The Fractional Half Dirac Operator Over Sobolev Spaces

Dejenie Alemayehu Lakew

doi:doi:10.11648/j.pamj.20261504.11

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The Fractional Half Dirac Operator Over Sobolev Spaces

Dejenie Alemayehu Lakew^*

Published in Pure and Applied Mathematics Journal (Volume 15, Issue 4)

Received: 10 August 2026 Accepted: 22 August 2026 Published: 14 October 2026

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Abstract

The classical Dirac operator of a regular order reduces smoothness of functions by the amount of the order and that property is valid when the order is fractional. What is interesting is that fractional order derivatives offer more practical framework on studying physical phenomena such as motion in viscous or resistive media. This article introduces a fractional half order Dirac operator and develops theoretical results in Sobolev spaces of an inner product which permit orthogonality. Using domain and boundary integrals, the author establishes that solutions to first and second order boundary value problems of the half order Dirac operator decompose in a unique way as an orthogonal sum of functions, one that evolves from the interior domain data and the other from the boundary trace value data. These results extend successfully classical decomposition results of spaces by a regular order Dirac operator to results of a fractional half order. At the end we demonstrate the practical use of Dirac operators of fractional orders and orthogonal decompositions through motions in electromagnetic fields and viscous media where a solution vector of a boundary value problem of fractional half order Dirac equation splits as an orthogonal sum of solution parts called irrotational and solenoidal.

Published in	Pure and Applied Mathematics Journal (Volume 15, Issue 4)
DOI	10.11648/j.pamj.20261504.11
Page(s)	56-72
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Fractional Half Dirac Operator, Orthogonal Sum, Boundary Value Problems (BVPs), Hilbert Spaces, Sobolev Spaces, Caputo Fractional Derivative, Riemann-Lioville Fractional Derivative

References

[1]	Dejenie A. Lakew, The Fractional Half Dirac Operator Over Sobolev Spaces, arXiv:2608.02871. https://doi.org/10.48550/arXiv.2608.02871
[2]	Dejenie Alemayehu Lakew and Rawal Nar, Orthogonal Parts of a Solution to a Cauchy BVP of Higher Order Over Sobolev Spaces, Parana J. Sci. Ed., Vol. 9, No. 7 (1-11), Dec. 1, 2023.
[3]	Dejenie A. Lakew, On Some Discrete Differential Equations, Parana J. Sci. Edu., Viol. 7, No. 9 (1-6) Nov. 12, 2021. https://doi.org/10.48550/arxiXiv.0805.1744
[4]	Dejenie A. Lakew, On Orthogonal Decomposition of a Sobolev Space, Adv. Oper. Theory, Vol. 2 (2017) No. 4, 419-427. https://doi.org/10.48550/arXiv.1611.04249
[5]	Dejenie A. Lakew, On Orthogonal Decomposition of the Hilbert Space ℒ ²(Ω), Int. J. Math. Comp. Sci. 10(2015), No. 1, 27-37. https://doi.org/10.48550/arXiv.1503.01209
[6]	Dejenie A. Lakew, New Proofs on Properties of Orthogonal Decomposition of a Hilbert Space, arXiv:1510.07944v1. https://doi.org/1048550/arXiv.1510.07944
[7]	Griffiths, D. J. , Introduction to Electrodynamics, 2013, 4th ed, Pearson.
[8]	Keith B. Oldham, Jerome Spanier, The Fractional Calculus, Theory and Applications, 2006, Dover Pub.
[9]	Dejenie A. Lakew and John Ryan, The Intrinsic π - Operator on Domain Manifolds in ℂ ⁽ⁿ⁺¹⁾, Compl. Anal. Oper. Theory, Vol. 4, No. 2 (2010) 271-280. https://doi:10.1007/s11785-009-0004-6
[10]	Dejenie A. Lakew and John Ryan, Clifford Analytic Complete Function Systems for Unbounded Domains, Math. Meth. Appl. Sci., Vol. 25, No. 16-18 (2002) 1527-1739. https://doi:10.1002/mma.389
[11]	John B. Conway, A Course in Functional Analysis, 2nd Ed, Graduate Texts in Mathematics, Springer.
[12]	L. A. Liusternik and V. J. Sobolev, Elements of Functional Analysis, Frederick Ungar Publishing Company, New York, 1961.
[13]	Robert McOwen, Partial Differential Equations, Methods and Applications, Prentice Hall, 1996
[14]	Lawrence Evans, Partial Differential Equations, American Mathematical Society, 1998.
[15]	F. Brackx, R. Delanghe and F. Sommen, Clifford Analysis, Research Notes in Mathematics, No. 76, Pitman, London 1982 Book Review: https://doi.org/10.1090/50273-0979-1984-15281-9
[16]	Di. Nezza et al, Hitchhiker's Guide to the Fractional Sobolev Spaces, Bull. Sci. Math (2012).
[17]	S. G. Mikhlin, S. Prossdrof, Singular Integral Operators, Aca. Verl. Berlin (1980).
[18]	K. Gurlebeck, U. Kahler, J. Ryan and W. Sproessig, Clifford Analysis over Unbounded Domain, Adv. Appl. Math. 19 (1997) 216-239. https://doi:10.1006/aama.1997.0541

