Juan J. Trujillo

University of La Laguna | ULL · Faculty of Mathematics

The authors study the following Cauchy-type problem for the nonlinear differential equation of fractional order α∈ℂ, Re(α)>0, (D a+ α y)(x)=f[x,y(x)],n-1<Re(α)≤n,n=-[-Re(α)], (D a+ α-k y)(a+)=b k ,b k ∈ℂ,k=1,2,⋯,n, containing the Riemann-Liouville fractional derivative D a+ α y, on a finite interval [a,b] of the real axis ℝ=(-∞,∞) in the space of s...

In this paper, we obtain an equivalent nonlinear integral equation to the stochastic neutral fractional system with bounded operator. Using the integral equation, the sufficient conditions for ensuring the complete controllability of the stochastic fractional neutral systems with Wiener and Lévy noise are obtained. Banach’s fixed point theorem is u...

We study the existence and uniqueness of a mild solution of fractional im- pulsive differential equations with nonlocal conditions. Here we consider fractional derivative in the non-instantaneous impulsive conditions. We use fixed point techniques and resolvent operators to prove our existence results.

This paper concerns the null controllability of fractional dynamical systems with constrained control. We assume that the linear system is controllable with square integrable control and provide sufficient conditions for the null controllability with constrained control. Further, sufficient conditions for the null controllability of nonlinear syste...

Sufficient conditions for relative controllability of stochastic fractional neutral systems with bounded operator and multiple time varying delay in the control are obtained. The result is proved using an equivalent nonlinear integral equation to the system and Banach contraction principle. The controllability results are derived for systems with b...

This paper studies the periodic functions in the perspective of fractional calculus application. It is shown that the fractional derivative of a periodic signal is periodic if it is defined on the whole real line. Several common fractional derivative formulations are considered, namely the Grünwald–Letnikov, Liouville and Caputo on \(\mathbb {R}\),...

The main purpose of this paper is to study the existence of solutions for the nonlinear fractional partial integrodifferential equations with Dirichlet boundary condition. Under suitable assumption the results are established by using the Leray-Schauder fixed point theorem and Arzela-Ascoli theorem. An example is provided to illustrate the main res...

In this paper, by means of upper and lower solutions, we develop monotone iterative method for the existence of extremal solutions for coupled system of nonlinear fractional integro-differential equations with advanced arguments. We illustrate this technique with the help of an example.

In this paper a method based on Sinc approximation is developed for the numerical solution of a nonlinear fractional pantograph equation. In order to use Sinc approximation, the problem needs to have an analytic solution. So we investigated the existence and uniqueness of analytic solutions in the proposed domain. Single and double exponential tran...

We study the linear fractional Schr\"odinger equation on a Hilbert space, with a fractional time derivative of order $0<\alpha<1,$ and a self-adjoint generator $A.$ Using the spectral theorem we prove existence and uniqueness of strong solutions, and we show that the solutions are governed by an operator solution family $\{U_{\alpha}(t)\}_{t\geq 0}...

This paper is devoted to investigate the basic results such as existence, uniqueness and continuous dependence of solutions of system of fractional differential equations involving the Caputo fractional derivative. Validity and convergence of Picard’s successive approximations for the solutions of the system of fractional differential equations hav...

We study the linear fractional Schr\"odinger equation on a Hilbert space, with a fractional time derivative of order $0<\alpha<1,$ and a self-adjoint generator $A.$ Using the spectral theorem we prove existence and uniqueness of strong solutions, and we show that the solutions are governed by an operator solution family $\{U_{\alpha}(t)\}_{t\geq 0}...

In this paper, the control problem of nonlinear neutral fractional Volterra integrodifferential systems with implicit fractional derivative is established. Such kind of problems involve a number of problems on complex media. Sufficient conditions for controllability are obtained through the notions of a condensing map and measure of noncompactness...

This document presents a sliding window algorithm for the calculation of the empirical mode decomposition for long signals. The spline calculation of very long signals requires a long computation time. Our aim is to improve the calculation time of the empirical mode decomposition for Long signals. Some authors have used sliding windows for the whol...

In this article, we obtain the solutions of some fractional differential systems with different orders using the inverse operator method and Mittag–Leffler function. Moreover, the controllability of time-invariant linear fractional system is investigated. Some examples are provided to illustrate the theory.

In this paper, we study boundary value problems for fractional integrodifferential equations involving Caputo derivative in Banach spaces. A generalized singular type Gronwall inequality is given to obtain an important priori bounds. Some sufficient conditions for the existence solutions are established by virtue of fractional calculus and fixed po...

We study the boundedness and compactness of Riemann-Liouville integral operators on the so-called Morrey spaces which are nonseparable spaces. There are no approximation or contractive skills in this kind of spaces. Moreover, unlike the use of dual or maximal point of view in integrable function spaces, the idea of our proof proceeds from the compa...

