Luis Rodríguez-Germá

Fractional differential equations as alternative models to nonlinear differential equations

The main objective of this paper is to demonstrate the possibility of using fractional differential equations to simulate the dynamics of anomalous processes whose analytical representations are continuous but strongly not differentiable, like Weierstrass-type functions. This allows for the possibility of modeling phenomena which traditional differential modeling cannot accomplish. To this end we shall see how some functions of this kind have a fractional derivative at every point in a real interval, and are therefore solutions to fractional differential equations.

Controllability Results for Nonlinear Fractional-Order Dynamical Systems

This paper establishes a set of sufficient conditions for the controllability of nonlinear fractional dynamical system of order 1<α<2 in finite dimensional spaces. The main tools are the Mittag–Leffler matrix function and the Schaefer’s fixed-point theorem. An example is provided to illustrate the theory.

Linear fractional differential equations with variable coefficients

Abstract This work is devoted to the study of solutions around an α -singular point x 0 ∈ [ a , b ] for linear fractional differential equations of the form [ L n α ( y ) ] ( x ) = g ( x , α ) , where [ L n α ( y ) ] ( x ) = y ( n α ) ( x ) + ∑ k = 0 n − 1 a k ( x ) y ( k α ) ( x ) with α ∈ ( 0 , 1 ] . Here n ∈ N , the real functions g ( x ) and a k ( x ) ( k = 0 , 1 , … , n − 1 ) are defined on the interval [ a , b ] , and y ( n α ) ( x ) represents sequential fractional derivatives of order k α of the function y ( x ) . This study is, in some sense, a generalization of the classical Frobenius method and it has applications, for example, in obtaining generalized special functions. These new special functions permit us to obtain the explicit solution of some fractional modeling of the dynamics of many anomalous phenomena, which until now could only be solved by the application of numerical methods. 1

Controllability of Nonlinear Fractional Delay Dynamical Systems

This paper is concerned with controllability of nonlinear fractional delay dynamical systems with delay in state variables. The solution representations of fractional delay differential equations have been established by using the Laplace transform technique and the Mittag—Leffler function. Necessary and sufficient conditions for the controllability criteria of linear fractional delay systems are established. Further sufficient condition for the controllability of nonlinear fractional delay dynamical system are obtained by using the fixed point argument. Examples and numerical simulation are presented to illustrate the results.

Stirling functions of first kind in the setting of fractional calculus and generalized differences

The purpose of this paper is to present a new approach to generalizations of Stirling numbers of the first kind by the application of differential and integration operators of fractional order and generalized, infinite differences. Such an approach allows us to extend the classical Stirling numbers of the first kind, s(n, k), to functions s(α, β), where both parameters n, k have been extended to complex α, β. Under such a construction the s(α, β) turn out to have the series representation—a major result of this paper for , with for any when β = 0. Various properties of the new Stirling functions are established, most generalize those known for the numbers s(n, k); some are new, i.e. a multiple sum formula for s(α, k), and an interesting connection between the s(α, β) and the Riemann zeta function for complex β with . Several connections between the s(α, β) and the Stirling functions of second kind, s(α, β), studied earlier by the authors, are deduced. Thus the s( − n, β) coincide with the Stirling functions S( − β, n) of second kind, apart from a multiplicative constant. Of fundamental importance is the orthogonality property of the s(α, k) and S(k, m). The basic tool here is the Shannon sampling theorem of signal analysis. The Riemann–Liouville fractional derivative is expressed in terms of Hadamard derivatives, which involve the powers of the operator δ = x(d/dx). The sampling representation of the Mittag–Leffler function as a function of α is one of the many new results. Finally, a new “infinite” or fractional order difference operator, Δα, is defined in terms of the s(α, k); it involves the powers of the operator Θ = xΔ. This calculus of “infinite” differences is applied to representative examples, including the factorial and exponential functions.

CAPUTO LINEAR FRACTIONAL DIFFERENTIAL EQUATIONS

Abstract This paper is devoted to the study of nonsequential linear fractional differential equations with constant coefficients involving the Caputo fractional derivatives. The Laplace transform is applied to obtain the general explicit solutions for the equations being studied in terms of Mittag-Leffler functions and generalized Wright functions. Conditions are given for obtaining linearly independent solutions which form a fundamental system of solutions. Some examples are presented.

Stabilizability of fractional dynamical systems

In this paper, we establish that the controllability and observability properties of fractional dynamical systems in a finite dimensional space are dual. Using this duality result and the Mittag-Leffler matrix function, we propose the stabilizability of fractional MIMO (Multiple-input Multipleoutput) systems. Some numerical examples are provided to show the effectiveness of the obtained results.

Fractional operators and some special functions

This paper considers the Riemann-Liouville fractional operator as a tool to reduce linear ordinary equations with variable coefficients to simpler problems, avoiding the singularities of the original equation. The main result is that this technique allow us to obtain an extension of the classical integral representation of the special functions related with the original differential equations. In particular, we will use as examples the cases of the well-known Generalized, Gauss and Confluent Hypergeometric equations, Laguerre equation, Hermite equation, Legendre equation and Airy equation.