Issue
Section
License
Copyright (c) 2023 Auwalu Sa'idu, Hikmet Koyunbakan
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
All the published articles are licensed under the terms of the Creative Commons Attribution Non-Commercial License (CC BY-NC 4.0) which permits unrestricted, non-commercial use, distribution and reproduction in any medium, provided the work is properly cited.