In this paper, we establish sufficient conditions for the global relative controllability of nonlinear neutral fractional Volterra integro-differential systems with distributed delays in control. The results are obtained by using the Mittag–Leffler functions and the Schauder fixed-point theorem. Examples are presented to illustrate the results. Cop...

We formulate a coherent approach to signals and systems theory on time scales. The two derivatives from the time-scale calculus are used, i.e., nabla (forward) and delta (backward), and the corresponding eigenfunctions, the so-called nabla and delta exponentials, computed. With these exponentials, two generalised discrete-time Laplace transforms ar...

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 controllabilit...

This paper deals with some existence and Ulam stability results for a class of partial integral equations via Hadamard’s fractional integral, by applying Schauder’s fixed-point theorem.

In this paper, we use a global implicit function theorem for the investigation of the existence and uniqueness of a solution as well as the sensitivity of a Cauchy problem for a general integro-differential system of order ∝ ε(lunate) of Volterra type, involving two functional parameters nonlinearly.

In this paper, we establish sufficient conditions for the existence of local solutions for a class of Cauchy type problems with arbitrary fractional order. The results are established by the application of the contraction mapping principle and Schaefer’s fixed point theorem. An example is provided to illustrate the applicability of the results.

Purpose: To compare the ordinary monoexponential model with three anomalous relaxation models-the stretched Mittag-Leffler, stretched exponential, and biexponential functions-using both simulated and experimental cartilage relaxation data. Methods: Monte Carlo simulations were used to examine both the ability of identifying a given model under h...

In this paper, we present some results concerning the existence and global asymptotic stability of solutions for a functional integral equation of fractional order. We use Schauder's fixed point theorem for the existence of solutions, and we prove that all these solutions are globally asymptotically stable.

The main objective of this paper is to propose a new generalisation of the Helmholtz decomposition theorem for both fractional time and space, which leads to four equations generalising the Maxwell equations that emerge as particular case. To get these results the well-known classical vectorial operators, gradient, divergence, curl, and laplacian a...

The non local fractional Laplacian plays a relevant role when modeling the dynamics of many processes through complex media. From 1933 to 1949, within the framework of potential theory, the Hungarian mathematician Marcel Riesz discovered the well known Riesz potential operators, a generalization of the Riemann-Liouville fractional integral to dimen...

In this paper, we use the representation of the generalized hypergeometric function p F q in terms of the known Meijer G-function to extend the range of parameters of such a special function so it be convergent. Also, we establish the corresponding series representation for such an extension. In particular, we extend the hypergeometric functions F...

The problem of steady state output of the discrete-time fractional differential systems is studied in this paper. Based on the fact that the exponentials are the eigenfunctions of such systems, a general algorithm for the output computation when the input is the product “rising factorial.exponential is presented. The singular case is studied and so...

In this paper, we investigate the existence of solutions for the fractional neutral differential equations with random impulses. The results are obtained by using Krasnoselskii’s fixed point theorem. Examples are added to show applications of the main results.

The decomposition of a fractional linear system is discussed in this paper. It is shown that it can be decomposed into an integer order part, corresponding to possible existing poles, and a fractional part. The first and second parts are responsible for the short and long memory behaviors of the system, respectively, known as characteristic of frac...

The goal of this short note is to let the readers of the “FCAA” journal to know that the “counterexamples” in the paper mentioned in the title of this note and referred in the sequel as [Bai et al.] are wrong in the sense that they contradict to nothing in authors' references, in contrast to what they state.

In this paper, the control problem of nonlinear neutral fractional Volterra integrodifferential systems with implicit fractional derivative is established. Such kind of problems involve a number of problems on complex media. Sufficient conditions for controllability are obtained through the notions of a condensing map and measure of noncompactness...

In this paper, we study the controllability of linear and nonlinear fractional damped dynamical systems, which involve fractional Caputo derivatives, with different order in finite dimensional spaces using the Mittag-Leffler matrix function and the iterative technique. A numerical example is provided to illustrate the theory.

In this paper we use the upper and lower solutions method combined with a fixed point theorem for condensing multivalued maps due toMartelli to investigate the existence of solutions of a class of partial hyperbolic differential inclusions with not instantaneous impulses.

A certain function space C α , 0≤α≤1, larger than the space of continuous functions, is introduced in order to study new properties and the extension of some already known results on the Riemann-Liouville fractional integral and derivative operators. Sufficient conditions for the continuity of I a 1-α f are given. Furthermore, necessary conditions...

A generalized Taylor’s formula of the form f(x)=∑ i=0 n a i (x-a) iα +T n (x), where a i ∈ℝ, x≥a, 0<α≤1, is established. In the particular case α=1 this expression is precisely the classic Taylor’s formula. In addition, detailed expressions for T n (x) and a i involving the Riemann-Liouville fractional derivative and some applications are also give...

