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Ambarzumyan-Type Theorem for a Conformable Fractional Diffusion Operator

Yıl 2023, Cilt: 6 Sayı: 3, 142 - 147, 17.09.2023

Yaşar Çakmak

https://doi.org/10.33434/cams.1281434

Öz

In this paper, we prove an Ambarzumyan-type theorem for a Conformable fractional diffusion operator, i.e. we show that $q(x)$ and $p(x)$ functions are zero if the eigenvalues are the same as the eigenvalues of zero potentials.

Anahtar Kelimeler

Ambarzumyan-type theorem Conformable fractional derivative Diffusion operator Inverse problem

Kaynakça

  • [1] V.A. Ambarzumian, Uber eine frage der eigenwerttheorie, Zeitschrift f¨ur Physik, 53 (1929), 690-695.
  • [2] H.H. Chern, C.K. Law, H.J. Wang, Extensions of Ambarzumyan’s theorem to general boundary conditions, J. Math. Anal. Appl., 263 (2001), 333-342. Corrigendum: J. Math. Anal. Appl., 309 (2) (2005), 764-768.
  • [3] E.M. Harrell, On the extension of Ambarzunyan’s inverse spectral theorem to compact symmetric spaces, Amer. J. Math., 109 (1987), 787-795.
  • [4] M. Horvath, On a theorem of Ambarzumyan, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 131 (2001), 899-907.
  • [5] M. Kiss, An n-dimensional Ambarzumyan type theorem for Dirac operators, Inverse Problems, 20 (5)(2004), 1593-1597.
  • [6] C.F. Yang, X.P. Yang, Some Ambarzumyan-type theorems for Dirac operators, Inverse Problems, 25 (9) (2009), 095012, 13 pages.
  • [7] C.F. Yang, Z.Y. Huang, X.P. Yang, Ambarzumyan’s theorems for vectorial Sturm-Liouville systems with coupled boundary conditions, Taiwanese J. Math., 14 (4) (2010), 1429-1437.
  • [8] C.L. Shen, On some inverse spectral problems related to the Ambarzumyan problem and the dual string of the string equation, Inverse Problems, 23 (6) (2007), 2417-2436.
  • [9] N.V. Kuznetsov, Generalization of a theorem of V.A. Ambarzumyan, Doklady Akademii Nauk SSSR., 146 (1962), 1259- 1262, (in Russian).
  • [10] H.H. Chern, C.L. Shen, On the n-dimensional Ambarzumyan’s theorem, Inverse Problems, 13 (1) (1997), 15-18.
  • [11] C.F. Yang, X.P. Yang, Ambarzumyan’s theorem with eigenparameter in the boundary conditions, Acta Math. Sci., 31 B(4) (2011), 1561-1568.
  • [12] V.A. Yurko, On Ambarzumyan-type theorems, Appl. Math. Lett., 26 (4) (2013), 506-509.
  • [13] K. Marton, An n-dimensional Ambarzumyan type theorem for Dirac operators, Inverse Problems, 20 (5) (2004), 1593-1597.
  • [14] A.A. Kırac, Ambarzumyan’s theorem for the quasi-periodic boundary conditions, Anal. Math. Phys., 6 (2016), 297-300.
  • [15] H. Koyunbakan, D. Lesnic, E.S. Panakhov, Ambarzumyan type theorem for a quadratic Sturm-Liouville operator, Turkish Journal of Sciences and Technology, 8 (1) (2013), 1-5.
  • [16] G. Freiling, V.A. Yurko, Inverse Sturm-Liouville Problems and Their Applications, Nova Science Publishers, New York, 2001.
  • [17] E. Yilmaz, H. Koyunbakan, Some Ambarzumyan type theorems for Bessel operator on a finite interval, Differ. Equ. Dyn. Syst., 27 (4) (2019), 553–559.
