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- A Time-Fractional Borel–Pompeiu Formula and a Related Hypercomplex Operator CalculusPublication . Ferreira, M.; Rodrigues, M. M.; Vieira, N.In this paper, we develop a time-fractional operator calculus in fractional Clifford analysis. Initially, we study the $L_p$-integrability of the fundamental solutions of the multi-dimensional time-fractional diffusion operator and the associated time-fractional parabolic Dirac operator. Then we introduce the time-fractional analogs of the Teodorescu and Cauchy-Bitsadze operators in a cylindrical domain, and we investigate their main mapping properties. As a main result, we prove a time-fractional version of the Borel-Pompeiu formula based on a time-fractional Stokes' formula. This tool in hand allows us to present a Hodge-type decomposition for the forward time-fractional parabolic Dirac operator with left Caputo fractional derivative in the time coordinate. The obtained results exhibit an interesting duality relation between forward and backward parabolic Dirac operators and Caputo and Riemann-Liouville time-fractional derivatives. We round off this paper by giving a direct application of the obtained results for solving time-fractional boundary value problems.
- A higher dimensional fractional Borel‐Pompeiu formula and a related hypercomplex fractional operator calculusPublication . Ferreira, M.; Kraußhar, R. S.; Rodrigues, M. M.; Vieira, N.In this paper, we develop a fractional integro-differential operator calculus for Clifford-algebra valued functions. To do that we introduce fractional analogs of the Teodorescu and Cauchy-Bitsadze operators and we investigate some of their mapping properties. As a main result, we prove a fractional Borel-Pompeiu formula based on a fractional Stokes formula. This tool in hand allows us to present a Hodge-type decomposition for the fractional Dirac operator. Our results exhibit an amazing duality relation between left and right operators and between Caputo and Riemann-Liouville fractional derivatives. We round off this paper by presenting a direct application to the resolution of boundary value problems related to Laplace operators of fractional order.