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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.
- Some representations for the eigenfunctions of the time-fractional wave operatorPublication . Rodrigues, M.M.; Ferreira, M.; Vieira, NelsonIn this work we present some new representations for the eigenfunctions of the time-fractional wave operator with the time-fractional derivative in the Caputo sense.
- 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.
- Time-fractional telegraph equation of distributed order in higher dimensions with Hilfer fractional derivativesPublication . Vieira, Nelson; Rodrigues, M. Manuela; Ferreira, MiltonIn this paper, we consider time-fractional telegraph equations of distributed order in higher spatial dimensions, where the time derivatives are in the sense of Hilfer, thus interpolating between the Riemann-Liouville and the Caputo fractional derivatives. By employing the techniques of the Fourier, Laplace, and Mellin transforms, we obtain a representation of the solution of the Cauchy problem associated with the equation in terms of convolutions involving functions that are Laplace integrals of Fox H-functions. Fractional moments of the first fundamental solution are computed and for the special case of double-order distributed it is analyzed in detail the asymptotic behavior of the second-order moment, by application of the Tauberian Theorem. Finally, we exhibit plots of the variance showing its behavior for short and long times, and for different choices of the parameters along small dimensions.
- First and second fundamental solutions of the time-fractional telegraph equation of order 2αPublication . Ferreira, Milton; Rodrigues, M. Manuela; Vieira, NelsonIn this work we obtain the first and second fundamental solutions of the multidimensional time-fractional equation of order 2α, α ∈]0, 1], where the two time-fractional derivatives are in the Caputo sense. We obtain representations of the fundamental solutions in terms of Hankel transform, double Mellin-Barnes integral, and H-functions of two variables. As an application, the fundamental solutions are used to solve a Cauchy problem and to study telegraph process with Brownian time.
- A Fractional Analysis in Higher Dimensions for the Sturm-Liouville ProblemPublication . Ferreira, Milton; Rodrigues, M. Manuela; Vieira, NelsonIn this work, we consider the n-dimensional fractional Sturm-Liouville eigenvalue problem, by using fractional versions of the gradient operator involving left and right Riemann-Liouville fractional derivatives. We study the main properties of the eigenfunctions and the eigenvalues of the associated fractional boundary problem. More precisely, we show that the eigenfunctions are orthogonal and the eigenvalues are real and simple. Moreover, using techniques from fractional variational calculus, we prove in the main result that the eigenvalues are separated and form an infinite sequence, where the eigenvalues can be ordered according to increasing magnitude. Finally, a connection with Clifford analysis is established.
- Time-fractional diffusion equation with psi-Hilfer derivativePublication . Vieira, Nelson; Rodrigues, M. Manuela; Ferreira, MiltonIn this work, we consider the multidimensional time-fractional diffusion equation with the $\psi$-Hilfer derivative. This fractional derivative enables the interpolation between Riemann-Liouville and Caputo fractional derivatives and its kernel depends on an arbitrary positive monotone increasing function $\psi$ thus encompassing several fractional derivatives in the literature. This allows us to obtain general results for different families of problems that depend on the function $\psi$ selected. By employing techniques of Fourier, $\psi$-Laplace, and Mellin transforms, we obtain a solution representation in terms of convolutions involving Fox H-functions for the Cauchy problem associated with our equation. Series representations of the first fundamental solution are explicitly obtained for any dimension as well as the fractional moments of arbitrary positive order. For the one-dimensional case, we show that the series representation reduces to a Wright function, and we prove that it corresponds to a probability density function for any admissible $\psi$. Finally, some plots of the fundamental solution are presented for particular choices of the function $\psi$ and the order of differentiation.
- 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 $n$.
- Fundamental solution of the time-fractional telegraph Dirac operatorPublication . Ferreira, Milton; Rodrigues, M. Manuela; Vieira, Nelsonn this work, we obtain the fundamental solution (FS) of the multidimensionaltime-fractional telegraph Dirac operator where the 2 time-fractional derivatives oforders𝛼∈]0,1]and𝛽∈]1,2]are in the Caputo sense. Explicit integral and seriesrepresentation of the FS are obtained for any dimension. We present and discusssome plots of the FS for some particular values of the dimension and of the frac-tional parameters𝛼and𝛽. Finally, using the FS, we study some Poisson and Cauchyproblems
- Time-fractional telegraph equation of distributed order in higher dimensionsPublication . Vieira, Nelson; Rodrigues, M. Manuela; Ferreira, MiltonIn this work, the Cauchy problem for the time-fractional telegraph equation of distributed order in Rn is considered. By employing the technique of the Fourier, Laplace, and Mellin transforms, a representation of the fundamental solution of this equation in terms of convolutions involving the Fox H-function is obtained. Some particular choices of the density functions in the form of elementary functions are studied. Fractional moments of the fundamental solution are computed in the Laplace domain. Finally, by application of the Tauberian theorems we study the asymptotic behaviour of the second-order moment (variance) in the time domain.