### 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.

Original language | English |
---|---|

Pages (from-to) | 417-438 |

Number of pages | 22 |

Journal | East Asian Journal on Applied Mathematics |

Volume | 7 |

Issue number | 2 |

DOIs | |

Publication status | Published - May 1 2017 |

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### Keywords

- Caputo fractional operator
- Inverse-source problem
- Mixed type equation

### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

**Inverse source non-local problem for a mixed type equation with a caputo fractional differential operator.** / Karimov, E.; Al-Salti, N.; Kerbal, S.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Inverse source non-local problem for a mixed type equation with a caputo fractional differential operator

AU - Karimov, E.

AU - Al-Salti, N.

AU - Kerbal, S.

PY - 2017/5/1

Y1 - 2017/5/1

N2 - 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.

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.

KW - Caputo fractional operator

KW - Inverse-source problem

KW - Mixed type equation

UR - http://www.scopus.com/inward/record.url?scp=85018730397&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85018730397&partnerID=8YFLogxK

U2 - 10.4208/eajam.051216.280217aS2079736217000268

DO - 10.4208/eajam.051216.280217aS2079736217000268

M3 - Article

VL - 7

SP - 417

EP - 438

JO - East Asian Journal on Applied Mathematics

JF - East Asian Journal on Applied Mathematics

SN - 2079-7362

IS - 2

ER -