53 documents found, page 1 of 6
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...
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 operators on conformal manifolds
Orelma, H.; Vieira, N.Conformal manifolds $M_\lambda$ are open subsets of $\mathbb{R}^n$ endowed with the metric \begin{align*} g_\lambda =\frac{dx_1^2+\ldots+dx_n^2}{\lambda^2} \end{align*} where $\lambda$ is called the conformal function. We show that there exists the $\alpha$-Dirac operator $D_\alpha$, with $\alpha\in \BR$, acting on functions valued by the Clifford algebra on $M_\lambda$. The operator behaves similarly to the us...
Bessel-type activation functions in neural networks
Martins, D.; Vieira, N.; Slavov, T.; Georgieva, P.This paper introduces a novel family of activation functions using hypergeometric functions with trainable parameters. Hypergeometric functions possess a complex series structure that may create numerical issues. Therefore, we focus on Bessel-type functions of the first kind. The Bessel functions are a subfamily of hypergeometric functions, with characteristics similar to ReLU and sinusoidal activation function...
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...