Skip to main content
×
Home
    • Aa
    • Aa

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)
Abstract
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.

Copyright
Corresponding author
*Corresponding author. Email addresses: erkinjon@gmail.com (E. Karimov), nalsalti@gmail.com (N. Al-Salti), skerbal@hotmail.com (S. Kerbal)
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[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.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

East Asian Journal on Applied Mathematics
  • ISSN: 2079-7362
  • EISSN: 2079-7370
  • URL: /core/journals/east-asian-journal-on-applied-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 7 *
Loading metrics...

Abstract views

Total abstract views: 43 *
Loading metrics...

* Views captured on Cambridge Core between 2nd May 2017 - 22nd May 2017. This data will be updated every 24 hours.