On some boundary-value problems of Sturm-Liouville for a second order differential equation with fractional derivatives in less significant members

Fluid, gas and plasma mechanics


Аuthors

Aleroev T. S.1*, Khasambiev M. V.1**, Khamzatova Z. U.2***

1. Moscow State University of Civil Engineering, MSUCE, 26, Yaroslavskoe shosse, Moscow, 129337, Russia
2. Chechen State University, 32, Sheripov Str., Grozny, 364907, Russia

*e-mail: aleroev@mail.ru
**e-mail: hasambiev@mail.ru
***e-mail: zura-hamzatova2014@yandex.ru

Abstract

In this paper, we study boundary value problems of Sturm-Liouville type the fractional differential equations. Such kind of problems stay in the spotlight of many authors in the first place due to their activities in various physical processes modeling, such as heat and mass transfer within an ambient of fractal structure and memory; random straying point particle, that starts its movement from the origin according to self-similar fractal set; oscillator movement under elastic forces, peculiar to viscoelastic medium, etc.

It is also worth noting that many direct and inverse problems associated with degenerate hyperbolic equations and equations of mixed hyperbolic-parabolic type are easily reduced to Sturm-Liouville problem for second-order equations with fractional derivations in less significant members.

This work consists of two parts. The first part is devoted to the study of boundary value problems for fractional differential equations, and the second — to the study of the of Sturm-Liouville problem for a second order differential equation with fractional derivatives in the less significant members.

In the first part the paper explores non-selfadjoint integral operators, induced by differential equations of fractional order and boundary conditions of Sturm-Liouville type. It should be noted that the spectral structure of these operators is poorly studied. The method, sated in the first part, allows obtaining estimates for eigenfunctions and eigenvalues.

In the second part of the work, we show that the operator generated by the second order differential expression with fractional derivatives (in the sense of Caputo) in the less siginificant members and boundary conditions of the Sturm-Liouville is an operator of Keldysh type. All the eigenfunctions of this operator are written out. We proved by methods of perturbation of linear operators theory that this operator possesses the main oscillation properties, ie all its eigenvalues are simple and of the same sign.

The obtained results show that the second order differential equations with fractional derivative in the less significant term can be applied to study the motion of the oscillator with viscoelastic damping.

Keywords:

operator of Keldysh type, perturbation, discrete spectrum, fractional differentiation, eigenvalues, resolution, the core of the integral operator

References

  1. Aleroev T.S. Sibirskie elektronnye matematicheskie izvestiya, 2013, no. 10, pp. 41–55.
  2. Dzhrbashyan M.M. Izvestiya Akademii nauk Armyanskoi SSR. Matematika., 1970, vol. 5, no. 2, pp. 71-96.

  3. Malamud M. M., Oridoroga L. L. The analogue of Birkhoff theorem and the completeness of eigenfunctions for differential equations of fractional order. Russian Journal of Mathematical Physics. 2001, vol. 8, no. 3, pp. 287-308.

  4. Aleroev T.S. Differentsial’nye uravneniya, 1982, vol. 18, no. 2, pp. 123-126.

  5. Aleroev T.S., Aleroeva Kh.T. Izvestiya Vuzov. Matematika., 2014, no. 10, pp. 3-12.

  6. Aleroev T.S. Differentsial’nye uravneniya, 1998, vol. 34, no.1, C. 123-125.

  7. Kaufmann E.R. Existence and nonexistence of positive solutions for a nonlinear fractional boundary value problem. Discrete and continuous dynamical systems. 2009, pp. 416-423.

  8. Nahushev A.M. Drobnoe ischislenie i ego primenenie (Fractional calculus and its application), Moscow, FIZMATLIT, 2003, 271 p.

  9. Aleroev T.S. Differentsial’nye uravneniya, 2000, vol. 36, no. 6, pp. 829-830.

  10. Aleroev T.S. Kraevye zadachi dlya differentsial’nykh uravnenii s drobnymi proizvodnymi (Boundary value problems for differential equations with fractional derivatives), Doctor’s thesis, Moscow, MGU, 2000, 120 p.

  11. Aleroev T.S., Aleroeva H.T., Ning-Ming Nie, and Yi-Fa Tang. Boundary Value Problems for Diferential Equations of Fractional Order. Mem. Diferential Equations Math. Phys., 2010, no. 49, pp. 19-82.

  12. Aleroev T.S., Aleroeva H.T. A problem on the zeros of the Mittag-Lefer function and the spectrum of a fractional-order diferential operator. Electron. J. Qual. Theory Difer. Equ., 2009, no. 25, pp. 1-18.

  13. Mainardi F. Fractional Relaxation-Oscillation and fractional Difusion-wave Phe-nomena Chaos. Solutions and Fractals, 1996, vol. 7, no. 9, pp. 1461-1477.

  14. Aleroev T.S. Sibirskii matematicheskii zhurnal, 2005, vol. 46, no. 6. pp. 1201-1207.

  15. Kato T. Teoriya vozmushchenii lineinykh operatorov (Perturbation theory of linear operators), Moscow, Mir, 1972, 740 p.

  16. Loginov B.V. Izvestiya AN UzSSR. Fiziko-matematicheskie nauki., 1963, no. 6, pp. 14-19.

  17. Popov A.Ju. Fundamental’naya i prikladnaya matematika, 2006, vol. 12, no. 6. pp. 137–155.

  18. Keldysh M.V. Doklady AN SSSR, 1951, vol. 77, no. 1, pp. 11–14.

  19. Ingman D., Suzdalnitsky J. Iteration method for equation of viscoelastic motion with fractional differential operator of damping. Comp. Methods Appl. Mech. Engrg. 190 (2001) 5027–5036.

  20. Larionov E.A., Zverjaev E.M., Aleroev T.S K teorii slabogo vozmushcheniya normal’nykh operatorov (The theory of weak perturbations of normal operators), Moscow, IPM RAN, 2014, no. 14, 26 p.

  21. Gohberg I.C., Krejn M.G. Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gil’bertovom prostranstve. (Introduction to the theory of linear nonselfadjoint operators in Hilbert space), Moscow, Nauka, 1965, 448 p.

  22. Titchmarsh Je.Ch. Razlozheniya po sobstvennym funktsiyam, svyazannye s differentsial’nymi uravneniyami vtorogo poryadka. Tom1. (Expansions in eigenfunctions connected with differential equations of the second order.Vol.1.), Moscow, Izdatel’stvo inostrannoj literatury, 1960, 278 p.


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