On two backward problems with Dzherbashian-Nersesian operator

Anwar Ahmad; Dumitru Baleanu; Anwar Ahmad; Dumitru Baleanu

doi:10.3934/math.2023043

AIMS Mathematics

2023, Volume 8, Issue 1: 887-904. doi: 10.3934/math.2023043

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Research article

On two backward problems with Dzherbashian-Nersesian operator

Anwar Ahmad ¹,
Dumitru Baleanu ^{2,3,4
,
,}

1.
Department of Mathematics, COMSATS University Islamabad, Park Road, Tarlai Kalan, Islamabad 45550, Pakistan
2.
Department of Mathematics, Cankaya University, Ankara, Turkey
3.
Institute of Space Sciences, Magurele-Bucharest, Romania
4.
Department of Medical Research, China Medical University, Taichung, Taiwan

Received: 30 July 2022 Revised: 24 September 2022 Accepted: 30 September 2022 Published: 12 October 2022
MSC : 33E12, 34A08, 65M32, 65N21, 80A23

We investigate the initial-boundary value problems for a fourth-order differential equation within the powerful fractional Dzherbashian-Nersesian operator (FDNO). Boundary conditions considered in this manuscript are of the Samarskii-Ionkin type. The solutions obtained here are based on a series expansion using Riesz basis in a space corresponding to a non-self-adjoint spectral problem. Conditional to some regularity, consistency, alongside orthogonality dependence, the existence and uniqueness of the obtained solutions are exhibited by using Fourier method. Acquired results here are more general than those obtained by making use of conventional fractional operators such as fractional Riemann-Liouville derivative (FRLD), fractional Caputo derivative (FCD) and fractional Hilfer derivative (FHD).
- Dzherbashian-Nersesian operator,
- nonlocal boundary conditions,
- backward problem,
- Mittag-Leffler function
Citation: Anwar Ahmad, Dumitru Baleanu. On two backward problems with Dzherbashian-Nersesian operator[J]. AIMS Mathematics, 2023, 8(1): 887-904. doi: 10.3934/math.2023043

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Abstract

We investigate the initial-boundary value problems for a fourth-order differential equation within the powerful fractional Dzherbashian-Nersesian operator (FDNO). Boundary conditions considered in this manuscript are of the Samarskii-Ionkin type. The solutions obtained here are based on a series expansion using Riesz basis in a space corresponding to a non-self-adjoint spectral problem. Conditional to some regularity, consistency, alongside orthogonality dependence, the existence and uniqueness of the obtained solutions are exhibited by using Fourier method. Acquired results here are more general than those obtained by making use of conventional fractional operators such as fractional Riemann-Liouville derivative (FRLD), fractional Caputo derivative (FCD) and fractional Hilfer derivative (FHD).

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