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On the Inverse Problems for Conformable Fractional Integro-Dirac Differential System with Parameter Dependent Boundary Conditions

Year 2024, Volume: 45 Issue: 4, 789 - 795, 30.12.2024

Hediye Dilara Tel Baki Keskin

https://doi.org/10.17776/csj.1423665

Abstract

This study considers a conformable fractional Dirac-type integral differential system, focusing on its mathematical properties and practical implications. Asymptotic formulas have been derived for the solutions, eigenvalues, and nodes of the problem, providing a deeper understanding of the behavior of the system under varying conditions. These asymptotic results form the basis for analyzing the spectral characteristics and node distribution of the system. In addition, an algorithm is developed that effectively solves the inverse nodal problem and reconstructs the system coefficients from the nodal data.

Keywords

Conformable Fractional Dirac System intego-differential operators inverse nodal problem

References

  • [1] Dirac PAM, The quantum theory of the electron, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 117( 778) (1928) 610-624.
  • [2] Levitan BM., IS. Sargsyan, Sturm Liouville and Dirac operators. Kluver Academic Publishers: Dudrecht/Boston/London; 1991.
  • [3] Albeverio S., R. Hryniv, Mykytyuk Ya., Reconstruction of radial Dirac and Schrödinger operators from two spectra, J. Math. Anal. Appl. 339 (2008) 45-57.
  • [4] Gasymov MG., Inverse problem of the scattering theory for Dirac system of order 2n, Tr. Mosk Mat. Obshch, 19 (1968) 41-112.
  • [5] Horvath M., On the inverse spectral theory of Schrödinger and Dirac operators, Trans. Amer. Math. Soc., 353 (2001) 4155-4171..
  • [6] Miller K. S., An Introduction to Fractional Calculus and Fractional Differential Equations, J. Wiley and Sons, New York, NY, USA, 1993.
  • [7] Kilbas A., Srivastava H., and Trujillo J., “Theory and applications of fractional differential equations,” in Math. Studies, North-Holland, New York, NY, USA, 2006.
  • [8] Oldham K., Spanier J., The Fractional Calculus, Theory and Applications of Differentiation and Integration of Arbitrary Order, Academic Press, Cambridge, MA, USA, 1974.
  • [9] Podlubny I., Fractional Differential Equations, Academic Press, Cambridge, MA, USA, 1999.
  • [10] Khalil R., M. Horani Al, Yousef A., Sababheh M., A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264 (2014) 65-70.
  • [11] Abdeljawad T., On conformable fractional calculus, Journal of Computational and Applied Mathematics, 279 (2015) 57–6
  • [12] Hammad M. Abu, , Khalil R., Abel’s formula and Wronskian for conformable fractional differential equations, Int. J. Differ. Equ. Appl., 13(3) (2014) 177–183.
  • [13] Al-Refai M., Abdeljawad T., Fundamental results of conformable Sturm–Liouville eigenvalue problems, Complexity, (2017) 3720471.
  • [14] Çakmak Y., Inverse nodal problem for a conformable fractional diffusion operator, Inverse Problems in Science and Engineering, 29-9 (2021) 1308-1322.
  • [15] Çakmak Y., Trace Formulae for a Conformable Fractional Diffusion Operator, Filomat, 36-14 (2022) 4665-4674.
  • [16] Keskin B., Inverse problems for one dimensional conformable fractional Dirac type integro differential system, Inverse Problems, 36 (2020) 065001.
  • [17] Allahverdiev, B.P., Tuna, H., One-dimensional conformable fractional Dirac system, Bol. Soc. Mat. Mex., (2019).
  • [18] Erdal B., Fundamental spectral theory of fractional singular Sturm-Liouville operatör, J Funct Space, 1 (2013) 113-129.
  • [19] Gulsen T., Yilmaz E., Goktas S., Conformable fractional Dirac system on time scales, J. Inequal.Appl., (2017) 161,2017.
  • [20] Adalar İ., Özkan A. S., Inverse problems for a conformable fractional Sturm-Liouville operator, Journal of ill-posed problems, 28(6) (2020) 775-782.
  • [21] Zhaowen Z. , Huixi L. , Jinming C., Yanwei Z., Criteria of limit-point case for conformable fractional Sturm-Liouville operators, Math Meth Appl Sci., 43 (2020) 2548–2557.
