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An Inverse Source Non-local Problem for a Mixed Type Equation with a Caputo Fractional Differential Operator

Part of: Equations and systems of special type Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  02 May 2017

E. Karimov ,
N. Al-Salti  and
S. Kerbal
E. Karimov*
Affiliation:
Institute of Mathematics named after V.I.Romanovskiy, Academy of Sciences of the Republic of Uzbekistan, Tashkent 125, Uzbekistan
N. Al-Salti*
Affiliation:
Department of Mathematics and Statistics, Sultan Qaboos University, Al-Khoudh 123, Muscat, Oman
S. Kerbal*
Affiliation:
Department of Mathematics and Statistics, Sultan Qaboos University, Al-Khoudh 123, Muscat, Oman
*
*Corresponding author. Email addresses:erkinjon@gmail.com (E. Karimov), nalsalti@gmail.com (N. Al-Salti), skerbal@hotmail.com (S. Kerbal)
*Corresponding author. Email addresses:erkinjon@gmail.com (E. Karimov), nalsalti@gmail.com (N. Al-Salti), skerbal@hotmail.com (S. Kerbal)
*Corresponding author. Email addresses:erkinjon@gmail.com (E. Karimov), nalsalti@gmail.com (N. Al-Salti), skerbal@hotmail.com (S. Kerbal)
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Abstract

We consider the unique solvability of an inverse-source problem with integral transmitting condition for a time-fractional mixed type equation in rectangular domain where the unknown source term depends only on the space variable. The solution is based on a series expansion using a bi-orthogonal basis in space, corresponding to a non-self-adjoint boundary value problem. Under certain regularity conditions on the given data, we prove the uniqueness and existence of the solution for the given problem. The influence of the transmitting condition on the solvability of the problem is also demonstrated. Two different transmitting conditions are considered — viz. a full integral form and a special case. In order to simplify the bulky expressions appearing in the proof of our main result, we establish a new property of the recently introduced Mittag-Leffler type function in two variables.

Keywords

Inverse-source problemmixed type equationCaputo fractional operator

MSC classification

Secondary: 35M10: Equations of mixed type 35R30: Inverse problems

Information

Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 2 , May 2017 , pp. 417 - 438
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Su, N: Mass-time and space-time fractional partial differential equations of water movement in soils: Theoretical framework and application to infiltration. Journal of Hydrology. 519, 1792–1803 (2014).Google Scholar
[2] Baleanu, D, Güvenç, ZB, Machado, JT: New Trends in Nanotechnology and Fractional Calculus Applications, Springer (2010).CrossRefGoogle Scholar
[3] Mainardi, F: Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press (2010).Google Scholar
[4] Li, G, Zhang, D, Jia, X, Yamamoto, M: Simultaneous inversion for the space-dependent diffusion coefficient and the fractional order in the time-fractional diffusion equation. Inverse Problems. 29 (2013) doi:10.1088/0266-5611/29/6/065014.Google Scholar
[5] Kirane, M, Malik, SA, Al-Gwaiz, MA: An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions. Math. Methods Appl. 36, 1056–1069 (2013). doi: 10.1002/mma.2661.Google Scholar
[6] Aleroev, TS, Kirane, M, Malik, SA: Determination of a source term for a time fractional diffusion equation with an integral type over-determining condition. Electr. J. Differ. Equ. 270, 1–16 (2013).Google Scholar
[7] Sabitov, KB, Martem’yanova NV: A nonlocal inverse problem for a mixed-type equation. Russian Mathematics (Izv.VUZ). 55(2), 61–74 (2011).CrossRefGoogle Scholar
[8] Ashyralyev, A, Ashyralyeva, MA: On source identification problem for a parabolic-hyperbolic equation. Contemporary Analysis and Applied Mathematics. 3(1), 88–103 (2015).CrossRefGoogle Scholar
[9] Feng, P, Karimov, ET: Inverse source problems for time-fractional mixed parabolic-hyperbolic-type equations. J. Inverse Ill-Posed Probl. 23(4), 339–353 (2015).Google Scholar
[10] Berdyshev, AS, Cabada, A, Karimov, ET: On a non-local boundary problem for a parabolic-hyperbolic equation involving Riemann-Liouville fractional differential operator. Nonlinear Analysis. 75, 3268–3273 (2012).CrossRefGoogle Scholar
[11] Berdyshev, AS, Cabada, A, Karimov, ET, Akhtaeva NS: On the Volterra property of a boundary problem with integral gluing condition for a mixed parabolic-hyperbolic equation. Boundary Value Problems. 94, 1–14 (2013).Google Scholar
[12] Furati, KM, Iyiola, OS, Kirane M: An inverse problem for a generalized fractional diffusion. Applied Mathematics and Computation. 249, 24–31 (2014).Google Scholar
[13] Kilbas, AA, Srivastava, HM, Trujillo, JJ. Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006).Google Scholar
[14] Podlubny, I: Fractional Differential Equations. Academic Press, San Diego (1999).Google Scholar
[15] Garg, M, Manohar, P, Kalla, SL: A Mittag-Leffler-type function of two variables. Integral Transforms and Special Functions. 24(11) (2013). doi:10.1080/10652469.2013.789872.Google Scholar
[16] Il’in, VA: Existence of a reduced system of eigen- and associated functions for a nonself adjoint ordinary differential operator. Trudy MIAN. 142, 148–155 (1976).Google Scholar
[17] Uflyand, YS: On the question of the distribution of fluctuations in composite electrical lines. Injenerno-fizicheskiy jurnal 7(1), 89–92 (1964).Google Scholar
[18] Agarwal, P, Karimov, ET, Berdyshev, AS: Solvability of a non-local problem with integral transmitting condition for mixed type equation with caputo fractional derivative. Results in Mathematics. Online First. DOI 10.1007/s00025-016-0620-1.Google Scholar
[19] Pskhu, AV: Uravneniya v chastnykh proizvodnykh drobnogo poryadka, in: Partial Differential Equations of Fractional Order, Nauka, Moscow (2005) (in Russian).Google Scholar
[20] Salakhitdinov, MS, Karimov, ET: Uniqueness of an inverse-source non-local problem for fractional order mixed type equations. Euroasian Journal of Mathematics. 7(1), 74–83 (2016).Google Scholar