Nelson Vieira

A numerical method for the fractional Schrödinger type equation of spatial dimension two

This work focuses on an investigation of the (n+1)-dimensional time-dependent fractional Schrödinger type equation. In the early part of the paper, the wave function is obtained using Laplace and Fourier transform methods and a symbolic operational form of the solutions in terms of Mittag-Leffler functions is provided. We present an expression for the wave function and for the quantum mechanical probability density. We introduce a numerical method to solve the case where the space component has dimension two. Stability conditions for the numerical scheme are obtained.

Factorization of the Non-Stationary Schrödinger Operator

Abstract.We consider a factorization of the non-stationary Schrödinger operator based on the parabolic Dirac operator introduced by Cerejeiras, Kähler and Sommen. Based on the fundamental solution for the parabolic Dirac operators, we shall construct appropriated Teodurescu and Cauchy-Bitsadze operators. Afterwards we will describe how to solve the nonlinear Schrödinger equation using Banach fixed point theorem.

Fractional Clifford Analysis

In this paper we present the basic tools of a fractional function theory in higher dimensions by means of a fractional correspondence to the Weyl relations. A Fischer decomposition, Almansi decomposition, fractional Euler and Gamma operators, monogenic projection, and basic fractional homogeneous powers will be constructed.

Eigenfunctions and Fundamental Solutions of the Caputo Fractional Laplace and Dirac Operators

In this paper, by using the method of separation of variables, we obtain eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator defined via fractional Caputo derivatives. The solutions are expressed using the Mittag-Leffler function and we show some graphical representations for some parameters. A family of fundamental solutions of the corresponding fractional Dirac operator is also obtained. Particular cases are considered in both cases.

Fundamental solution of the multi-dimensional time fractional telegraph equation

Abstract In 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.

Integral Formulas for k-hypermonogenic Functions in ℝ3

We consider harmonic functions with respect to the Laplace–Beltrami operator of the Riemannian metric \( ds^{2} = x_{2}^{-2k}(\sum \limits{_{i=0}^{2}}{dx_{i}^{2}}) \) and their quaternion function theory in ℝ3. Leutwiler noticed around 1990 that if the usual Euclidean metric is changed to the hyperbolic one, that is k=1, then the power function \( (x_{0}+x_{1}e_{1}+x_{2}e_{2})^{n} \), calculated using quaternions, is the conjugate gradient of a hyperbolic harmonic function. We study generalized holomorphic functions, called k-hypermonogenic functions satisfying the modified Dirac equation. Note that 0-hypermonogenic are monogenic and 1-hypermonogenic functions are hypermonogenic defined by H. Leutwiler and the first author.

Fundamental Solutions of the Instationary Schrodinger Difference Operator

In this paper, we study the fundamental solutions for the explicit and implicit time dependent Schrödinger operator, via the discrete Fourier transform and the arising symbol for the Laplace operator. In both cases, we prove the convergence of the obtained discrete fundamental solutions to the continuous ones in the l 1 norm.

Some Properties of the Fractional Circle Zernike Polynomials

In this paper, we present a fractional extension of the classical circle Zernike polynomials defined via g-Jacobi functions. Some properties of this new class of functions are studied, such as recurrence relations for consecutive and distant neighborhoods, and differential relations. A graphic representation for the proposed fractional circle Zernike polynomials will be presented in the final section of the paper.

Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators using Caputo derivatives

Abstract In this paper, we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator where and the fractional derivatives , , are in the Caputo sense. Applying integral transform methods, we describe a complete family of eigenfunctions and fundamental solutions of the operator 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.

Fischer Decomposition in Generalized Fractional Clifford Analysis

In this paper we present the basic tools of a fractional function theory in higher dimensions by means of a fractional correspondence to the Weyl relations via Gelfond–Leontiev operators of generalized differentiation. A Fischer decomposition is established. Furthermore, we give an algorithm for the construction of monogenic homogeneous polynomials of arbitrary degree.