  • [22] McLaughlin JR., Inverse spectral theory using nodal points as data – a uniqueness result, J. Diff. Eq., 73 (1988) 354–362.
  • [23] Hald OH., McLaughlin JR., Solutions of inverse nodal problems, Inverse Problems, 5 (1989) 307–347.
  • [24] Yang XF., A solution of the nodal problem, Inverse Problems, 13 (1997) 203-213.
  • [25] Browne PJ., Sleema B.D., Inverse nodal problem for Sturm–Liouville equation with eigenparameter depend boundary conditions, Inverse Problems, 12 (1996) 377–381.
  • [26] Cheng Y., Law CK. and Tsay J., Remarks on a new inverse nodal problem, J. Math. Anal. Appl., 248 (2000) 145–155.
  • [27] Guo Y., Wei Y., Inverse problems: Dense nodal subset on an interior subinterval, J. Diff. Eq., 255 (2002) 2017.
  • [28] Law CK., Shen CL., Yang CF., The Inverse Nodal Problem on the Smoothness of the Potential Function, Inverse Problems, 15 (1999), no.1, 253-263 (Erratum, Inverse Problems 2001; 17: 361-363).
  • [29] Ozkan AS., Keskin B., Inverse Nodal Problems for Sturm–Liouville Equation with Eigenparameter Dependent Boundary and Jump Conditions, Inverse Problems in Science and Engineering, 23(8) (2015) 1306-1312.
  • [30] Wang YP., Yurko V., On the inverse nodal problems for discontinuous Sturm Liouville operators, J. Differential Equations, 260 (2016) 4086-4109.
  • [31] Wang YP., Lien KY., Shieh CT., Inverse problems for the boundary value problem with the interior nodal subsets, Applicable Analysis, 96 (2017) 1229-1239.
  • [32] Wei Z., Guo Y., Wei G., Incomplete inverse spectral and nodal problems for Dirac operatör, Adv. Difference Equ., 2015 ( 2015) 88.
  • [33] Shieh CT., Yurko VA., Inverse nodal and inverse spectral problems for discontinuous boundary value problems, J. Math. Anal. Appl., 347 (2008) 266-272.
  • [34] Yang XF., A new inverse nodal problem, J. Differential Equations, 169 (2001) 633-653.
  • [35] Guo Y., Wei Y., Inverse Nodal Problem for Dirac Equations with Boundary Conditions Polynomially Dependent on the Spectral Parameter, Results in Math., 67 (2015) 95–110.
  • [36] [36] Yang CF., Huang ZY., Reconstruction of the Dirac operator from nodal data, Integr. Equ. Oper. Theory, 66 (2010) 539–551.
  • [37] Yang CF., Pivovarchik VN., Inverse nodal problem for Dirac system with spectral parameter in boundary conditions, Complex Anal. Oper. Theory, 7 (2013) 1211–1230
  • [38] Fulton C. T., Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A , 77 (3–4) (1977) 293–308.
  • [39] Fulton C. T., Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A , 87 (1–2) (1980) 1–34. 10.1017.
  • [40] Guliyev N. J., Spectral identities for Schrodinger operators , Canad. Math. Bull., (to appear)
  • [41] Guliyev N. J., Essentially isospectral transformations and their applications, Ann. Mat. Pura Appl., 199(4) (2020) 1621–1648.
  • [42] Guliyev N. J., Schrödinger operators with distributional potentials and boundary conditions dependent on the eigenvalue parameter, J. Math. Phys., 60(6) (2019) 063501.
  • [43] Bondarenko NP., An inverse problem for the integro-differential Dirac system with partial information given on the convolution kernel, J. Inverse Ill-Posed Probl., 27 (2) (2018) 151-157.
  • [44] Buterin SA., On an Inverse Spectral Problem for a Convolution Integro-Differential Operator, Results in Mathematics, 50 (2007) 173-181.
  • [45] Kuryshova YV., Shieh CT., An Inverse Nodal Problem for Integro-Differential Operators, Journal of Inverse and III-posed Problems, 18 (2010) 357–369.
  • [46] Wu B., Yu J., Uniqueness of an Inverse Problem for an Integro-Differential Equation Related to the Basset Problem, Boundary Value Problems, 229 (2014).
  • [47] Keskin B., Ozkan A. S., Inverse nodal problems for Dirac-type integro-differential operators, J. Differential Equations, 263 (2017) 8838–8847
  • [48] Keskin B., Tel H. D., Reconstruction of the Dirac-Type Integro-Differential Operator From Nodal Data, Numerical Functional Analysis and Optimization, 39-11 (2018) 1208–1220 .

