Research Article
Year 2023,
Issue: 44, 43 - 51, 30.09.2023
Abstract
In this paper, we consider a diffusion operator with discrete boundary conditions, which include the conformable fractional derivatives of order $\alpha$ such that $0<\alpha\leq1$ instead of the ordinary derivatives in the classical diffusion operator. We prove that the coefficients of the given operator are uniquely determined by the Weyl function and spectral data, which consist of a spectrum and normalizing numbers. Moreover, using the well-known Hadamard's factorization theorem, we prove that the characteristic function $\Delta_{\alpha}\left(\rho\right)$ is determined by the specification of its zeros for each fixed $\alpha$. The obtained results in this paper can be regarded as partial $\alpha$-generalizations of similar findings obtained for the classical diffusion operator.
Keywords
Inverse problem diffusion operator conformable fractional derivative Weyl function spectral data
References
- V. A. Ambartsumyan, \"{U}ber Eine Frage Der Eigenwerttheorie, Zeitschrift f\"{u}r Physik 53 (1929) 690--695.
- G. Borg, Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe: Bestimmung der Differentialgleichung durch die Eigenwerte, Acta Mathematica 78 (1946) 1--96.
- N. Levinson, The Inverse Sturm-Liouville Problem, Matematisk Tidsskrift B 25 (1949) 25--30.
- V. A. Marchenko, Concerning the Theory of a Differential Operator of the Second Order, Doklady Akademii Nauk SSSR 72 (1950) 457--460.
- E. L. Isaacson, E. Trubowitz, The Inverse Sturm-Liouville Problem I, Communications on Pure and Applied Mathematics 36 (1983) 767--783.
- M. G. Gasymov, G. Sh. Guseinov, Determining of the Diffusion Operator from Spectral Data, Doklady Akademii Nauk Azerbaijan SSR 37 (2) (1981) 19--23.
- G. Freiling, V.A. Yurko, Inverse Sturm-Liouville Problems and Their Applications, Nova Science Publishers, New York, 2001.
- R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A New Definition of Fractional Derivative, Journal of Computational and Applied Mathematics 264 (2014) 65--70.
- T. Abdeljawad, On Conformable Fractional Calculus, Journal of Computational and Applied Mathematics 279 (2015) 57--66.
- A. Atangana, D. Baleanu, A. Alsaedi, New Properties of Conformable Derivative, Open Mathematics 13 (2015) 889--898.
- M. Abu Hammad, R.Khalil, Abel's Formula and Wronskian for Conformable Fractional Differential Equations, International Journal of Differential Equations and Applications 13 (3) (2014) 177--183.
- O. T. Birgani, S. Chandok, N. Dedovic, S. Radenovi\c{c}, A Note on Some Recent Results of the Conformable Derivative, Advances in the Theory of Nonlinear Analysis and its Applications 3 (1) (2019) 11--17.
- D. Zhao, M. Luo, General Conformable Fractional Derivative and its Physical Interpretation, Calcolo 54 (2017) 903--917.
- Y. Wang, J. Zhou, Y. Li, Fractional Sobolev's Spaces on Time Scales via Conformable Fractional Calculus and Their Application to a Fractional Differential Equation on Time Scales, Advances in Mathematical Physics 2016 (2016) Article ID 963649121 21 pages.
- F. Jarad, E. U\u{g}urlu, T. Abdeljawad, D. Baleanu, On a New Class of Fractional Operators, Advances in Difference Equations 2017 (2017) Article Number 247 16 pages.
- H. Mortazaasl, A. Jodayree Akbarfam, Trace Formula and Inverse Nodal Problem for a Conformable Fractional Sturm-Liouville Problem, Inverse Problems in Science and Engineering 28 (4) (2020) 524--555.
- B. Keskin, Inverse Problems for One Dimentional Conformable Fractional Dirac Type Integro Differential System, Inverse Problems 36 (6) (2020) 065001 10 pages.
- İ. Adalar, A. S. Özkan, Inverse Problems for a Conformable Fractional Sturm-Liouville Operators, Journal of Inverse and Ill-posed Problems 28 (6) (2020) 775--782.
- Y. \c{C}akmak, Inverse Nodal Problem for a Conformable Fractional Diffusion Operator, Inverse Problems in Science and Engineering 29 (9) (2021) 1308--1322.
- Y. \c{C}akmak, Inverse Nodal Problem for a Conformable Fractional Diffusion Operator with Parameter-Dependent Nonlocal Boundary Condition, Cumhuriyet Science Journal 44 (2) (2023) 356--363
- Y. \c{C}akmak, Trace Formulae for a Conformable Fractional Diffusion Operator, Filomat 36 (14) (2022) 4665--4674.
- S. A. Buterin, On Half Inverse Problem for Differential Pencils with the Spectral Parameter in the Boundary Conditions, Tamkang Journal of Mathematics 42 (3) (2011) 355--364.