  • [18] R. Khalil, M. Al Horania, A. Yousefa, et al., A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70.
  • [19] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66.
  • [20] A. Atangana, D. Baleanu, A. Alsaedi, New properties of conformable derivative, Open Math., 13 (2015), 889-898.
  • [21] M. Abu Hammad, R. Khalil, Abel’s formula and Wronskian for conformable fractional differential equations, International J. Differ. Equ. Appl., 13 (3) (2014), 177-183.
  • [22] O.T. Birgani, S. Chandok, N. Dedovic, S. Radenoviç, 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.
  • [23] D. Zhao, M. Luo, General conformable fractional derivative and its physical interpretation, Calcolo, 54 (3) (2017), 903-917.
  • [24] H.W. Zhou, S. Yang, S.Q. Zhang, Conformable derivative approach to anomalous diffusion, Physica A: Statistical Mechanics and its Applications, 491 (2018), 1001-1013.
  • [25] F. Jarad, E. Uğurlu, T. Abdeljawad, D. Baleanu, On a new class of fractional operators, Adv. Difference Equ., 2017 (2017), Article ID 247, 16 pages.
  • [26] H. Mortazaasl, A. Jodayree Akbarfam, Trace formula and inverse nodal problem for a conformable fractional Sturm-Liouville problem, Inverse Probl. Sci. Eng., 28 (4) (2020), 524-555.
  • [27] B. Keskin, Inverse problems for one dimensional conformable fractional Dirac type integro differential system, Inverse Problems, 36 (6) (2020), 065001.
  • [28] E. Yilmaz, T. Gulsen, E.S. Panakhov, Existence results for a conformable type Dirac system on time scales in quantum physics, Appl. Comput. Math., 21 (3) (2022), 279-291.
  • [29] I. Adalar, A.S. Ozkan, Inverse problems for a conformable fractional Sturm-Liouville operators, J. Inverse III Posed Probl., 28 (6) (2020), 775-782.
  • [30] Y. Çakmak, Inverse nodal problem for a conformable fractional diffusion operator, Inverse Probl. Sci. Eng., 29 (9) (2021), 1308-1322.
  • [31] Y. Çakmak, Trace formulae for a conformable fractional diffusion operator, Filomat, 36 (14) (2022), 4665-4674.
  • [32] E. Koç, Y. Çakmak, a-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.
  • [33] B.P. Allahverdiev, H. Tuna, Y. Yalçınkaya, Conformable fractional Sturm-Liouville equation, Math. Methods Appl. Sci., 42 (10) (2019), 3508-3526.
  • [34] D. Baleanu, Z.B. Guvenc, J.T. Machado, New Trends in Nanotechnology and Fractional Calculus Applications, Springer, New York, 2010.
  • [35] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010.
  • [36] C.A. Monje, Y. Chen, B.M. Vinagre, et al., Fractional-Order Systems and Controls: Fundamentals and Applications, Springer, London, 2010.
  • [37] A. Palfalvi, Efficient solution of a vibration equation involving fractional derivatives, Internat. J. Non-Linear Mech., 45 (2010), 169–175.
  • [38] M.F. Silva, J.A.T. Machado, Fractional order PDa joint control of legged robots, J. Vib. Control, 12 (12) (2006), 1483–1501

