Research Article
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Year 2023, Volume: 44 Issue: 1, 170 - 180, 26.03.2023

Esengül Koç Yaşar Çakmak

https://doi.org/10.17776/csj.1208016
Cited By: 2

Abstract

Keywords

Diffusion operator Integral representation Conformable fractional derivative Spectral data

References

  • [1] Khalil R., Al Horania M., Yousefa A., et al., A New Definition of Fractional Derivative, J. Comput. Appl. Math., 264 (2014) 65-70.
  • [2] Abdeljawad T., On Conformable Fractional Calculus, J. Comput. Appl. Math., 279 (2015) 57-66.
  • [3] Atangana A., Baleanu D., Alsaedi A., New Properties of Conformable Derivative, Open Math., 13 (2015) 889-898.
  • [4] Abu Hammad M., Khalil R., Abel’s Formula and Wronskian for Conformable Fractional Differential Equations, Int. J. Differ. Equ. Appl., 13 (3) (2014) 177-183.
  • [5] Birgani O.T., Chandok S., Dedovic N., Radenoviç S., A Note on Some Recent Results of the Conformable Derivative, Adv. Theory Nonlinear Anal. Appl., 3 (1) (2019) 11-17.
  • [6] Zhao D., Luo M., General Conformable Fractional Derivative and its Physical Interpretation, Calcolo, 54 (2017) 903–917
  • [7] Zhou H.W., Yang S., Zhang S.Q., Conformable Derivative Approach to Anomalous Diffusion, Physica A, 491 (2018) 1001-1013.
  • [8] Chung W.S., Fractional Newton Mechanics with Conformable Fractional Derivative, J. Comput. Appl. Math., 290 (2015) 150-158.
  • [9] Benkhettou N., Hassani S., Torres D.F.M., A Conformable Fractional Calculus on Arbitrary Time Scales, J. King Saud Univ. Sci., 28 (2016) 93–98.
  • [10] Jarad F., Uğurlu E., Abdeljawad T., Baleanu D., On a New Class of Fractional Operators, Adv. in Differ. Equ., 2017 (2017) Article ID 247 .
  • [11] Mortazaasl H., Jodayree Akbarfam A,. Trace Formula and Inverse Nodal Problem for a Conformable Fractional Sturm-Liouville Problem, Inverse Problems in Science and Engineering, 28 (4) (2020) 524-555.
  • [12] Keskin B., Inverse Problems for one Dimentional Conformable Fractional Dirac Type Integro Differential System, Inverse Problems, 36 (6) (2020) 065001.
  • [13] Adalar I., Ozkan A.S., Inverse Problems for a Conformable Fractional Sturm-Liouville Operators, Journal of Inverse and Ill-posed Problems, 28 (6) (2020).
  • [14] Çakmak Y., Inverse Nodal Problem for a Conformable Fractional Diffusion Operator, Inverse Problems in Science and Engineering, 29 (9) (2021) 1308-1322.
  • [15] Allahverdiev B.P., Tuna H., Yalçinkaya Y., Conformable Fractional Sturm-Liouville Equation, Mathematical Methods in the Applied Sciences, 42 (10) (2019) 3508-3526.
  • [16] Gasymov M.G., Gusejnov G.S., Determination of a Diffusion Operator from Spectral Data, Akad. Nauk Azerb. SSR. Dokl., 37 (2) (1981) 19-23.
  • [17] Koyunbakan H., Panakhov E.S., Half-Inverse Problem for Diffusion Operators on the Finite Interval, J. Math. Anal. Appl., 326 (2) (2007) 1024-1030.
  • [18] Yang C.F., New Trace Formulae for a Quadratic Pencil of the Schrodinger Operator, Journal of Mathematical Physics, 51 (3) (2010) 33506-33506-10.
  • [19] Wang Y., Zhou J., Li Y., Fractional Sobolev’s Spaces on Time Scales via Conformable Fractional Calculus and Their Application to a Fractional Differential Equation on Time Scales, Adv. Math. Phys., 2016 (2016) 21.
  • [20]Guseinov G.Sh., Inverse Spectral Problems for a Quadratic Pencil of Sturm-Liouville Operators on a Finite Interval, Spectral Theory of Operators and Its Applications, Elm, Baku, Azerbaijan, 7, (1986) 51–101. (In Russian)
  • [21] Freiling G., Yurko V.A., Inverse Sturm–Liouville Problems and Their Applications, New York: Nova Science Publishers, (2001).

α-Integral Representation of The Solution for A Conformable Fractional Diffusion Operator and Basic Properties of The Operator

Year 2023, Volume: 44 Issue: 1, 170 - 180, 26.03.2023

Esengül Koç Yaşar Çakmak

https://doi.org/10.17776/csj.1208016
Cited By: 2

Abstract

In this paper, we consider a diffusion operator which includes conformable fractional derivatives of order α (0<α≤1) instead of the ordinary derivatives in a traditional diffusion operator. We give an α-integral representation for the solution of this operator and obtain the conditions provided by the kernel functions in this representation. Also, by investigating the basic properties of this operator, we obtain the asymptotics of the data {λ_n,α_n }, which are called the spectral data of the operator.

