Abstract
We introduce a fractional version of the d-bar derivative operator acting on quaternionic functions. Some natural properties of this new operator are shown, including composition properties and its action on certain functions such as Fueter polynomials, which help to demonstrate its suitability compared with other attempts to define fractional quaternionic and Clifford derivatives.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
A. Fernandez, C. Bouzouina, Fractionalisation of complex d-bar derivatives. Complex Variables Elliptic Equ. 66(3), 437–475 (2021)
A.S. Fokas, D.A. Pinotsis, Quaternions, evaluation of integrals and boundary value problems. Comput. Methods Funct. Theor. 7, 443–476 (2007)
G.R. Franssens, Introducing Clifford analysis as the natural tool for electromagnetic research, in Proceedings of the Progress in Electromagnetics Research Symposium, Moscow (2012)
G. Frobenius, Über lineare Substitutionen und bilineare Formen, J. für die Reine und Angew. Math. 84, 1–63 (1878)
K. Gürlebeck, K. Habetha, W. Sprößig, Holomorphic Functions in the Plane andn-Dimensional Space (Birkhäuser, Berlin, 2008)
U. Kähler, N. Vieira, Fractional Clifford analysis, in Hypercomplex Analysis: New Perspectives and Applications, ed. by S. Bernstein, U. Kähler, I. Sabadini, F. Sommen (Springer International Publishing, Cham, 2014)
P. Lounesto, Clifford Algebras and Spinors, 2nd edn. (Cambridge University Press, Cambridge, 2001)
P.A. Nekrassov, On general differentiation. Mat. Sb. 14(1), 45–168 (1888)
S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach, Yverdon, 1993)
H.G. Sun, Y. Zhang, D. Baleanu, W. Chen, Y.Q. Chen, A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlinear Sci. Numer. Simul. 64, 213–231 (2018)
D. Xu, C. Jahanchahi, C.C. Took, D.P. Mandic, Enabling quaternion derivatives: the generalized HR calculus. R. Soc. Open Sci. 2(8), 150255 (2015)
Acknowledgements
This research is financially supported by Eastern Mediterranean University via a BAP-C grant with project number BAPC-04-22-03. The first two authors are also grateful to the Ghent Analysis & PDE group for hosting them during the period this research began.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Fernandez, A., Güder, C., Yasin, W. (2024). On Fractional Quaternionic d-Bar Derivatives. In: Ruzhansky, M., Torebek, B. (eds) Extended Abstracts MWCAPDE 2023. MWCAPDE 2023. Trends in Mathematics(), vol 1. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-41665-1_3
Download citation
DOI: https://doi.org/10.1007/978-3-031-41665-1_3
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-031-41664-4
Online ISBN: 978-3-031-41665-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)