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APA Style

Lakew, D. A. (2026). The Fractional Half Dirac Operator Over Sobolev Spaces. Pure and Applied Mathematics Journal, 15(4), 56-72. https://doi.org/10.11648/j.pamj.20261504.11

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ACS Style

Lakew, D. A. The Fractional Half Dirac Operator Over Sobolev Spaces. Pure Appl. Math. J. 2026, 15(4), 56-72. doi: 10.11648/j.pamj.20261504.11

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AMA Style

Lakew DA. The Fractional Half Dirac Operator Over Sobolev Spaces. Pure Appl Math J. 2026;15(4):56-72. doi: 10.11648/j.pamj.20261504.11

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@article{10.11648/j.pamj.20261504.11,
  author = {Dejenie Alemayehu Lakew},
  title = {The Fractional Half Dirac Operator Over Sobolev Spaces},
  journal = {Pure and Applied Mathematics Journal},
  volume = {15},
  number = {4},
  pages = {56-72},
  doi = {10.11648/j.pamj.20261504.11},
  url = {https://doi.org/10.11648/j.pamj.20261504.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20261504.11},
  abstract = {The classical Dirac operator of a regular order reduces smoothness of functions by the amount of the order and that property is valid when the order is fractional. What is interesting is that fractional order derivatives offer more practical framework on studying physical phenomena such as motion in viscous or resistive media. This article introduces a fractional half order Dirac operator and develops theoretical results in Sobolev spaces of an inner product which permit orthogonality. Using domain and boundary integrals, the author establishes that solutions to first and second order boundary value problems of the half order Dirac operator decompose in a unique way as an orthogonal sum of functions, one that evolves from the interior domain data and the other from the boundary trace value data. These results extend successfully classical decomposition results of spaces by a regular order Dirac operator to results of a fractional half order. At the end we demonstrate the practical use of Dirac operators of fractional orders and orthogonal decompositions through motions in electromagnetic fields and viscous media where a solution vector of a boundary value problem of fractional half order Dirac equation splits as an orthogonal sum of solution parts called irrotational and solenoidal.},
 year = {2026}
}

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TY  - JOUR
T1  - The Fractional Half Dirac Operator Over Sobolev Spaces
AU  - Dejenie Alemayehu Lakew
Y1  - 2026/10/14
PY  - 2026
N1  - https://doi.org/10.11648/j.pamj.20261504.11
DO  - 10.11648/j.pamj.20261504.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 56
EP  - 72
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20261504.11
AB  - The classical Dirac operator of a regular order reduces smoothness of functions by the amount of the order and that property is valid when the order is fractional. What is interesting is that fractional order derivatives offer more practical framework on studying physical phenomena such as motion in viscous or resistive media. This article introduces a fractional half order Dirac operator and develops theoretical results in Sobolev spaces of an inner product which permit orthogonality. Using domain and boundary integrals, the author establishes that solutions to first and second order boundary value problems of the half order Dirac operator decompose in a unique way as an orthogonal sum of functions, one that evolves from the interior domain data and the other from the boundary trace value data. These results extend successfully classical decomposition results of spaces by a regular order Dirac operator to results of a fractional half order. At the end we demonstrate the practical use of Dirac operators of fractional orders and orthogonal decompositions through motions in electromagnetic fields and viscous media where a solution vector of a boundary value problem of fractional half order Dirac equation splits as an orthogonal sum of solution parts called irrotational and solenoidal.
VL  - 15
IS  - 4
ER  -