Year 2024, Volume: 45 Issue: 4, 789 - 795, 30.12.2024

Hediye Dilara Tel Baki Keskin

https://doi.org/10.17776/csj.1423665

Abstract

References

  • [1] Dirac PAM, The quantum theory of the electron, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 117( 778) (1928) 610-624.
  • [2] Levitan BM., IS. Sargsyan, Sturm Liouville and Dirac operators. Kluver Academic Publishers: Dudrecht/Boston/London; 1991.
  • [3] Albeverio S., R. Hryniv, Mykytyuk Ya., Reconstruction of radial Dirac and Schrödinger operators from two spectra, J. Math. Anal. Appl. 339 (2008) 45-57.
  • [4] Gasymov MG., Inverse problem of the scattering theory for Dirac system of order 2n, Tr. Mosk Mat. Obshch, 19 (1968) 41-112.
  • [5] Horvath M., On the inverse spectral theory of Schrödinger and Dirac operators, Trans. Amer. Math. Soc., 353 (2001) 4155-4171..
  • [6] Miller K. S., An Introduction to Fractional Calculus and Fractional Differential Equations, J. Wiley and Sons, New York, NY, USA, 1993.
  • [7] Kilbas A., Srivastava H., and Trujillo J., “Theory and applications of fractional differential equations,” in Math. Studies, North-Holland, New York, NY, USA, 2006.
  • [8] Oldham K., Spanier J., The Fractional Calculus, Theory and Applications of Differentiation and Integration of Arbitrary Order, Academic Press, Cambridge, MA, USA, 1974.
  • [9] Podlubny I., Fractional Differential Equations, Academic Press, Cambridge, MA, USA, 1999.
  • [10] Khalil R., M. Horani Al, Yousef A., Sababheh M., A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264 (2014) 65-70.
  • [11] Abdeljawad T., On conformable fractional calculus, Journal of Computational and Applied Mathematics, 279 (2015) 57–6
  • [12] Hammad M. Abu, , Khalil R., Abel’s formula and Wronskian for conformable fractional differential equations, Int. J. Differ. Equ. Appl., 13(3) (2014) 177–183.
  • [13] Al-Refai M., Abdeljawad T., Fundamental results of conformable Sturm–Liouville eigenvalue problems, Complexity, (2017) 3720471.
  • [14] Çakmak Y., Inverse nodal problem for a conformable fractional diffusion operator, Inverse Problems in Science and Engineering, 29-9 (2021) 1308-1322.
  • [15] Çakmak Y., Trace Formulae for a Conformable Fractional Diffusion Operator, Filomat, 36-14 (2022) 4665-4674.
  • [16] Keskin B., Inverse problems for one dimensional conformable fractional Dirac type integro differential system, Inverse Problems, 36 (2020) 065001.
  • [17] Allahverdiev, B.P., Tuna, H., One-dimensional conformable fractional Dirac system, Bol. Soc. Mat. Mex., (2019).
  • [18] Erdal B., Fundamental spectral theory of fractional singular Sturm-Liouville operatör, J Funct Space, 1 (2013) 113-129.
  • [19] Gulsen T., Yilmaz E., Goktas S., Conformable fractional Dirac system on time scales, J. Inequal.Appl., (2017) 161,2017.
  • [20] Adalar İ., Özkan A. S., Inverse problems for a conformable fractional Sturm-Liouville operator, Journal of ill-posed problems, 28(6) (2020) 775-782.
  • [21] Zhaowen Z. , Huixi L. , Jinming C., Yanwei Z., Criteria of limit-point case for conformable fractional Sturm-Liouville operators, Math Meth Appl Sci., 43 (2020) 2548–2557.
  • [22] McLaughlin JR., Inverse spectral theory using nodal points as data – a uniqueness result, J. Diff. Eq., 73 (1988) 354–362.
  • [23] Hald OH., McLaughlin JR., Solutions of inverse nodal problems, Inverse Problems, 5 (1989) 307–347.
  • [24] Yang XF., A solution of the nodal problem, Inverse Problems, 13 (1997) 203-213.
  • [25] Browne PJ., Sleema B.D., Inverse nodal problem for Sturm–Liouville equation with eigenparameter depend boundary conditions, Inverse Problems, 12 (1996) 377–381.
  • [26] Cheng Y., Law CK. and Tsay J., Remarks on a new inverse nodal problem, J. Math. Anal. Appl., 248 (2000) 145–155.
  • [27] Guo Y., Wei Y., Inverse problems: Dense nodal subset on an interior subinterval, J. Diff. Eq., 255 (2002) 2017.