Yıl 2023, Cilt: 6 Sayı: 3, 142 - 147, 17.09.2023

Yaşar Çakmak

https://doi.org/10.33434/cams.1281434

Öz

Kaynakça

  • [1] V.A. Ambarzumian, Uber eine frage der eigenwerttheorie, Zeitschrift f¨ur Physik, 53 (1929), 690-695.
  • [2] H.H. Chern, C.K. Law, H.J. Wang, Extensions of Ambarzumyan’s theorem to general boundary conditions, J. Math. Anal. Appl., 263 (2001), 333-342. Corrigendum: J. Math. Anal. Appl., 309 (2) (2005), 764-768.
  • [3] E.M. Harrell, On the extension of Ambarzunyan’s inverse spectral theorem to compact symmetric spaces, Amer. J. Math., 109 (1987), 787-795.
  • [4] M. Horvath, On a theorem of Ambarzumyan, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 131 (2001), 899-907.
  • [5] M. Kiss, An n-dimensional Ambarzumyan type theorem for Dirac operators, Inverse Problems, 20 (5)(2004), 1593-1597.
  • [6] C.F. Yang, X.P. Yang, Some Ambarzumyan-type theorems for Dirac operators, Inverse Problems, 25 (9) (2009), 095012, 13 pages.
  • [7] C.F. Yang, Z.Y. Huang, X.P. Yang, Ambarzumyan’s theorems for vectorial Sturm-Liouville systems with coupled boundary conditions, Taiwanese J. Math., 14 (4) (2010), 1429-1437.
  • [8] C.L. Shen, On some inverse spectral problems related to the Ambarzumyan problem and the dual string of the string equation, Inverse Problems, 23 (6) (2007), 2417-2436.
  • [9] N.V. Kuznetsov, Generalization of a theorem of V.A. Ambarzumyan, Doklady Akademii Nauk SSSR., 146 (1962), 1259- 1262, (in Russian).
  • [10] H.H. Chern, C.L. Shen, On the n-dimensional Ambarzumyan’s theorem, Inverse Problems, 13 (1) (1997), 15-18.
  • [11] C.F. Yang, X.P. Yang, Ambarzumyan’s theorem with eigenparameter in the boundary conditions, Acta Math. Sci., 31 B(4) (2011), 1561-1568.
  • [12] V.A. Yurko, On Ambarzumyan-type theorems, Appl. Math. Lett., 26 (4) (2013), 506-509.
  • [13] K. Marton, An n-dimensional Ambarzumyan type theorem for Dirac operators, Inverse Problems, 20 (5) (2004), 1593-1597.
  • [14] A.A. Kırac, Ambarzumyan’s theorem for the quasi-periodic boundary conditions, Anal. Math. Phys., 6 (2016), 297-300.
  • [15] H. Koyunbakan, D. Lesnic, E.S. Panakhov, Ambarzumyan type theorem for a quadratic Sturm-Liouville operator, Turkish Journal of Sciences and Technology, 8 (1) (2013), 1-5.
  • [16] G. Freiling, V.A. Yurko, Inverse Sturm-Liouville Problems and Their Applications, Nova Science Publishers, New York, 2001.
  • [17] E. Yilmaz, H. Koyunbakan, Some Ambarzumyan type theorems for Bessel operator on a finite interval, Differ. Equ. Dyn. Syst., 27 (4) (2019), 553–559.
  • [18] R. Khalil, M. Al Horania, A. Yousefa, et al., A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70.
  • [19] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66.
  • [20] A. Atangana, D. Baleanu, A. Alsaedi, New properties of conformable derivative, Open Math., 13 (2015), 889-898.
  • [21] M. Abu Hammad, R. Khalil, Abel’s formula and Wronskian for conformable fractional differential equations, International J. Differ. Equ. Appl., 13 (3) (2014), 177-183.
  • [22] O.T. Birgani, S. Chandok, N. Dedovic, S. Radenoviç, 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.
  • [23] D. Zhao, M. Luo, General conformable fractional derivative and its physical interpretation, Calcolo, 54 (3) (2017), 903-917.
  • [24] H.W. Zhou, S. Yang, S.Q. Zhang, Conformable derivative approach to anomalous diffusion, Physica A: Statistical Mechanics and its Applications, 491 (2018), 1001-1013.
  • [25] F. Jarad, E. Uğurlu, T. Abdeljawad, D. Baleanu, On a new class of fractional operators, Adv. Difference Equ., 2017 (2017), Article ID 247, 16 pages.
  • [26] H. Mortazaasl, A. Jodayree Akbarfam, Trace formula and inverse nodal problem for a conformable fractional Sturm-Liouville problem, Inverse Probl. Sci. Eng., 28 (4) (2020), 524-555.
  • [27] B. Keskin, Inverse problems for one dimensional conformable fractional Dirac type integro differential system, Inverse Problems, 36 (6) (2020), 065001.
  • [28] E. Yilmaz, T. Gulsen, E.S. Panakhov, Existence results for a conformable type Dirac system on time scales in quantum physics, Appl. Comput. Math., 21 (3) (2022), 279-291.
  • [29] I. Adalar, A.S. Ozkan, Inverse problems for a conformable fractional Sturm-Liouville operators, J. Inverse III Posed Probl., 28 (6) (2020), 775-782.
  • [30] Y. Çakmak, Inverse nodal problem for a conformable fractional diffusion operator, Inverse Probl. Sci. Eng., 29 (9) (2021), 1308-1322.
  • [31] Y. Çakmak, Trace formulae for a conformable fractional diffusion operator, Filomat, 36 (14) (2022), 4665-4674.
  • [32] E. Koç, Y. Çakmak, a-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.
  • [33] B.P. Allahverdiev, H. Tuna, Y. Yalçınkaya, Conformable fractional Sturm-Liouville equation, Math. Methods Appl. Sci., 42 (10) (2019), 3508-3526.
  • [34] D. Baleanu, Z.B. Guvenc, J.T. Machado, New Trends in Nanotechnology and Fractional Calculus Applications, Springer, New York, 2010.
  • [35] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010.
  • [36] C.A. Monje, Y. Chen, B.M. Vinagre, et al., Fractional-Order Systems and Controls: Fundamentals and Applications, Springer, London, 2010.
  • [37] A. Palfalvi, Efficient solution of a vibration equation involving fractional derivatives, Internat. J. Non-Linear Mech., 45 (2010), 169–175.
  • [38] M.F. Silva, J.A.T. Machado, Fractional order PDa joint control of legged robots, J. Vib. Control, 12 (12) (2006), 1483–1501
Toplam 38 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Articles
Yazarlar

