Proportional fractional Dirac dynamic system

Tuba Gulsen; Emrah Yilmaz; Ayse Çiğdem Yar; Tuba Gulsen; Emrah Yilmaz; Ayse Çiğdem Yar

doi:10.3934/math.2024487

AIMS Mathematics

2024, Volume 9, Issue 4: 9951-9968. doi: 10.3934/math.2024487

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Proportional fractional Dirac dynamic system

1.
Department of Mathematics, Faculty of Science, Firat University, Elazığ, Turkey
2.
Department of Mathematics, Institute of Science, Firat University, Elazığ, Turkey

Received: 22 January 2024 Revised: 04 March 2024 Accepted: 07 March 2024 Published: 12 March 2024
MSC : 35Q41, 34N05, 26A33

In this study, considering the proportional fractional derivative, which is a generalization of the conformable fractional derivative, we provided some important spectral properties such as the reality of eigenvalues, the orthogonality of eigenfunctions, the self-adjointness of the operator, the asymptotic estimations of eigenfunctions, and Picone's identity for a proportional Dirac system on an arbitrary time scale. We also presented graphics representing the eigenfunctions of the Dirac system on a time scale, produced by taking advantage of the proportional fractional derivative with some special cases. The main purpose of presenting these graphics was to examine the effect of the proportional fractional derivative on the Dirac system on a time scale, as well as the effect of the eigenvalues, which are meaningful for the subject we were studying for the solution functions.
- Dirac operator,
- time scales,
- proportional fractional derivative
Citation: Tuba Gulsen, Emrah Yilmaz, Ayse Çiğdem Yar. Proportional fractional Dirac dynamic system[J]. AIMS Mathematics, 2024, 9(4): 9951-9968. doi: 10.3934/math.2024487

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Abstract

In this study, considering the proportional fractional derivative, which is a generalization of the conformable fractional derivative, we provided some important spectral properties such as the reality of eigenvalues, the orthogonality of eigenfunctions, the self-adjointness of the operator, the asymptotic estimations of eigenfunctions, and Picone's identity for a proportional Dirac system on an arbitrary time scale. We also presented graphics representing the eigenfunctions of the Dirac system on a time scale, produced by taking advantage of the proportional fractional derivative with some special cases. The main purpose of presenting these graphics was to examine the effect of the proportional fractional derivative on the Dirac system on a time scale, as well as the effect of the eigenvalues, which are meaningful for the subject we were studying for the solution functions.

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