# IJPAM: Volume 113, No. 4 (2017)

# Title

A SAMARSKII-IONKIN PROBLEM FOR TWO-DIMENSIONALPARABOLIC EQUATION WITH THE CAPUTO FRACTIONAL

DIFFERENTIAL OPERATOR

# Authors

Abdumauvlen S. Berdyshev, Bakhtiyor J. KadirkulovAbay Kazakh National Pedagogical University

Almaty, KAZAKHSTAN

Tashkent State Institute of Oriental Studies

Tashkent, UZBEKISTAN

# Abstract

In the work, the authors consider a Samarskii-Ionkin type non-local problem for a fourth-order partial differential equation with the Caputo fractional differential operator in a spatial domain. Applying the method of separation of variables the authors prove the theorem of the existence and uniqueness of the regular solution of these problems.# History

**Received: **December 15, 2016
**Revised: **January 30, 2017
**Published: **March 30, 2017

# AMS Classification, Key Words

**AMS Subject Classification: **35M10, 35R11, 35R30
**Key Words and Phrases: **Samarskii-Ionkin type non-local problem, Caputo fractional differential operator, fractional differential equation, eigenfunction, associated function, completeness, biorthogonal property, Riesz basis

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## Bibliography

- 1
- K.A. Anatoly, S.M. Hari, T.J. Juan,
*Theory and applications of fractional differential equations. North-Holland Mathematics Studies*, Elsevier Science B.V., Amsterdam, 204, (2006). - 2
- S.J. Isabel, J.A. Tenreiro Machado, Fractional control of heat diffusion systems,
*Nonlinear Dyn.*, 54, (2008), 263-282. - 3
- R.C. Koeller, Application of fractional calculus to the theory of viscoelasticity,
*J. Appl. Mech.*, 51, (1984), 299-307. - 4
- N.I. Ionkin, Solution of a boundary-value problem of the theory of heat conduction with non-classical boundary condition,
*Differ. Equ.*, 13, No. 2 (1977), 294-304. - 5
- Z.A. Nakhusheva, The Samarskii modified problem for the nonlocal diffrential equation,
*Doklady AMAN*, 2, No. 2 (1997), 36-41 (in Russian). - 6
- E.I. Moiseev, On solving of non-local boundary-value problem by spectral method,
*Differ. Equ.*, 35, No. 8 (1999), 1094-1100. - 7
- A.S. Berdyshev, A. Cabada, B.J. Kadirkulov, The Samarskii-Ionkin type problem for the fourth order parabolic equation with fractional differential operator,
*Computers and Mathematics with Applications*, 62, No. 10 (2011), 3884-3893. - 8
- I.N. Ionkin, A.B. Morozova, The two-dimensional heat equation with nonlocal boundary conditions,
*Differential Equations*, 36, No. 7 (2000), 982-987. - 9
- M. Kirane, S.A. Malik, M.A. Al-Gwaiz, An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions,
*Mathematical Methods in the Applied Sciences*, 36 (2013), 1056-1069. - 10
- T.X. Xiong, Q. Zhou, C.Y. Hon, An inverse problem for fractional diffusion equation in 2-dimensional case: stability analysis and regularization,
*Journal of Mathematical Analysis and Applications*, 393, No. 1 (2012), 185�-199. - 11
- E. Kamke,
*Handbook of ordinary differential equations*, Nauka, Moscow (1965). - 12
- A.S. Berdyshev, B.J. Kadirkulov, On a Nonlocal Problem for a Fourth-Order Parabolic Equation with the Fractional Dzhrbashyan-Nersesyan Operator,
*Differential Equations*, 52, No. 1 (2016), 122-127. - 13
- K. Bari, Biorthogonal systems and bases in Hilbert spaces,
*Uch.Zap., MGU*, 148, No. 4 (1951), 69-107. - 14
- V.A. Il'in, M. Barnovska, Riesz basis of a spectral problem with an infinite-to-one eigenvalues,
*Mathematica Slovaca*, 35, No. 2 (1985), 161-167. - 15
- M.M. Dzhrbashian,
*Integral transformation and representation of functions in complex domain*, Moscow (1966). - 16
- V.A. Ilyin, E.G. Poznyak,
*Basics of mathematical analysis. Part II*, Moscow (1973). - 17
- A.S. Berdyshev, B.J. Kadirkulov, A Samarskii-Ionkin problem for two-dimensional parabolic equation with the caputo fractional differential operator,
*Kazakh Mathematical Journal*, 17, No. 1 (2017), 31-32.

# How to Cite?

**DOI: 10.12732/ijpam.v113i4.6**

International Journal of Pure and Applied Mathematics

**How to cite this paper?****Source:****ISSN printed version:**1311-8080

**ISSN on-line version:**1314-3395

**Year:**2017

**Volume:**113

**Issue:**4

**Pages:**53 - 64

Google Scholar; DOI (International DOI Foundation); WorldCAT.

**This work is licensed under the Creative Commons Attribution International License (CC BY).**