M. M. Rodrigues

Aveiro Discretization Method in Mathematics: A New Discretization Principle

We found a very general discretization method for solving wide classes of mathematical problems by applying the theory of reproducing kernels. An illustration of the generality of the method is here performed by considering several distinct classes of problems to which the method is applied. In fact, one of the advantages of the present method—in comparison to other well-known and well-established methods—is its global nature and no need of special or very particular data conditions. Numerical experiments have been made, and consequent results are here exhibited. Due to the powerful results which arise from the application of the present method, we consider that this method has everything to become one of the next-generation methods of solving general analytical problems by using computers. In particular, we would like to point out that we will be able to solve very global linear partial differential equations satisfying very general boundary conditions or initial values (and in a somehow independent way of the boundary and domain). Furthermore, we will be able to give an ultimate sampling theory and an ultimate realization of the consequent general reproducing kernel Hilbert spaces. The general theory is here presented in a constructive way, and contains some related historical and concrete examples.

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.

Fundamental solutions of the fractional two-parameter telegraph equation

This paper is intended to investigate a fractional telegraph equation of the form with positive real parameters a, b and c. Here , and are operators of the Riemann–Liouville fractional derivative, where 0<α≤1 and 0<β≤1. A symbolic operational form of the solutions in terms of the Mittag–Leffler functions is exhibited. Using the Banach fixed point theorem, the existence and uniqueness of solutions are studied for this kind of fractional differential equations.

Numerical analysis of a two-parameter fractional telegraph equation

Abstract In this paper we consider the two-parameter fractional telegraph equation of the form − C D t 0 + α + 1 u ( t , x ) + C D x 0 + β + 1 u ( t , x ) − C D t 0 + α u ( t , x ) − u ( t , x ) = 0 . Here C D t 0 + α , C D t 0 + α + 1 , C D x 0 + β + 1 are operators of the Caputo-type fractional derivative, where 0 ≤ α 1 and 0 ≤ β 1 . A numerical method is introduced to solve this fractional telegraph equation and stability conditions for the numerical method are obtained. Numerical examples are given in the final section of the paper.

Operational Calculus for Bessel’s Fractional Equation

This paper is intended to investigate a fractional differential Bessel’s equation of order 2α with \( \alpha \in]0,1] \) involving the Riemann–Liouville derivative. We seek a possible solution in terms of power series by using operational approach for the Laplace and Mellin transform. A recurrence relation for coefficients is obtained. The existence and uniqueness of solutions is discussed via Banach fixed point theorem.

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.

Some operational properties of the Laguerre transform

This paper is devoted to the study of some properties of the Laguerre transform. We define new properties of the Laguerre transform in a weighted L2–space. Moreover, we present some results concerning the action of this integral transform over some class of polynomials.

Stability analysis of Raman propagation equations

Raman Fiber Amplifiers (RFA) are assuredly one of the most seriously renewed research subjects in the field of optical fiber communication systems. The amplification is based on stimulated Raman scattering (SRS), which occurs when there is enough pumping power within the fiber transferred to the signals. In the simplest situation, only the interaction between the pumping lasers and the probe signals needs to be accounted being the system power evolution along the distance given by a set of ordinary nonlinear ordinary differential equations (ODE). Although some consistent work has been done to provide numerical solutions for both forward and backward pumping configurations, the nonlinearity of the equations disable the computation of analytical solutions. However, efforts that lead to qualitative understanding of the solution rather than detailed quantitative information are quite valuable. This approach is geometrical and deals with the topic of stability. In this paper, we present a qualitative study of Raman equations valid for forward and backward pumping. The origin was proved to be an asymptotically stable node. The determination of the second equilibrium point is not straightforward for a generic number of equations, but some simplifications can be done.

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.

Reproducing Kernels and Discretization

We give a short survey of a general discretization method based on the theory of reproducing kernels. We believe our method will become the next generation method for solving analytical problems by computers. For example, for solving linear PDEs with general boundary or initial value conditions, independently of the domains. Furthermore, we give an ultimate sampling formula and a realization of reproducing kernel Hilbert spaces.