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DOI: 10.23671/VNC.2016.3.5873
Neumann Problem for an Ordinary Differential Equation of Fractional Order
Gadzova L. H.
Vladikavkaz Mathematical Journal 2016. Vol. 18. Issue 3.
Abstract: A linear ordinary differential equation of fractional order with constant coefficients is considered in the paper. Such equation should be subsumed into the class
of discretely distributed order, or multi-term differential equations. The fractional differentiation is given by the Caputo derivative. We solve The Nuemann problem for the equation under study, prove the existence and uniqueness of the solution, find an explicit representation for solution in terms of the Wright function, and construct the respective Green function. It is also proveâ that the real part of the spectrum of the problem may consist at most of a finite number of eigenvalues.
Keywords: boundary value problem, operator of fractional differentiation, Riemann--Liouville operator, Caputo operator
Nakhushev A. M. Drobnoe ischislenie i ego primenenie [Fractional
calculus and its application]. M.: Fizmatlit, 2003. 272 p. [in
Russian]
Barrett J. H. Differential equations of non-integer order. Canadian
J.Math. 1954. vol. 6, no 4. pp. 529-541.
Dzhrbashyan M. M., Nersesyan A.B. Fractional Derivatives and the
Cauchy Problem for Fractional Differential Equations. Izv. Akad.
Nauk Armjan. SSR Ser. Mat., 1968, vol. 3, no. 1, pp. 3-28. [in
Russian]
Dzhrbashyan M. M. A Boundary Value Problem for a Fractional
Sturm-Liouville Type Differential Operator.Izv. Akad. Nauk Armjan.
SSR, 1970, vol. 5, no. 2, pp. 71-96. [in Russian]
Nakhushev A. M. The Sturm-Liouville Problem for a Second Order
Ordinary Differential Equation with Fractional Derivatives in the
Lower-Order Terms. Dokl. Akad. Nauk SSSR, 1977, vol. 234, no. 2, pp.
308-311. [in Russian]
Ozturk I. On the Theory of Fractional Differential Equation. Dokl.
Adygsk. (Cherkessk) Mezhdunar. Akad. Nauk, 1998, vol. 3, no. 2, pp.
35-39. [in Russian]
Pskhu A. V. On the Theory Cauchy Problem for Linear Ordinary
Differential Equation of Fractional Order. Dokl. Adygsk.
(Cherkessk.) Mezhdunar. Akad. Nauk, 2009, vol. 11, no. 1, pp. 61-
65. [in Russian]
Pskhu A. V. Initial-value problem for a linear ordinary differential
equation of noninteger order. Sbornik: Mathematics, 2011, vol. 202,
no 4. pp. 571-582.
Gadzova L. H. Generalized Dirichlet problem for a fractional linear
differential equation with constant coeffcients. Differential
Equations, 2014, vol. 50, no 1, pp. 122-127.
Kilbas A. A., Srivastava H. M., Trujillo J. J. Theory and
applications of fractional differential equations. North-Holland
Math. Stud., Elsevier, Amsterdam. 2006. vol. 204.
Gadzova L. Kh. Dirichlet problem for an ordinary differential
equation of fractional order. Dokl. Adygsk. (Cherkessk.) Mezhdunar.
Akad. Nauk, 2013, vol. 15, no. 2, pp. 36-39. [in Russian]
Gadzova L. Kh. On the theory of boundary value problems for
differential equations with fractional order derivative of Caputo.
Dokl. Adygsk. (Cherkessk.) Mezhdunar. Akad. Nauk. 2014. vol. 16, no
2. pp. 34-40. [in Russian]
Bagley R. L., Torvik P. J. Fractional Calculus in the Transient
Analysis of Viscoelastically Damped Structures. AIAA Journal, 1985.
vol. 23, no 6. pp. 918-925.
Samko S. G., Kilbas A. A., Marichev O. I. Integraly i proizvodnye
drobnogo poryadka i nekotorye ikh prilozheniya [Fractional Integral
and Derivatives]. Minsk.: Nauka i tekhnika, 1987. 688 p. [in
Russian]
Nakhushev A. M., Tkhakakhov R. B. On the Continual Analogues of
Reologie Equations of Condition and Logistic Law of Polimers
Viscus-Elastic Feature Chanqing. Dokl. Adygsk. (Cherkessk.)
Mezhdunar. Akad. Nauk., 1995. vol. 1, no 2. pp. 6-11. (in Russian)
Wright E. M. On the coeffcients of power series having exponential
singularities. J.London Math. Soc., 1933. vol. 8, no 29. pp. 71-79.
Pskhu A. V. Uravneniya v chastnykh proizvodnykh drobnogo poryadka
[Partial Differential Equations of Fractional Order]. M.: Nauka,
2005. 199 p. [in Russian]
Shtokala I. Z. Operatsionnoe ischislenie (obobshchenie i
prilozheniya) [Operational Calculus (Generalization and
Application)]. Kiev: Naukova dumka, 1972. 304 p. [in Russian]
Kudryavtsev L. D. Kurs matematicheskogo analiza [Course of
Mathematical Analysis], vol. II. M.: Vysshaya shkola, 1981. 584 p.
[in Russian]
