Percorrer por autor "Ferreira, Milton"
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- 3D deformations by means of monogenic functionsPublication . Ferreira, Milton; Morais, JoãoIn this paper, the authors compute the coefficient of quasiconformality for monogenic functions in an arbitrary ball of the Euclidean space $\mathbb{R}^3$. This quantification may be needed in applications but also appear to be of intrinsic interest. The main tool used is a 3D Fourier series development of monogenic functions in terms of a special set of solid spherical monogenics. Ultimately, we present some examples showing the applicability of our approach.
- Complex Boosts: A Hermitian Clifford Algebra ApproachPublication . Ferreira, Milton; Sommen, FranciscusThe aim of this paper is to study complex boosts in complex Minkowski space-time that preserves the Hermitian norm. Starting from the spin group Spin$^+(2n,2m,\bkR)$ in the real Minkowski space $\bkR^{2n,2m}$ we construct a Clifford realization of the pseudo-unitary group U$(n,m)$ using the space-time Witt basis in the framework of Hermitian Clifford algebra. Restricting to the case of one complex time direction we derive a general formula for a complex boost in an arbitrary complex direction and its $KAK-$decomposition, generalizing the well-known formula of a real boost in an arbitrary real direction. In the end we derive the complex Einstein velocity addition law for complex relativistic velocities, by the projective model of hyperbolic $n-$space.
- Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators using Caputo derivativesPublication . Ferreira, Milton; Vieira, Nelson Felipe LoureiroIn this paper we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator ${}^C\!\Delta_+^{(\alpha,\beta,\gamma)}:= {}^C\!D_{x_0^+}^{1+\alpha} +{}^C\!D_{y_0^+}^{1+\beta} +{}^C\!D_{z_0^+}^{1+\gamma},$ where $(\alpha, \beta, \gamma) \in \,]0,1]^3$ and the fractional derivatives ${}^C\!D_{x_0^+}^{1+\alpha}$, ${}^C\!D_{y_0^+}^{1+\beta}$, ${}^C\!D_{z_0^+}^{1+\gamma}$ are in the Caputo sense. Applying integral transform methods we describe a complete family of eigenfunctions and fundamental solutions of the operator ${}^C\!\Delta_+^{(\alpha,\beta,\gamma)}$ in classes of functions admitting a summable fractional derivative. The solutions are expressed using the Mittag-Leffler function. From the family of fundamental solutions obtained we deduce a family of fundamental solutions of the corresponding fractional Dirac operator, which factorizes the fractional Laplace operator introduced in this paper.
- Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators: the Riemann-Liouville casePublication . Ferreira, Milton; Vieira, Nelson Felipe LoureiroIn this paper, we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator $\Delta_+^{(\alpha,\beta,\gamma)}:= D_{x_0^+}^{1+\alpha} +D_{y_0^+}^{1+\beta} +D_{z_0^+}^{1+\gamma},$ where $(\alpha, \beta, \gamma) \in \,]0,1]^3$, and the fractional derivatives $D_{x_0^+}^{1+\alpha}$, $D_{y_0^+}^{1+\beta}$, $D_{z_0^+}^{1+\gamma}$ are in the Riemann-Liouville sense. Applying operational techniques via two-dimensional Laplace transform we describe a complete family of eigenfunctions and fundamental solutions of the operator $\Delta_+^{(\alpha,\beta,\gamma)}$ in classes of functions admitting a summable fractional derivative. Making use of the Mittag-Leffler function, a symbolic operational form of the solutions is presented. From the obtained family of fundamental solutions we deduce a family of fundamental solutions of the fractional Dirac operator, which factorizes the fractional Laplace operator. We apply also the method of separation of variables to obtain eigenfunctions and fundamental solutions.
- Factorizations of Möbius gyrogroupsPublication . Ferreira, MiltonIn this paper, we consider a Möbius gyrogroup on a real Hilbert space (of finite or infinite dimension) and we obtain its factorization by gyrosubgroups and subgroups. It is shown that there is a duality relation between the quotient spaces and the orbits obtained. As an example, we present the factorization of the Möbius gyrogroup of the unit ball in R^n linked to the proper Lorentz group Spin+(1, n).
- 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.
- First and second fundamental solutions of the time-fractional telegraph equation with Laplace or Dirac operatorsPublication . Ferreira, Milton; Rodrigues, Manuela M.; Vieira, NelsonIn this work we obtain the first and second fundamental solutions (FS) of the multidimensional time-fractional equation with Laplace or Dirac operators, where the two time-fractional derivatives of orders α ∈]0, 1] and β ∈]1, 2] are in the Caputo sense. We obtain representations of the FS in terms of Hankel transform, double Mellin- Barnes integrals, and H-functions of two variables. As an application, the FS are used to solve Cauchy problems of Laplace and Dirac type.
- 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.
- Fundamental Solution of the Multi-Dimensional Time Fractional Telegraph EquationPublication . Ferreira, Milton; Rodrigues, M. Manuela; Vieira, NelsonIn this paper we study the fundamental solution (FS) of the multidimensional time-fractional telegraph equation where the time-fractional derivatives of orders α ∈]0,1] and β ∈]1,2] are in the Caputo sense. Using the Fourier transform we obtain an integral representation of the FS in the Fourier domain expressed in terms of a multivariate Mittag-Leffler function. The Fourier inversion leads to a double Mellin-Barnes type integral representation and consequently to a H-function of two variables. An explicit series representation of the FS, depending on the parity of the dimension, is also obtained. As an application, we study a telegraph process with Brownian time. Finally, we present some moments of integer order of the FS, and some plots of the FS for some particular values of the dimension and of the fractional parameters α and β.
- 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
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