Post-Doctoral (Operational methods for solving fractional partial differential equations - FCT - Post-Doctoral fellowship: SFRH/BPD/73537/2010), University of Porto.
Ph.D. - Doctor in Mathematics, University of Aveiro.
Master's degree in Mathematics (Positive Semidefinite Programming), University of Lisboa.
Graduate Diploma in Mathematics, University of Coimbra.
RESEARCH INTERESTS
Fractional Calculus;
Linear and Non-Linear Fractional ODEs and PDEs;
Fractional Boundary Value Problems;
Numerical Methods for Fractional ODEs and PDEs;
Special Functions;
Integral Equations and Integral Transforms;
Mathematical Modeling;
Partial Differential Equations.
Factorization à la Dirac applied to the time-fractional telegraph equation M. Ferreira, M.M. Rodrigues, N. Vieira Communications in Nonlinear Science and Numerical Simulation, 2026 • Coupled time-fractional diffusion Dirac system derived via factorization method • Novel Fourier transform pairs with bivariate Mittag-Leffler and Fox H-functions • Explicit solutions, enabling analysis in both frequency and time-space, are derived • The long and short-term (asymptotic) behaviour of solutions is rigorously analysed • Factorization allows alternative triplets of Pauli matrices, yielding related solutions This paper examines a coupled system of two-term time-fractional diffusion Dirac-type equations. The system is derived by factorizing the multi-dimensional time-fractional telegraph equation with Hilfer fractional derivatives, using the Dirac method and a triplet of Pauli matrices. Solutions are obtained using operational methods provided by the combination of the Fourier transform in the space variable and the Laplace transform in the time variable. Key results include the discovery of novel Fourier transform pairs. These pairs relate specific Fourier kernels of bivariate Mittag-Leffler functions to Fox H-functions of two variables. This allows to obtain explicit solutions of the system in both Fourier-time and space-time domains. The asymptotic behaviour of these solutions is rigorously analysed, and graphical representations are generated. Further, we show that the factorization allows for the use of alternative triplets of Pauli matrices yielding related solutions. The results obtained can be generalised to the case of ψ -Hilfer derivatives.
Hypercomplex operator calculus for the fractional Helmholtz equation Nelson Vieira, Milton Ferreira, M. Manuela Rodrigues, Rolf Sören Kraußhar Mathematical Methods in the Applied Sciences, 2024 In this paper, we develop a hypercomplex operator calculus to treat fully analytically boundary value problems for the homogeneous and inhomogeneous fractional Helmholtz equation where fractional derivatives in the sense of Caputo and Riemann–Liouville are applied. Our method extends the recently proposed fractional reduced differential transform method (FRDTM) by using fractional derivatives in all directions. For the special separable case in three dimensions, we obtain completely explicit representations for the fundamental solution. This allows us to interpret and to understand the appearance of spatial steady‐state solutions or spatial blow‐ups of the fractional Helmholtz equation in a better way. More precisely, we were able to present explicit conditions for the parameters in the representation formulas of the fundamental solutions under which we obtain bounded or spatial decreasing steady‐solutions and when spatial blow‐ups occur. We also illustrate this with some representative numerical examples. Furthermore, we show that it is possible to recover the recently studied cases as well as the classical cases as particular limit cases within our more general setting. Using the hypercomplex operator approach also allows us to factorize the fractional Helmholtz operator and obtain some interesting duality relations between left and right derivatives, Caputo and Riemann–Liouville derivatives, and eigensolutions of antipodal eigenvalues in terms of a generalized Borel–Pompeiu formula. This factorization, in turn, allows us to tackle inhomogeneous fractional Helmholtz problems.
Distributed-order relaxation-oscillation equation M. M. Rodrigues, M. Ferreira, N. Vieira Aip Conference Proceedings, 2024 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.
Dirac’s Method Applied to the Time-Fractional Diffusion-Wave Equation M. Ferreira, N. Vieira, M. M. Rodrigues Aip Conference Proceedings, 2024 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.
On a Fractional Sturm-Liouville Problem in Higher Dimensions N. Vieira, M. M. Rodrigues, M. Ferreira Aip Conference Proceedings, 2023 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.
Uniformly Distributed-Order Wave Equation in Higher Dimensions N. Vieira, M. M. Rodrigues, M. Ferreira Aip Conference Proceedings, 2023 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.
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