Speaker: Nelson Vieira, CIDMA - Department of Mathematics, University of Aveiro

Time : 16:00 - 17:00 CEST (Rome/Paris)

Hosted at: SISSA, International School of Advanced Studies, Trieste, Italy

Zoom : A Zoom link will appear here, an hour before the talk

Organizers : Pavan Pranjivan Mehta* (pavan.mehta@sissa.it) and Arran Fernandez** (arran.fernandez@emu.edu.tr)

* SISSA, International School of Advanced Studies, Italy

** Eastern Mediterranean University, Northern Cyprus

Keywords: Fractional Helmholtz equations; Fundamental solutions; Fractional Clifford analysis; Stokes formula; Borel-Pompeiu formula; Leray-Hodge decomposition.

Abstract: This talk explores the development of a hypercomplex operator calculus designed to address boundary value problems for fractional Helmholtz equations. By employing a framework that integrates Clifford analysis with fractional calculus, we investigate both homogeneous and inhomogeneous cases involving Caputo and Riemann-Liouville fractional derivatives. The proposed methodology extends existing transform methods by considering fractional differentiation across all spatial directions.

A central feature of this approach is the factorization of the fractional Helmholtz operator, which facilitates the derivation of integral representations and duality relations between different types of fractional operators. Furthermore, we discuss the construction of fundamental solutions and the analytical conditions governing the behavior of steady-state solutions, including criteria for spatial decay and potential blow-up phenomena. The theoretical results are complemented by illustrative examples, demonstrating how this general setting recovers classical and recently studied models as particular cases.

Biography: Nelson Vieira is an Assistant Professor in the Department of Mathematics at the University of Aveiro and a researcher at CIDMA – Center for Research and Development in Mathematics and Applications. He received his PhD in Mathematics from the University of Aveiro in 2009. His research centers on Clifford and hypercomplex analysis, fractional calculus, special functions, and their applications to differential equations, signal processing, and machine learning. He has published extensively in these fields and held competitive FCT research positions (CEECIND 2018, Investigador FCT 2014). He participates in funded projects and serves on editorial boards while reviewing for international journals.

Bibliography

[1] N. Vieira, M. Ferreira, M.M. Rodrigues, and R.S. Kraußhar. “Hypercomplex operator calculus for the fractional Helmholtz equation”. In: Mathematical Methods in the Applied Sciences 47.14 (2024), pp. 11439–11472.
[2] M. Ferreira, R. Kraußhar, M.M. Rodrigues, and N. Vieira. “A higher dimensional fractional Borel-Pompeiu formula and a related hypercomplex fractional operator calculus”. In: Mathematical Methods in the Applied Sciences 42.10 (2019), pp. 3633–3653.

[3] M. Ferreira and N. Vieira. “Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators using Caputo derivatives”. In: Complex Variables and Elliptic Equations 62.9 (2017), pp. 1237–1253.
[4] M. Ferreira and N. Vieira. “Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators: the Riemann-Liouville case”. In: Complex Analysis and Operator Theory 10.5 (2016), pp. 1081–1100.