Keywords

Diffusion operator Integral representation Conformable fractional derivative Spectral data

References

  • [1] Khalil R., Al Horania M., Yousefa A., et al., A New Definition of Fractional Derivative, J. Comput. Appl. Math., 264 (2014) 65-70.
  • [2] Abdeljawad T., On Conformable Fractional Calculus, J. Comput. Appl. Math., 279 (2015) 57-66.
  • [3] Atangana A., Baleanu D., Alsaedi A., New Properties of Conformable Derivative, Open Math., 13 (2015) 889-898.
  • [4] Abu Hammad M., Khalil R., Abel’s Formula and Wronskian for Conformable Fractional Differential Equations, Int. J. Differ. Equ. Appl., 13 (3) (2014) 177-183.
  • [5] Birgani O.T., Chandok S., Dedovic N., Radenoviç S., A Note on Some Recent Results of the Conformable Derivative, Adv. Theory Nonlinear Anal. Appl., 3 (1) (2019) 11-17.
  • [6] Zhao D., Luo M., General Conformable Fractional Derivative and its Physical Interpretation, Calcolo, 54 (2017) 903–917
  • [7] Zhou H.W., Yang S., Zhang S.Q., Conformable Derivative Approach to Anomalous Diffusion, Physica A, 491 (2018) 1001-1013.
  • [8] Chung W.S., Fractional Newton Mechanics with Conformable Fractional Derivative, J. Comput. Appl. Math., 290 (2015) 150-158.
  • [9] Benkhettou N., Hassani S., Torres D.F.M., A Conformable Fractional Calculus on Arbitrary Time Scales, J. King Saud Univ. Sci., 28 (2016) 93–98.
  • [10] Jarad F., Uğurlu E., Abdeljawad T., Baleanu D., On a New Class of Fractional Operators, Adv. in Differ. Equ., 2017 (2017) Article ID 247 .
  • [11] Mortazaasl H., Jodayree Akbarfam A,. Trace Formula and Inverse Nodal Problem for a Conformable Fractional Sturm-Liouville Problem, Inverse Problems in Science and Engineering, 28 (4) (2020) 524-555.
  • [12] Keskin B., Inverse Problems for one Dimentional Conformable Fractional Dirac Type Integro Differential System, Inverse Problems, 36 (6) (2020) 065001.
  • [13] Adalar I., Ozkan A.S., Inverse Problems for a Conformable Fractional Sturm-Liouville Operators, Journal of Inverse and Ill-posed Problems, 28 (6) (2020).
  • [14] Çakmak Y., Inverse Nodal Problem for a Conformable Fractional Diffusion Operator, Inverse Problems in Science and Engineering, 29 (9) (2021) 1308-1322.
  • [15] Allahverdiev B.P., Tuna H., Yalçinkaya Y., Conformable Fractional Sturm-Liouville Equation, Mathematical Methods in the Applied Sciences, 42 (10) (2019) 3508-3526.
  • [16] Gasymov M.G., Gusejnov G.S., Determination of a Diffusion Operator from Spectral Data, Akad. Nauk Azerb. SSR. Dokl., 37 (2) (1981) 19-23.
  • [17] Koyunbakan H., Panakhov E.S., Half-Inverse Problem for Diffusion Operators on the Finite Interval, J. Math. Anal. Appl., 326 (2) (2007) 1024-1030.
  • [18] Yang C.F., New Trace Formulae for a Quadratic Pencil of the Schrodinger Operator, Journal of Mathematical Physics, 51 (3) (2010) 33506-33506-10.
  • [19] Wang Y., Zhou J., Li Y., Fractional Sobolev’s Spaces on Time Scales via Conformable Fractional Calculus and Their Application to a Fractional Differential Equation on Time Scales, Adv. Math. Phys., 2016 (2016) 21.
  • [20]Guseinov G.Sh., Inverse Spectral Problems for a Quadratic Pencil of Sturm-Liouville Operators on a Finite Interval, Spectral Theory of Operators and Its Applications, Elm, Baku, Azerbaijan, 7, (1986) 51–101. (In Russian)
  • [21] Freiling G., Yurko V.A., Inverse Sturm–Liouville Problems and Their Applications, New York: Nova Science Publishers, (2001).
There are 21 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Esengül Koç 0000-0002-9565-5266

Yaşar Çakmak 0000-0002-6820-1322

Submission Date November 21, 2022
Acceptance Date March 6, 2023
Publication Date March 26, 2023
Published in Issue Year 2023 Volume: 44 Issue: 1

Cite

APA Koç, E., & Çakmak, Y. (2023). α-Integral Representation of The Solution for A Conformable Fractional Diffusion Operator and Basic Properties of The Operator. Cumhuriyet Science Journal, 44(1), 170-180. https://doi.org/10.17776/csj.1208016

Cited By

Inverse Problems for a Conformable Fractional Diffusion Operator
Journal of New Theory
https://doi.org/10.53570/jnt.1335702
Ambarzumyan-Type Theorem for a Conformable Fractional Diffusion Operator
Communications in Advanced Mathematical Sciences
https://doi.org/10.33434/cams.1281434

Editor