Juan J. Trujillo

Fractional calculus: models and numerical methods

Survey of Numerical Methods to Solve Ordinary and Partial Fractional Differential Equations Specific and Efficient Methods to Solve Ordinary and Partial Fractional Differential Equations Fractional Variational Principles Continuous-Time Random Walks (CTRWs) Applications to Finance and Economics Generalized Stirling Numbers of First and Second Kind in the Framework of Fractional Calculus.

Review: On the fractional signals and systems

A look into fractional calculus and its applications from the signal processing point of view is done in this paper. A coherent approach to the fractional derivative is presented, leading to notions that are not only compatible with the classic but also constitute a true generalization. This means that the classic are recovered when the fractional domain is left. This happens in particular with the impulse response and transfer function. An interesting feature of the systems is the causality that the fractional derivative imposes. The main properties of the derivatives and their representations are presented. A brief and general study of the fractional linear systems is done, by showing how to compute the impulse, step and frequency responses, how to test the stability and how to insert the initial conditions. The practical realization problem is focussed and it is shown how to perform the input-ouput computations. Some biomedical applications are described.

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.

Mellin transform analysis and integration by parts for Hadamard-type fractional integrals

Abstract This paper is devoted to the study of four integral operators that are basic generalizations and modifications of fractional integrals of Hadamard, in the space Xpc of Lebesgue measurable functions f on R + =(0,∞) such that ∫ 0 ∞ u c f(u) p du u ess sup u>0 u c f(u) for c∈ R =(−∞,∞) , in particular in the space Lp(0,∞) (1⩽p⩽∞). Formulas for the Mellin transforms of the four Hadamard-type fractional integral operators are established as well as relations of fractional integration by parts for them.

Fractional calculus in the Mellin setting and Hadamard-type fractional integrals

Abstract The purpose of this paper and some to follow is to present a new approach to fractional integration and differentiation on the half-axis R + =(0,∞) in terms of Mellin analysis. The natural operator of fractional integration in this setting is not the classical Liouville fractional integral Iα0+f but J α 0+,c f (x):= 1 Γ(α) ∫ 0 x u x c log x u α−1 f(u) du u (x>0) for α>0, c∈ R . The Mellin transform of this operator is simply (c−s) −α M [f](s) , for s=c+it, c,t∈ R . The Mellin transform of the associated fractional differentiation operator D α 0+,c f is similar: (c−s) α M [f](s) . The operator D α 0+,c f may even be represented as a series in terms of xkf(k)(x), k∈ N 0 , the coefficients being certain generalized Stirling functions Sc(α,k) of second kind. It turns out that the new fractional integral J α 0+,c f and three further related ones are not the classical fractional integrals of Hadamard (J. Mat. Pure Appl. Ser. 4, 8 (1892) 101–186) but far reaching generalizations and modifications of these. These four new integral operators are first studied in detail in this paper. More specifically, conditions will be given for these four operators to be bounded in the space Xcp of Lebesgue measurable functions f on (0,∞), for c∈(−∞,∞), such that ∫ ∞ 0 |u c f(u)| p du/u for 1⩽p ess sup u>0 [u c |f(u)|] for p=∞, in particular in the space Lp(0,∞) for 1⩽p⩽∞. Connections of these operators with the Liouville fractional integration operators are discussed. The Mellin convolution product in the above spaces plays an important role.

Compositions of Hadamard-type fractional integration operators and the semigroup property

Abstract This paper is devoted to the study of four integral operators that are basic generalizations and modifications of fractional integrals of Hadamard in the space Xcp of functions f on R + =(0,∞) such that ∫ 0 ∞ u c f(u) p du u ess sup u>0 u c |f(u)| for c∈ R =(−∞,∞) , in particular in the space Lp(0,∞) (1⩽p⩽∞). The semigroup property and its generalizations are established for the generalized Hadamard-type fractional integration operators under consideration. Conditions are also given for the boundedness in Xcp of these operators; they involve Kummer confluent hypergeometric functions as kernels.

On the existence of solutions of fractional integro-differential equations

Under some suitable conditions, we prove the solvability of a large class of nonlinear fractional integro-differential equations by establishing some fractional integral inequalities and using the nonlinear alternative Leray-Schauder type. The uniqueness of solutions is also proved in some situations.

On development of fractional calculus during the last fifty years

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.

Fractional dynamics of populations

Abstract Nature often presents complex dynamics, which cannot be explained by means of ordinary models. In this paper, we establish an approach to certain fractional dynamic systems using only deterministic arguments. The behavior of the trajectories of fractional non-linear autonomous systems around the corresponding critical points in the phase space is studied. In this work we arrive to several interesting conclusions; for example, we conclude that the order of fractional derivation is an excellent controller of the velocity how the mentioned trajectories approach to (or away from) the critical point. Such property could contribute to faithfully represent the anomalous reality of the competition among some species (in cellular populations as Cancer or HIV). We use classical models, which describe dynamics of certain populations in competition, to give a justification of the possible interest of the corresponding fractional models in biological areas of research.

Stability of fractional order systems

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 along the literature, and it becomes difficult for researchers to have a complete and systematic picture of the present day knowledge. This paper is an attempt to overcome this situation by reviewing the state of the art and putting this topic in a systematic form. While the problem is formulated with rigour, from the mathematical point of view, the exposition intends to be easy to read by the applied researchers. Different types of systems are considered, namely, linear/nonlinear, positive, with delay, distributed, and continuous/discrete. Several possible routes of future progress that emerge are also tackled.