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Author Information

Dejenie Alemayehu Lakew

Department of Mathematics, Hampton University, Hampton, USA

Contact Email

http://orcid.org/0000-0001-8151-0448

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APA Style

Lakew, D. A. (2026). The Fractional Half Dirac Operator Over Sobolev Spaces. Pure and Applied Mathematics Journal, 15(4), 56-72. https://doi.org/10.11648/j.pamj.20261504.11

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ACS Style

Lakew, D. A. The Fractional Half Dirac Operator Over Sobolev Spaces. Pure Appl. Math. J. 2026, 15(4), 56-72. doi: 10.11648/j.pamj.20261504.11

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AMA Style

Lakew DA. The Fractional Half Dirac Operator Over Sobolev Spaces. Pure Appl Math J. 2026;15(4):56-72. doi: 10.11648/j.pamj.20261504.11

Copy | Download

@article{10.11648/j.pamj.20261504.11,
  author = {Dejenie Alemayehu Lakew},
  title = {The Fractional Half Dirac Operator Over Sobolev Spaces},
  journal = {Pure and Applied Mathematics Journal},
  volume = {15},
  number = {4},
  pages = {56-72},
  doi = {10.11648/j.pamj.20261504.11},
  url = {https://doi.org/10.11648/j.pamj.20261504.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20261504.11},
  abstract = {The classical Dirac operator of a regular order reduces smoothness of functions by the amount of the order and that property is valid when the order is fractional. What is interesting is that fractional order derivatives offer more practical framework on studying physical phenomena such as motion in viscous or resistive media. This article introduces a fractional half order Dirac operator and develops theoretical results in Sobolev spaces of an inner product which permit orthogonality. Using domain and boundary integrals, the author establishes that solutions to first and second order boundary value problems of the half order Dirac operator decompose in a unique way as an orthogonal sum of functions, one that evolves from the interior domain data and the other from the boundary trace value data. These results extend successfully classical decomposition results of spaces by a regular order Dirac operator to results of a fractional half order. At the end we demonstrate the practical use of Dirac operators of fractional orders and orthogonal decompositions through motions in electromagnetic fields and viscous media where a solution vector of a boundary value problem of fractional half order Dirac equation splits as an orthogonal sum of solution parts called irrotational and solenoidal.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - The Fractional Half Dirac Operator Over Sobolev Spaces
AU  - Dejenie Alemayehu Lakew
Y1  - 2026/10/14
PY  - 2026
N1  - https://doi.org/10.11648/j.pamj.20261504.11
DO  - 10.11648/j.pamj.20261504.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 56
EP  - 72
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20261504.11
AB  - The classical Dirac operator of a regular order reduces smoothness of functions by the amount of the order and that property is valid when the order is fractional. What is interesting is that fractional order derivatives offer more practical framework on studying physical phenomena such as motion in viscous or resistive media. This article introduces a fractional half order Dirac operator and develops theoretical results in Sobolev spaces of an inner product which permit orthogonality. Using domain and boundary integrals, the author establishes that solutions to first and second order boundary value problems of the half order Dirac operator decompose in a unique way as an orthogonal sum of functions, one that evolves from the interior domain data and the other from the boundary trace value data. These results extend successfully classical decomposition results of spaces by a regular order Dirac operator to results of a fractional half order. At the end we demonstrate the practical use of Dirac operators of fractional orders and orthogonal decompositions through motions in electromagnetic fields and viscous media where a solution vector of a boundary value problem of fractional half order Dirac equation splits as an orthogonal sum of solution parts called irrotational and solenoidal.
VL  - 15
IS  - 4
ER  -

Copy | Download