- E. Ko\c{c}, Y. \c{C}akmak, $\alpha $-Integral Representation of the Solution for a Conformable Fractional Diffusion Operator and Basic Properties of the Operator, Cumhuriyet Science Journal 44 (1) (2023) 170--180.
Year 2023,
Issue: 44, 43 - 51, 30.09.2023
Abstract
References
- V. A. Ambartsumyan, \"{U}ber Eine Frage Der Eigenwerttheorie, Zeitschrift f\"{u}r Physik 53 (1929) 690--695.
- G. Borg, Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe: Bestimmung der Differentialgleichung durch die Eigenwerte, Acta Mathematica 78 (1946) 1--96.
- N. Levinson, The Inverse Sturm-Liouville Problem, Matematisk Tidsskrift B 25 (1949) 25--30.
- V. A. Marchenko, Concerning the Theory of a Differential Operator of the Second Order, Doklady Akademii Nauk SSSR 72 (1950) 457--460.
- E. L. Isaacson, E. Trubowitz, The Inverse Sturm-Liouville Problem I, Communications on Pure and Applied Mathematics 36 (1983) 767--783.
- M. G. Gasymov, G. Sh. Guseinov, Determining of the Diffusion Operator from Spectral Data, Doklady Akademii Nauk Azerbaijan SSR 37 (2) (1981) 19--23.
- G. Freiling, V.A. Yurko, Inverse Sturm-Liouville Problems and Their Applications, Nova Science Publishers, New York, 2001.
- R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A New Definition of Fractional Derivative, Journal of Computational and Applied Mathematics 264 (2014) 65--70.
- T. Abdeljawad, On Conformable Fractional Calculus, Journal of Computational and Applied Mathematics 279 (2015) 57--66.
- A. Atangana, D. Baleanu, A. Alsaedi, New Properties of Conformable Derivative, Open Mathematics 13 (2015) 889--898.
- M. Abu Hammad, R.Khalil, Abel's Formula and Wronskian for Conformable Fractional Differential Equations, International Journal of Differential Equations and Applications 13 (3) (2014) 177--183.
- O. T. Birgani, S. Chandok, N. Dedovic, S. Radenovi\c{c}, A Note on Some Recent Results of the Conformable Derivative, Advances in the Theory of Nonlinear Analysis and its Applications 3 (1) (2019) 11--17.
- D. Zhao, M. Luo, General Conformable Fractional Derivative and its Physical Interpretation, Calcolo 54 (2017) 903--917.
- Y. Wang, J. Zhou, Y. Li, Fractional Sobolev's Spaces on Time Scales via Conformable Fractional Calculus and Their Application to a Fractional Differential Equation on Time Scales, Advances in Mathematical Physics 2016 (2016) Article ID 963649121 21 pages.
- F. Jarad, E. U\u{g}urlu, T. Abdeljawad, D. Baleanu, On a New Class of Fractional Operators, Advances in Difference Equations 2017 (2017) Article Number 247 16 pages.
- H. Mortazaasl, A. Jodayree Akbarfam, Trace Formula and Inverse Nodal Problem for a Conformable Fractional Sturm-Liouville Problem, Inverse Problems in Science and Engineering 28 (4) (2020) 524--555.
- B. Keskin, Inverse Problems for One Dimentional Conformable Fractional Dirac Type Integro Differential System, Inverse Problems 36 (6) (2020) 065001 10 pages.
- İ. Adalar, A. S. Özkan, Inverse Problems for a Conformable Fractional Sturm-Liouville Operators, Journal of Inverse and Ill-posed Problems 28 (6) (2020) 775--782.
- Y. \c{C}akmak, Inverse Nodal Problem for a Conformable Fractional Diffusion Operator, Inverse Problems in Science and Engineering 29 (9) (2021) 1308--1322.
- Y. \c{C}akmak, Inverse Nodal Problem for a Conformable Fractional Diffusion Operator with Parameter-Dependent Nonlocal Boundary Condition, Cumhuriyet Science Journal 44 (2) (2023) 356--363
- Y. \c{C}akmak, Trace Formulae for a Conformable Fractional Diffusion Operator, Filomat 36 (14) (2022) 4665--4674.
- S. A. Buterin, On Half Inverse Problem for Differential Pencils with the Spectral Parameter in the Boundary Conditions, Tamkang Journal of Mathematics 42 (3) (2011) 355--364.
- E. Ko\c{c}, Y. \c{C}akmak, $\alpha $-Integral Representation of the Solution for a Conformable Fractional Diffusion Operator and Basic Properties of the Operator, Cumhuriyet Science Journal 44 (1) (2023) 170--180.
There are 23 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Ordinary Differential Equations, Difference Equations and Dynamical Systems |
| Journal Section | Research Article |
| Authors | |
| Publication Date | September 30, 2023 |
| Submission Date | August 1, 2023 |
| Published in Issue | Year 2023 Issue: 44 |
Cite
|