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

  • E. Karimov (a1), N. Al-Salti (a2) and S. Kerbal (a2)

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.

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*Corresponding author. Email addresses: (E. Karimov), (N. Al-Salti), (S. Kerbal)
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[1] N Su : Mass-time and space-time fractional partial differential equations of water movement in soils: Theoretical framework and application to infiltration. Journal of Hydrology. 519, 17921803 (2014).

[2] D Baleanu , ZB Güvenç , JT Machado : New Trends in Nanotechnology and Fractional Calculus Applications, Springer (2010).

[3] F Mainardi : Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press (2010).

[5] M Kirane , SA Malik , MA Al-Gwaiz : An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions. Math. Methods Appl. 36, 10561069 (2013). doi: 10.1002/mma.2661.

[7] KB Sabitov , Martem’yanova NV: A nonlocal inverse problem for a mixed-type equation. Russian Mathematics (Izv.VUZ). 55(2), 6174 (2011).

[9] P Feng , ET Karimov : Inverse source problems for time-fractional mixed parabolic-hyperbolic-type equations. J. Inverse Ill-Posed Probl. 23(4), 339353 (2015).

[10] AS Berdyshev , A Cabada , ET Karimov : On a non-local boundary problem for a parabolic-hyperbolic equation involving Riemann-Liouville fractional differential operator. Nonlinear Analysis. 75, 32683273 (2012).

[12] KM Furati , OS Iyiola , Kirane M: An inverse problem for a generalized fractional diffusion. Applied Mathematics and Computation. 249, 2431 (2014).

[15] M Garg , P Manohar , SL Kalla : A Mittag-Leffler-type function of two variables. Integral Transforms and Special Functions. 24(11) (2013). doi:10.1080/10652469.2013.789872.

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East Asian Journal on Applied Mathematics
  • ISSN: 2079-7362
  • EISSN: 2079-7370
  • URL: /core/journals/east-asian-journal-on-applied-mathematics
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