Center for Research and Development in Mathematics and Applications

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Publications

Fractional gradient methods via ψ-Hilfer derivative

Publication . 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.

2023-03Journal article

Open access

On a fractional Sturm-Liouville problem in higher dimensions

Publication . Vieira, N.; Rodrigues, M. M.; Ferreira, M.

In this short paper, we consider an n-dimensional fractional Sturm-Liouville eigenvalue problem, by using fractional versions of the gradient operator involving left Caputo and right Riemann-Liouville fractional derivatives. We study the main properties of the eigenfunctions and the eigenvalues of the associated fractional boundary problem.

2023-09Journal article

Open access

Uniformly distributed-order wave equation in higher dimensions

Publication . Vieira, N.; Rodrigues, M. M.; Ferreira, M.

In this short paper, we obtain the eigenfunctions of the uniformly distributed-order wave equation in Rn ×R+, as Laplace integral of Fox H-functions. For the particular case of the first fundamental solution, the fractional moment of second order of the fundamental solution is studied using the Tauberian Theorem.

2023-09Journal article

Open access

Distributed-order relaxation-oscillation equation

Publication . Rodrigues, M. M.; Ferreira, M.; Vieira, N.

In this short paper, we study the Cauchy problem associated with the forced time-fractional relaxation-oscillation equation with distributed order. We employ the Laplace transform technique to derive the solution. Additionally, for the scenario without external forcing, we focus on density functions characterized by a single order, demonstrating that under these conditions, the solution can be expressed using two-parameter Mittag-Leffler functions.

2024-06-07Journal article

Open access

Dirac’s method applied to the time-fractional diffusion-wave equation

Publication . Ferreira, M.; Vieira, N.; Rodrigues, M. M.

We compute the fundamental solution for time-fractional diffusion Dirac-like equations, which arise from the factorization of the multidimensional time-fractional diffusion-wave equation using Dirac’s factorization approach.

2024-06-07Conference object

Open access