Research Article
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Year 2021, , 1 - 5, 30.06.2021
https://doi.org/10.47000/tjmcs.823775

Abstract

References

  • [1] Abdeljawad, T., On conformable fractional calculus, J. Comput Appl. Math., 279(2015), 57-66.
  • [2] Allahverdiev, B. P., On dilation theory and spectral analysis of dissipative Schrödinger operators in Weyl’s limit-circle, Izv. Akad. Nauk. SSSR 54(2)(1990), English Transl. in: Math USSR Izv. 36(2(1991)), 247-262.
  • [3] Balachandran, K., Kiruthika, S., Rivero, M., Trujillo, J. J., Existence of Solutions for Fractional Delay Integrodifferential Equations, Journal of Applied Nonlinear Dynamics, 1(2012), 309-319.
  • [4] Bruk, V. M., On a class of boundary –value problemswith a spectral parameter in the boundary conditions, Mat. Sb., 100(1976), 210-216.
  • [5] Canoglu, A., Allahverdiev, B. P., Self-adjoint and dissipative extensions of a symmetric Schrödinger operator, Math. Balkanica, 17(1-2)(2003), 113-120.
  • [6] Canoglu, A., Allahverdiev, B. P., Extensions of a symmetric second-order differential operator, Math. Balkanica, 12(3-4)(1998), 271-275.
  • [7] Gorbachuk, V. I., Gorbachuk, M. L., Boundary Value Problems for Operator Differential Equaitions, Naukova Dumka, Kiev, 1984; English Trans.: Kluwer, Dordrecht, 1991.
  • [8] Khalil, R., Horani, M., Al., Yousef, A., Sababheh, M. A., New Definition of Fractional Derivative, J. Computat. Appl. Math., 264(2014), 65-70.
  • [9] Kilbas, A., Srivastava, H., Trujillo, J.,Theory and Applications of Fractional Differential Equations, in: Math. Studies. North-Holland, New York, 2006.
  • [10] Kochubei, A. N., Extensions of symmetric operators and symmetric binary relations, Mat. Zametki 17(1975), 41-48; English transl. in Math. Notes 17 (1975), 25-28.
  • [11] Kulish, V. V., Lage, J. L., Application of Fractional Calculus to Fluid Mechanics, J. Fluids Eng., 124(3)(2002), 803-806.
  • [12] Magin, R. L., Fractional Calculus Models of Complex Dynamics in Biological Tissues, Comput. Math. Appl., 59(5)(2010), 1586-1593.
  • [13] Miller, K. S., Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York, 1993.
  • [14] Naimark, M. A., Linear Differential Operators, 2nd Ed., Nauka, Moscow, 1969; English Trans. of 1st Ed., Parts I,II Ungar, N.York, 1967, 1968.
  • [15] Oldham, K. B., Spanier J., The Fractional Calculus, Academic Press, New York, 1974.
  • [16] Podlubny, I., Fractional Differential Equations, Academic Press, New York, 1999.
  • [17] Rofe-Beketov, F. S., Self-adjoint extensions of differential operators in a space of vector valued functions, Dokl. Akad. Nauk SSSR 184 (1969), 1034-1037, English transl. in Soviet Math. Dokl. 10 (1969), 188-192.
  • [18] Rossikhin, Y. A., Shitikova, M. V., Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl. Mech. Rev. 50(1997), 15-67.
  • [19] Samko, S. G., Kilbas, A. A., Marichev, O. I., Fractional Integrals and Derivatives-Theory and Applications, Gordon and Breach, Longhorne, PA, 1993.
  • [20] Schneider, W., Wyss, W., Fractional Diffusion and Wave Equations, J. Math. Phys., 30(1)(1989), 134-144.
  • [21] Von Neumann, J., Allgemeine Eigenwertheorie Hermitischer Functionaloperatoren, Math. Ann. 102(1929), 49-131.
  • [22] Wang, Y., Zhou, J., Li, Y., Fractional Sobolev.s space on time scale via comformable fractional calculus and their application to a fractional differential equation on time scale, Adv. Math. Phys, (2016), Art. ID 963491.

