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- Fractional gradient methods via ψ-Hilfer derivativePublication . Vieira, N.; Rodrigues, M. M.; Ferreira, M.Motivated by the increasing of practical applications in fractional calculus, we study the classical gradient method under the perspective of the ψ-Hilfer derivative. This allows us to cover in our study several definitions of fractional derivatives that are found in the literature. The convergence of the ψ-Hilfer continuous fractional gradient method is studied both for strongly and non-strongly convex cases. Using a series representation of the target function, we develop an algorithm for the ψ-Hilfer fractional order gradient method. The numerical method obtained by truncating higher-order terms was tested and analyzed using benchmark functions. Considering variable order differentiation and optimizing the step size, the ψ-Hilfer fractional gradient method shows better results in terms of speed and accuracy. Our results generalize previous works in the literature.
- 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.
- Eigenfunctions of the time‐fractional diffusion‐wave operatorPublication . Ferreira, M.; Luchko, Yu.; Rodrigues, M. M.; Vieira, N.In this paper, we present some new integral and series representations for the eigenfunctions of the multidimensional time-fractional diffusion-wave operator with the time-fractional derivative of order $\beta \in ]1,2[$ defined in the Caputo sense. The integral representations are obtained in form of the inverse Fourier-Bessel transform and as double contour integrals of the Mellin-Barnes type. Concerning series expansions, the eigenfunctions are expressed as the double generalized hypergeometric series for any $\beta \in ]1,2[$ and as Kamp\'{e} de F\'{e}riet and Lauricella series in two variables for the rational values of $\beta$. The limit cases $\beta=1$ (diffusion operator) and $\beta=2$ (wave operator) as well as an intermediate case $\beta=\frac{3}{2}$ are studied in detail. Finally, we provide several plots of the eigenfunctions to some selected eigenvalues for different particular values of the fractional derivative order $\beta$ and the spatial dimension $n$.
- 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.