Yaşar Çakmak 0000-0002-6820-1322

Erken Görünüm Tarihi 15 Eylül 2023
Yayımlanma Tarihi 17 Eylül 2023
Gönderilme Tarihi 11 Nisan 2023
Kabul Tarihi 10 Eylül 2023
Yayımlandığı Sayı Yıl 2023 Cilt: 6 Sayı: 3

Kaynak Göster

APA Çakmak, Y. (2023). Ambarzumyan-Type Theorem for a Conformable Fractional Diffusion Operator. Communications in Advanced Mathematical Sciences, 6(3), 142-147. https://doi.org/10.33434/cams.1281434
AMA Çakmak Y. Ambarzumyan-Type Theorem for a Conformable Fractional Diffusion Operator. Communications in Advanced Mathematical Sciences. Eylül 2023;6(3):142-147. doi:10.33434/cams.1281434
Chicago Çakmak, Yaşar. “Ambarzumyan-Type Theorem for a Conformable Fractional Diffusion Operator”. Communications in Advanced Mathematical Sciences 6, sy. 3 (Eylül 2023): 142-47. https://doi.org/10.33434/cams.1281434.
EndNote Çakmak Y (01 Eylül 2023) Ambarzumyan-Type Theorem for a Conformable Fractional Diffusion Operator. Communications in Advanced Mathematical Sciences 6 3 142–147.
IEEE Y. Çakmak, “Ambarzumyan-Type Theorem for a Conformable Fractional Diffusion Operator”, Communications in Advanced Mathematical Sciences, c. 6, sy. 3, ss. 142–147, 2023, doi: 10.33434/cams.1281434.
ISNAD Çakmak, Yaşar. “Ambarzumyan-Type Theorem for a Conformable Fractional Diffusion Operator”. Communications in Advanced Mathematical Sciences 6/3 (Eylül 2023), 142-147. https://doi.org/10.33434/cams.1281434.
JAMA Çakmak Y. Ambarzumyan-Type Theorem for a Conformable Fractional Diffusion Operator. Communications in Advanced Mathematical Sciences. 2023;6:142–147.
MLA Çakmak, Yaşar. “Ambarzumyan-Type Theorem for a Conformable Fractional Diffusion Operator”. Communications in Advanced Mathematical Sciences, c. 6, sy. 3, 2023, ss. 142-7, doi:10.33434/cams.1281434.
Vancouver Çakmak Y. Ambarzumyan-Type Theorem for a Conformable Fractional Diffusion Operator. Communications in Advanced Mathematical Sciences. 2023;6(3):142-7.
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