On the Dissipative Extensions of the Conformable Fractional Sturm-Liouville Operator

Year 2021, , 1 - 5, 30.06.2021
https://doi.org/10.47000/tjmcs.823775

Abstract

In this work, we consider singular conformable fractional Sturm-Liouville
operators defined by the expression
\[
\varrho (y)=-T_{\alpha }^{2}y(t)+\frac{\xi ^{2}-\frac{1}{4}}{t^{2}}y(t)+%
p(t)y(t),\
\]
where $0 < t < \infty ,\ \xi \geq1~$and$\ p(.)\ $is real-valued functions defined on $[0,\infty )$ and satisfy the condition$\ p\left( .\right) \in L_{\alpha, loc}^{1}(0,\infty )$. We construct a space of boundary values for minimal symmetric singular conformable fractional Sturm-Liouville operators in limit-circle case at singular end point. Finally, we give a description of all maximal dissipative, accumulative and self-adjoint extensions of conformable fractional Sturm-Liouville operators with the help of boundary conditions.

References

  • [1] Abdeljawad, T., On conformable fractional calculus, J. Comput Appl. Math., 279(2015), 57-66.
  • [2] Allahverdiev, B. P., On dilation theory and spectral analysis of dissipative Schrödinger operators in Weyl’s limit-circle, Izv. Akad. Nauk. SSSR 54(2)(1990), English Transl. in: Math USSR Izv. 36(2(1991)), 247-262.
  • [3] Balachandran, K., Kiruthika, S., Rivero, M., Trujillo, J. J., Existence of Solutions for Fractional Delay Integrodifferential Equations, Journal of Applied Nonlinear Dynamics, 1(2012), 309-319.
  • [4] Bruk, V. M., On a class of boundary –value problemswith a spectral parameter in the boundary conditions, Mat. Sb., 100(1976), 210-216.
  • [5] Canoglu, A., Allahverdiev, B. P., Self-adjoint and dissipative extensions of a symmetric Schrödinger operator, Math. Balkanica, 17(1-2)(2003), 113-120.
  • [6] Canoglu, A., Allahverdiev, B. P., Extensions of a symmetric second-order differential operator, Math. Balkanica, 12(3-4)(1998), 271-275.
  • [7] Gorbachuk, V. I., Gorbachuk, M. L., Boundary Value Problems for Operator Differential Equaitions, Naukova Dumka, Kiev, 1984; English Trans.: Kluwer, Dordrecht, 1991.
  • [8] Khalil, R., Horani, M., Al., Yousef, A., Sababheh, M. A., New Definition of Fractional Derivative, J. Computat. Appl. Math., 264(2014), 65-70.
  • [9] Kilbas, A., Srivastava, H., Trujillo, J.,Theory and Applications of Fractional Differential Equations, in: Math. Studies. North-Holland, New York, 2006.
  • [10] Kochubei, A. N., Extensions of symmetric operators and symmetric binary relations, Mat. Zametki 17(1975), 41-48; English transl. in Math. Notes 17 (1975), 25-28.
  • [11] Kulish, V. V., Lage, J. L., Application of Fractional Calculus to Fluid Mechanics, J. Fluids Eng., 124(3)(2002), 803-806.
  • [12] Magin, R. L., Fractional Calculus Models of Complex Dynamics in Biological Tissues, Comput. Math. Appl., 59(5)(2010), 1586-1593.
  • [13] Miller, K. S., Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York, 1993.
  • [14] Naimark, M. A., Linear Differential Operators, 2nd Ed., Nauka, Moscow, 1969; English Trans. of 1st Ed., Parts I,II Ungar, N.York, 1967, 1968.
  • [15] Oldham, K. B., Spanier J., The Fractional Calculus, Academic Press, New York, 1974.
  • [16] Podlubny, I., Fractional Differential Equations, Academic Press, New York, 1999.
  • [17] Rofe-Beketov, F. S., Self-adjoint extensions of differential operators in a space of vector valued functions, Dokl. Akad. Nauk SSSR 184 (1969), 1034-1037, English transl. in Soviet Math. Dokl. 10 (1969), 188-192.
  • [18] Rossikhin, Y. A., Shitikova, M. V., Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl. Mech. Rev. 50(1997), 15-67.
  • [19] Samko, S. G., Kilbas, A. A., Marichev, O. I., Fractional Integrals and Derivatives-Theory and Applications, Gordon and Breach, Longhorne, PA, 1993.
  • [20] Schneider, W., Wyss, W., Fractional Diffusion and Wave Equations, J. Math. Phys., 30(1)(1989), 134-144.
  • [21] Von Neumann, J., Allgemeine Eigenwertheorie Hermitischer Functionaloperatoren, Math. Ann. 102(1929), 49-131.
  • [22] Wang, Y., Zhou, J., Li, Y., Fractional Sobolev.s space on time scale via comformable fractional calculus and their application to a fractional differential equation on time scale, Adv. Math. Phys, (2016), Art. ID 963491.
There are 22 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Bilender Paşaoğlu 0000-0002-9315-4652

