Volume 7, Issue 2
An Inverse Source Non-Local Problem for a Mixed Type Equation with a Caputo Fractional Differential Operator

E. Karimov, N. Al-Salti & S. Kerbal

East Asian J. Appl. Math., 7 (2017), pp. 417-438.

Published online: 2018-02

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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 problem, mixed type equation, Caputo fractional operator.

  • AMS Subject Headings

35M10, 35R30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{EAJAM-7-417, author = {}, title = {An Inverse Source Non-Local Problem for a Mixed Type Equation with a Caputo Fractional Differential Operator}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {7}, number = {2}, pages = {417--438}, 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.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.051216.280217a}, url = {http://global-sci.org/intro/article_detail/eajam/10757.html} }
TY - JOUR T1 - An Inverse Source Non-Local Problem for a Mixed Type Equation with a Caputo Fractional Differential Operator JO - East Asian Journal on Applied Mathematics VL - 2 SP - 417 EP - 438 PY - 2018 DA - 2018/02 SN - 7 DO - http://doi.org/10.4208/eajam.051216.280217a UR - https://global-sci.org/intro/article_detail/eajam/10757.html KW - Inverse-source problem, mixed type equation, Caputo fractional operator. AB -

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.

E. Karimov, N. Al-Salti & S. Kerbal. (2020). An Inverse Source Non-Local Problem for a Mixed Type Equation with a Caputo Fractional Differential Operator. East Asian Journal on Applied Mathematics. 7 (2). 417-438. doi:10.4208/eajam.051216.280217a
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