Speaker: Milton Ferreira, Polytechnic University of Leiria, Portugal and CIDMA - University of Aveiro, Portugal

Time : 15:00 - 16:00 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 Clifford Analysis, Time-fractional parabolic Dirac operator, Fundamental solution, Borel–Pompeiu formula

Abstract: In this talk, we present a time-fractional operator calculus within the framework of fractional Clifford analysis. We begin by studying the L_p -integrability properties of the fundamental solutions associated with the multidimensional time-fractional diffusion operator and the parabolic Dirac operator. We then introduce time-fractional analogues of the Teodorescu and Cauchy–Bitsadze operators in a space–time setting and investigate their main mapping properties. As a principal result, we establish a time-fractional version of the Borel–Pompeiu formula, derived from an appropriate time-fractional Stokes’ theorem. This framework allows us to formulate a Hodge-type decomposition of the L^p -space. The results reveal an interesting duality between forward and backward parabolic Dirac operators and the Caputo and Riemann–Liouville time-fractional derivatives. Finally, we discuss applications of the developed theory to the solution of time-fractional boundary value problems. The results presented in this talk are based on [1].

Biography: Milton dos Santos Ferreira holds a Ph.D. in Mathematics from the University of Aveiro (2008), where his doctoral research focused on continuous wavelet transforms on the sphere. His scientific interests include gyrogroups and gyroharmonic analysis, quaternionic, Clifford, and octonionic analysis, fractional Laplace and Dirac operators, fractional calculus, and time-fractional differential equations. In these areas, he has authored more than 40 peer-reviewed journal articles.

Bibliography

[1] M. Ferreira, M.M. Rodrigues, and N. Vieira. “A time-fractional Borel-Pompeiu formula and a related hypercomplex operator calculus”. In: Complex Anal. Oper. Theory 13 (2019), pp. 2495—2526.
[2] K.H. Kim and S. Lim. “Asymptotic behaviours of fundamental solution and its derivatives to fractional diffusion-wave equations”. In: J. Korean Math. Soc. 53.4 (2016), pp. 929–967.

[3] M. Ferreira and N. Vieira. “Fundamental solutions of the time fractional diffusion-wave and parabolic Dirac operators”. In: J. Math. Anal. Appl. 447.1 (2017), pp.329–353.