Hüseyin Tuna 0000-0003-2483-1493

Yüksel Yalçınkaya 0000-0002-1633-8343

Publication Date June 30, 2021
Published in Issue Year 2021

Cite

APA Paşaoğlu, B., Tuna, H., & Yalçınkaya, Y. (2021). On the Dissipative Extensions of the Conformable Fractional Sturm-Liouville Operator. Turkish Journal of Mathematics and Computer Science, 13(1), 1-5. https://doi.org/10.47000/tjmcs.823775
AMA Paşaoğlu B, Tuna H, Yalçınkaya Y. On the Dissipative Extensions of the Conformable Fractional Sturm-Liouville Operator. TJMCS. June 2021;13(1):1-5. doi:10.47000/tjmcs.823775
Chicago Paşaoğlu, Bilender, Hüseyin Tuna, and Yüksel Yalçınkaya. “On the Dissipative Extensions of the Conformable Fractional Sturm-Liouville Operator”. Turkish Journal of Mathematics and Computer Science 13, no. 1 (June 2021): 1-5. https://doi.org/10.47000/tjmcs.823775.
EndNote Paşaoğlu B, Tuna H, Yalçınkaya Y (June 1, 2021) On the Dissipative Extensions of the Conformable Fractional Sturm-Liouville Operator. Turkish Journal of Mathematics and Computer Science 13 1 1–5.
IEEE B. Paşaoğlu, H. Tuna, and Y. Yalçınkaya, “On the Dissipative Extensions of the Conformable Fractional Sturm-Liouville Operator”, TJMCS, vol. 13, no. 1, pp. 1–5, 2021, doi: 10.47000/tjmcs.823775.
ISNAD Paşaoğlu, Bilender et al. “On the Dissipative Extensions of the Conformable Fractional Sturm-Liouville Operator”. Turkish Journal of Mathematics and Computer Science 13/1 (June 2021), 1-5. https://doi.org/10.47000/tjmcs.823775.
JAMA Paşaoğlu B, Tuna H, Yalçınkaya Y. On the Dissipative Extensions of the Conformable Fractional Sturm-Liouville Operator. TJMCS. 2021;13:1–5.
MLA Paşaoğlu, Bilender et al. “On the Dissipative Extensions of the Conformable Fractional Sturm-Liouville Operator”. Turkish Journal of Mathematics and Computer Science, vol. 13, no. 1, 2021, pp. 1-5, doi:10.47000/tjmcs.823775.
Vancouver Paşaoğlu B, Tuna H, Yalçınkaya Y. On the Dissipative Extensions of the Conformable Fractional Sturm-Liouville Operator. TJMCS. 2021;13(1):1-5.