Research Article
The Fractional Half Dirac Operator Over Sobolev Spaces
Dejenie Alemayehu Lakew*
Issue:
Volume 15, Issue 4, August 2026
Pages:
56-72
Received:
10 August 2026
Accepted:
22 August 2026
Published:
14 October 2026
DOI:
10.11648/j.pamj.20261504.11
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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.
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 introduc...
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