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

Year 2021, Volume: 13 Issue: 1, 1 - 5, 30.06.2021

Bilender Paşaoğlu , Hüseyin Tuna , Yüksel Yalçınkaya

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.

Keywords

Dissipative extansions, self adjoint extansion, a boundary value space, boundary condition

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.