44 documents found, page 1 of 5
Factorization à la Dirac applied to the time-fractional telegraph equation
Ferreira, M.; Rodrigues, M. M.; Vieira, N.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...
Dirac's method applied to the time-fractional diffusion-wave equation
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.
Distributed-order relaxation-oscillation equation
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 t...
Hypercomplex operator calculus for the fractional Helmholtz equation
Vieira, N.; Ferreira, M.; Rodrigues, M. M.; Kraußhar, R. S.In this paper, we develop a hypercomplex operator calculus to treat fully analytically boundary value problems for the homogeneous and inhomogeneous fractional Helmholtz equations 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...
The Teodorescu and the Π-operator in octonionic analysis and some applications
Kraußhar, R. S.; Ferreira, M.; Vieira, N.; Rodrigues, M. M.In the development of function theory in octonions, the non-associativity property produces an additional associator term when applying the Stokes formula. To take the non-associativity into account, particular intrinsic weight factors are implemented in the definition of octonion-valued inner products to ensure the existence of a reproducing Bergman kernel. This Bergman projection plays a pivotal role in the $...
Dirac’s method applied to the time-fractional diffusion-wave equation
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.
Distributed-order relaxation-oscillation equation
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 t...
On a fractional Sturm-Liouville problem in higher dimensions
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.
Uniformly distributed-order wave equation in higher dimensions
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.
Eigenfunctions of the time-fractional telegraph equation of distributed order
Vieira, N.; Rodrigues, M. M.; Ferreira, M.In this work, the eigenfunction problem for the time-fractional telegraph operator of distributed order in Rn ×R+ is considered. By employing the technique of the Fourier, Laplace and Mellin transforms, an integral representation of the eigenfunctions involving the Fox H-function is obtained.