  • [28] Law CK., Shen CL., Yang CF., The Inverse Nodal Problem on the Smoothness of the Potential Function, Inverse Problems, 15 (1999), no.1, 253-263 (Erratum, Inverse Problems 2001; 17: 361-363).
  • [29] Ozkan AS., Keskin B., Inverse Nodal Problems for Sturm–Liouville Equation with Eigenparameter Dependent Boundary and Jump Conditions, Inverse Problems in Science and Engineering, 23(8) (2015) 1306-1312.
  • [30] Wang YP., Yurko V., On the inverse nodal problems for discontinuous Sturm Liouville operators, J. Differential Equations, 260 (2016) 4086-4109.
  • [31] Wang YP., Lien KY., Shieh CT., Inverse problems for the boundary value problem with the interior nodal subsets, Applicable Analysis, 96 (2017) 1229-1239.
  • [32] Wei Z., Guo Y., Wei G., Incomplete inverse spectral and nodal problems for Dirac operatör, Adv. Difference Equ., 2015 ( 2015) 88.
  • [33] Shieh CT., Yurko VA., Inverse nodal and inverse spectral problems for discontinuous boundary value problems, J. Math. Anal. Appl., 347 (2008) 266-272.
  • [34] Yang XF., A new inverse nodal problem, J. Differential Equations, 169 (2001) 633-653.
  • [35] Guo Y., Wei Y., Inverse Nodal Problem for Dirac Equations with Boundary Conditions Polynomially Dependent on the Spectral Parameter, Results in Math., 67 (2015) 95–110.
  • [36] [36] Yang CF., Huang ZY., Reconstruction of the Dirac operator from nodal data, Integr. Equ. Oper. Theory, 66 (2010) 539–551.
  • [37] Yang CF., Pivovarchik VN., Inverse nodal problem for Dirac system with spectral parameter in boundary conditions, Complex Anal. Oper. Theory, 7 (2013) 1211–1230
  • [38] Fulton C. T., Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A , 77 (3–4) (1977) 293–308.
  • [39] Fulton C. T., Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A , 87 (1–2) (1980) 1–34. 10.1017.
  • [40] Guliyev N. J., Spectral identities for Schrodinger operators , Canad. Math. Bull., (to appear)
  • [41] Guliyev N. J., Essentially isospectral transformations and their applications, Ann. Mat. Pura Appl., 199(4) (2020) 1621–1648.
  • [42] Guliyev N. J., Schrödinger operators with distributional potentials and boundary conditions dependent on the eigenvalue parameter, J. Math. Phys., 60(6) (2019) 063501.
  • [43] Bondarenko NP., An inverse problem for the integro-differential Dirac system with partial information given on the convolution kernel, J. Inverse Ill-Posed Probl., 27 (2) (2018) 151-157.
  • [44] Buterin SA., On an Inverse Spectral Problem for a Convolution Integro-Differential Operator, Results in Mathematics, 50 (2007) 173-181.
  • [45] Kuryshova YV., Shieh CT., An Inverse Nodal Problem for Integro-Differential Operators, Journal of Inverse and III-posed Problems, 18 (2010) 357–369.
  • [46] Wu B., Yu J., Uniqueness of an Inverse Problem for an Integro-Differential Equation Related to the Basset Problem, Boundary Value Problems, 229 (2014).
  • [47] Keskin B., Ozkan A. S., Inverse nodal problems for Dirac-type integro-differential operators, J. Differential Equations, 263 (2017) 8838–8847
  • [48] Keskin B., Tel H. D., Reconstruction of the Dirac-Type Integro-Differential Operator From Nodal Data, Numerical Functional Analysis and Optimization, 39-11 (2018) 1208–1220 .
There are 48 citations in total.

Details

Primary Language English
Subjects Applied Mathematics (Other)
Journal Section Natural Sciences
Authors

Hediye Dilara Tel 0000-0003-1139-6146

Baki Keskin 0000-0003-1689-8954

Publication Date December 30, 2024
Submission Date January 22, 2024
Acceptance Date December 23, 2024
Published in Issue Year 2024Volume: 45 Issue: 4

Cite

APA Tel, H. D., & Keskin, B. (2024). On the Inverse Problems for Conformable Fractional Integro-Dirac Differential System with Parameter Dependent Boundary Conditions. Cumhuriyet Science Journal, 45(4), 789-795. https://doi.org/10.17776/csj.1423665
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