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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
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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