Symmetric differential operators of fractional order and their extensions

Tokmagambetov, N.; Torebek, B.

doi:10.1090/mosc/279

Symmetric differential operators of fractional order and their extensions
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by N. E. Tokmagambetov and B. T. Torebek
Translated by: Christopher D. Hollings PDF

Trans. Moscow Math. Soc. 2018, 177-185 Request permission

Abstract:

This paper is devoted to the description of symmetric operators and the justification of Green’s formula for a fractional analogue of the Sturm–Liouville operator of order $2\alpha$, where $\frac {1}{2}<\alpha <1$.

References

A. A. Dezin, Obshchie voprosy teorii granichnykh zadach, “Nauka”, Moscow, 1980 (Russian). MR 596223
V. I. Gorbachuk and M. L. Gorbachuk, Granichnye zadachi dlya differentsial′no-operatornykh uravneniĭ, “Naukova Dumka”, Kiev, 1984 (Russian). MR 776604
Qasem M. Al-Mdallal, On the numerical solution of fractional Sturm-Liouville problems, Int. J. Comput. Math. 87 (2010), no. 12, 2837–2845. MR 2728212, DOI 10.1080/00207160802562549
Tomasz Blaszczyk and Mariusz Ciesielski, Numerical solution of fractional Sturm-Liouville equation in integral form, Fract. Calc. Appl. Anal. 17 (2014), no. 2, 307–320. MR 3181056, DOI 10.2478/s13540-014-0170-8
Łukasz Płociniczak, Eigenvalue asymptotics for a fractional boundary-value problem, Appl. Math. Comput. 241 (2014), 125–128. MR 3223415, DOI 10.1016/j.amc.2014.05.029
Hassan Khosravian-Arab, Mehdi Dehghan, and M. R. Eslahchi, Fractional Sturm-Liouville boundary value problems in unbounded domains: theory and applications, J. Comput. Phys. 299 (2015), 526–560. MR 3384739, DOI 10.1016/j.jcp.2015.06.030
Jing Li and Jiangang Qi, Eigenvalue problems for fractional differential equations with right and left fractional derivatives, Appl. Math. Comput. 256 (2015), 1–10. MR 3316043, DOI 10.1016/j.amc.2014.12.146
M. M. Džrbašjan, A boundary value problem for a Sturm-Liouville type differential operator of fractional order, Izv. Akad. Nauk Armjan. SSR Ser. Mat. 5 (1970), no. 2, 71–96 (Russian, with Armenian and English summaries). MR 0414982
A. M. Nahušev, The Sturm-Liouville problem for a second order ordinary differential equation with fractional derivatives in the lower terms, Dokl. Akad. Nauk SSSR 234 (1977), no. 2, 308–311 (Russian). MR 0454145
T. S. Aleroev, The Sturm-Liouville problem for a second-order differential equation with fractional derivatives in the lower terms, Differentsial′nye Uravneniya 18 (1982), no. 2, 341–342, 367 (Russian). MR 649679
Anatoly M. Sedletskii, On zeros of Laplace transform of finite measure, Integral Transform. Spec. Funct. 1 (1993), no. 1, 51–59. MR 1421434, DOI 10.1080/10652469308819008
I. V. Ostrovskiĭ and I. N. Peresyolkova, Nonasymptotic results on distribution of zeros of the function $E_\rho (z,\mu )$, Anal. Math. 23 (1997), no. 4, 283–296 (English, with Russian summary). MR 1629981, DOI 10.1007/BF02789843
M. M. Malamud and L. L. Oridoroga, On some questions of the spectral theory of ordinary differential equations of fractional order, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki 9 (1998), 39–47 (English, with Ukrainian summary). MR 1704850
T. S. Aleroev, On the eigenvalues of a boundary value problem for a fractional-order differential operator, Differ. Uravn. 36 (2000), no. 10, 1422–1423, 1440 (Russian, with Russian summary); English transl., Differ. Equ. 36 (2000), no. 10, 1569–1570. MR 1838493, DOI 10.1007/BF02757400
A. Yu. Popov, On the number of real eigenvalues of a boundary value problem for a second-order equation with a fractional derivative, Fundam. Prikl. Mat. 12 (2006), no. 6, 137–155 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 151 (2008), no. 1, 2726–2740. MR 2314136, DOI 10.1007/s10948-008-0169-7