The paper proposes a new approach to compute the impulse response of fractional order linear time invariant systems. By applying a general approach to decompose a Laplace transform into a Laurent like series, we obtain a power series that generalizes the Taylor and MacLaurin series. The algorithm is applied to the computation of the impulse respons...

Our aim in this paper is to study the existence and the stability of solutions for Riemann–Liouville Volterra–Stieltjes quadratic integral equations of fractional order. Our results are obtained by using some fixed point theorems. Some examples are provided to illustrate the main results.

The paper is concerned with existence of mild solution of evolution equation with Hilfer fractional derivative which generalized the famous Riemann-Liouville fractional derivative. By noncompact measure method, we obtain some sufficient conditions to ensure the existence of mild solution. Our results are new and more general to known results.

In this paper, we consider a class of fractional integro-differential inclusions in Banach spaces. This paper deals with the controllability for fractional integro-differential control systems. First, we establishes a set of sufficient conditions for the controllability of fractional semilinear integro-differential inclusions in Banach spaces via r...

We study the existence of mild and classical solutions are proved for a class of impulsive integrodifferential equations with nonlocal conditions in Banach spaces. The main results are obtained by using measure of noncompactness and semigroup theory. An example is presented.

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...

Fractional order dynamics and chaotics systems have been recently combined, yielding interesting behaviours. In this paper, a novel integer order hyperchaotic system is considered. Then, a fractional order hyperchaotic representation of said system is proposed using a natural fractionalization. Two different linear control methodologies to deal wit...

Fractional calculus (fractional derivatives and fractional integrals together with their applications) is undergoing a rapid development, from both theoretical as well as applied viewpoints. Such a tool is an emergent topic, and within its framework new concepts and applications, which lead to a challenging insight, have appeared during the last fe...

a b s t r a c t In this paper we formulate a coherent discrete-time signals and systems theory taking derivative concepts as basis. Two derivatives – nabla (forward) and delta (backward) – are defined and generalized to fractional orders, obtaining two formulations that are discrete versions of the well-known Grünwald–Letnikov derivatives. The eige...

a b s t r a c t This paper analyzes the mathematical modeling of a two-region composite reservoir using the concepts of fractal geometry and fractional calculus. Heterogeneity of the reservoir is considered based on fractal geometry. Fractional calculus is used to consider production history in fractal reservoirs. An analytical solution is derived...

In this paper, we consider the concept of the so-called gener-alized metric space and give some results on the existence, uniqueness and estimation of the solutions of Fredholm type integro-differential equations in two variables using Perov's fixed point theorem. Furthermore, we give some illustrative examples to verify the effectiveness and appli...

In this paper, we obtain formulas of fractional integration (of Marichev– Saigo–Maeda type) of the generalized multi-index Mittag-Leffler functions E γ,κ [(α j ,β j )m ; z] generalizing 2m-parametric Mittag-Leffler functions studied by Saxena and Nishimoto (J. Fract. Calc. 37 (2010] 43–52). Some interesting special cases of our main results are con...

In this paper we investigate the existence and uniqueness of solutions on a compact interval for non-linear fractional integro-differential equations with state-dependent delay. Our results will be obtained using suitable fixed point theorems and the tech- nique of measures of noncompactness. Some application of the main result have been included.

The objective of this paper is to show an approach to the fractional version of the Sturm-Liouville problem, by using different fractional operators that return to the ordinary operator for integer order. For each fractional operator we study some of the basic properties of the Sturm-Liouville theory. We analyze a particular example that evidences...

This paper presents an extension of the classical elastic law. The main objective of this new law is to represent linear and non-linear behaviour for computational metal forming purposes. The extension of the model is built by means of a stress–strain relationship given by an integral equation, its kernel characterising the mentioned complex behavi...

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.

This paper is devoted to study the existence of solutions of a Cauchy type problem for a nonlinear fractional differential equation, via the techniques of measure of noncompactness. The investigation is based on a new fixed point result which is a generalization of the well known Darbo’s fixed point theorem. The main result is less restrictive than t...

This paper presents a survey of useful, established formulas in Fractional Calculus, systematically collected for reference purposes.

It is well known that a classical PI controller cannot accurately follow a ramp setpoint. A real world situation which shows this behavior could be a system composed of a direct current motor whose input setpoint is an acceleration/deceleration ramp. In this paper, we propose the design of a fractional PIλ controller, with a fractional-order parame...

This article deals with the existence of solutions of nonlinear fractional pantograph equations. Such a model can be considered suitable to be applied when the corresponding process occurs through strongly anomalous media. The results are obtained using fractional calculus and fixed point theorems. An example is provided to illustrate the main resu...