A. V. Agibalova, On the completeness of systems of root functions of a fractional-order differential operator with matrix coefficients, Mat. Zametki 88 (2010), no. 2, 317–320 (Russian); English transl., Math. Notes 88 (2010), no. 1-2, 287–290. MR 2867057, DOI 10.1134/S0001434610070266
T. S. Aleroev and Kh. T. Aleroeva, On a class of nonselfadjoint operators concomitant to differential equations of fractional order, Izv. Vyssh. Uchebn. Zaved. Mat. 10 (2014), 3–12 (Russian, with English and Russian summaries). MR 3379574
Małgorzata Klimek, Tatiana Odzijewicz, and Agnieszka B. Malinowska, Variational methods for the fractional Sturm-Liouville problem, J. Math. Anal. Appl. 416 (2014), no. 1, 402–426. MR 3182768, DOI 10.1016/j.jmaa.2014.02.009
L. M. Eneeva, A boundary value problem for a differential equation with derivatives of fractional order with different origins, Vestnik KRAUNTS. Fiz.-Mat. Nauki (2015), no. 2(11), 39–44 (Russian); English translation: Bulletin KRASEC. Phys. Math. Sci. 11(2) (2015), 36–40.
M. Klimek and O. P. Agrawal, Fractional Sturm-Liouville problem, Comput. Math. Appl. 66 (2013), no. 5, 795–812. MR 3089387, DOI 10.1016/j.camwa.2012.12.011
L. M. Isaeva and T. S. Aleroev, Qualitative properties of the one-dimensional fractional differential equation of advection-diffusion, Vestnik MGSU 7 (2014), 28–33 (Russian).
Mohsen Zayernouri and George Em Karniadakis, Discontinuous spectral element methods for time- and space-fractional advection equations, SIAM J. Sci. Comput. 36 (2014), no. 4, B684–B707. MR 3240858, DOI 10.1137/130940967
M. Klimek, 2D space-time fractional diffusion on bounded domain — Application of the fractional Sturm–Liouville theory, 2015 20th International Conference on Methods and Models in Automation and Robotics (MMAR), Miedzyzdroje, 2015, pp. 309–314.
Liangliang Qiu, Weihua Deng, and Jan S. Hesthaven, Nodal discontinuous Galerkin methods for fractional diffusion equations on 2D domain with triangular meshes, J. Comput. Phys. 298 (2015), 678–694. MR 3374572, DOI 10.1016/j.jcp.2015.06.022
Stefan G. Samko, Anatoly A. Kilbas, and Oleg I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. Theory and applications; Edited and with a foreword by S. M. Nikol′skiĭ; Translated from the 1987 Russian original; Revised by the authors. MR 1347689
A. M. Nakhushev, Fractional calculus and its application, Fizmatlit, Moscow, 2003 (Russian).
Anatoly A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006. MR 2218073
M. A. Naĭmark, Lineĭ nye differentsial′nye operatory, Izdat. “Nauka”, Moscow, 1969 (Russian). Second edition, revised and augmented; With an appendix by V. È. Ljance. MR 0353061
Michael Ruzhansky and Niyaz Tokmagambetov, Nonharmonic analysis of boundary value problems, Int. Math. Res. Not. IMRN 12 (2016), 3548–3615. MR 3544614, DOI 10.1093/imrn/rnv243
Julio Delgado, Michael Ruzhansky, and Niyaz Tokmagambetov, Schatten classes, nuclearity and nonharmonic analysis on compact manifolds with boundary, J. Math. Pures Appl. (9) 107 (2017), no. 6, 758–783 (English, with English and French summaries). MR 3650324, DOI 10.1016/j.matpur.2016.10.005
M. Ruzhansky and N. Tokmagambetov, Nonharmonic analysis of boundary value problems without WZ condition, Math. Model. Nat. Phenom. 12 (2017), no. 1, 115–140. MR 3614589, DOI 10.1051/mmnp/201712107
Baltabek Kanguzhin and Niyaz Tokmagambetov, The Fourier transform and convolutions generated by a differential operator with boundary condition on a segment, Fourier analysis, Trends Math., Birkhäuser/Springer, Cham, 2014, pp. 235–251. MR 3362022
Baltabek Kanguzhin, Niyaz Tokmagambetov, and Kanat Tulenov, Pseudo-differential operators generated by a non-local boundary value problem, Complex Var. Elliptic Equ. 60 (2015), no. 1, 107–117. MR 3295092, DOI 10.1080/17476933.2014.896351