During the last fifty years the area of Fractional Calculus verified a considerable progress. This paper analyzes and measures the evolution that occurred since 1966.

The aim of this paper is to derive a set of sufficient conditions for controllability of nonlinear fractional dynamical system of order 1<α<2 in finite dimensional spaces. The results are obtained using the Schauder fixed point theorem. Examples are included to verify the result.

In this paper, we establish the controllability for a class of abstract impulsive mixed-type functional integro-differential equations with finite delay in a Banach space. Some sufficient conditions for controllability are obtained by using the Mönch fixed point theorem via measures of noncompactness and semigroup theory. Particularly, we do not as...

A derivative based discrete-time signal processing is presented. Both nabla (forward) and delta (backward) derivatives are studied and generalised including the fractional case. The corresponding exponentials are introduced as eigenfunctions of such derivatives.

In this paper we study the observability and controllability of fractional linear dynamical systems in finite dimensional spaces. Examples are included to illustrate the theoretical results proved in this manuscript.

In this paper, we establish sufficient conditions for the existence of solutions for a class of initial value problems with integral condition for impulsive fractional integro-differential equations. The results are established by the application of the contraction mapping principle and the Krasnoselskii fixed point theorem. An example is provided...

Stability as an extremely important property of dynamical systems can be investigated in various domains (Bellman, 1953; Dorf and Bishop, 1990The usual concept of the bounded input-bounded output (BIBO) or external stability in time domain can be defined via the following general stability conditions (Matignon, 1998): A causal LTI system with impul...

Fractional calculus generalizes integer order derivatives and integrals. During the last half century a considerable progress took place in this scientific area. This paper addresses the evolution and establishes an assertive measure of the research development.

We study the observability of linear and nonlinear fractional differential systems of order 0 < α < 1 by using the Mittag-Leffler matrix function and the application of Banach’s contraction mapping theorem. Several examples illustrate the concepts.

The theory and applications of fractional calculus (FC) had a considerable progress during the last years. Dynamical systems and control are one of the most active areas, and several authors focused on the stability of fractional order systems. Nevertheless, due to the multitude of efforts in a short period of time, contributions are scattered alon...

This paper is concerned with the controllability problem for a class of mixed type impulsive fractional integro-differential equations in Banach spaces. Sufficient conditions for the controllability result are established by using suitable fixed point theorem combined with the fractional calculus theory and solution operator under some weak conditi...

A new fractional derivative of the Grüwald–Letnikov type is proposed and some properties are studied. The new definition incorporates both the forward and backward Grüwald–Letnikov and the two-sided fractional derivatives. Several properties of such generalized operator are presented, and some particular cases deduced.

In this article, we study the existence of solutions for a class of fractional integrodifferential equations with time varying delay by using the resolvent operators and fixed point technique. The main results present in this paper improve some results on this issue that have been studied recently. An example is provided to illustrate our main theo...

In this paper, we study existence and uniqueness of fractional integrodifferential equations with boundary value conditions. A new generalized singular type Gronwall inequality is given to obtain an important a priori bounds. Existence and uniqueness results of solutions are established by virtue of fractional calculus and fixed point method under...

This paper is concerned with the global relative controllability of fractional dynamical systems with multiple delays in control for finite dimensional spaces. Sufficient conditions for controllability results are obtained using Schauder's fixed point theorem and the controllability Grammian matrix which is defined by the Mittag-Leffler matrix func...

This paper deals with the global relative controllability of linear and nonlinear fractional dynamical systems with distributed delays in control for finite dimensional spaces. Sufficient conditions for controllability results are obtained using Schauder’s fixed point theorem and the controllability Grammian matrix which is defined by Mittag Leffle...

A real regularised integral formulation of the fractional derivative is obtained from the generalised Grünwald–Letnikov derivative without using the Cauchy derivative. This new approach is based on the properties of the Mellin transform. The usual Riemann–Liouville and Caputo derivatives are expressed in a similar way emphasising their regularising...

The generalized incremental ratio fractional derivative is revised and its main properties deduced. It is shown that in the case of analytic functions, it enjoys some interesting properties like: linearity and causality and has a semi-group structure. Some simple examples are presented. The enlargement of the set of functions for which the group pr...

In this paper we establish a set of sufficient conditions for the controllability of nonlinear fractional dynamical systems. The results are obtained by using the recently derived formula for solution representation of systems of fractional differential equations and the application of the Schauder fixed point theorem. Examples are provided to illu...

This book will give readers the possibility of finding very important mathematical tools for working with fractional models and solving fractional differential equations, such as a generalization of Stirling numbers in the framework of fractional calculus and a set of efficient numerical methods. Moreover, we will introduce some applied topics, in...