Showing papers by "Juan J. Trujillo published in 2003"

Hadamard-type integrals as G-transforms

Abstract: This paper is devoted to the study of four integral operators, which are generalizations and modifications of integrals of Hadamard, in the space X c p of Lebesgue measurable functions f on R + = (0, ∞) such that for c ∈ R = (−∞, ∞) [Formula: See Text] Representations for the operators are given in the form of integral transforms involving the Meijer G-function in the kernels. The mapping properties such as the boundedness, representation and range are established.

Stirling Functions of the Second Kind in the Setting of Difference and Fractional Calculus

Abstract: The purpose of this article and companion ones is to present a new approach to generalizations of Stirling numbers of the first and the second kind in terms of fractional calculus analysis by using differences and differentiation operators of fractional order. Such an approach allows us to extend the classical Stirling numbers of the first and the second kind in a natural way not only to any positive, but also to any negative order. Moreover, an application of the fractional approach gives us the opportunity to extend the classical Stirling numbers to more general complex functions. In the present article we extend the classical Stirling numbers of the second kind, S(n, k), for the first parameter from a nonnegative integer number n to any complex α. Such constructions, S(α, k), will be defined for any complex α and by when k >0, while S(0, 0) = 1 and when k = 0. We show that S(α, k) with positive α can be represented by the Liouville and Marchaud fractional derivatives of the exponential functio...

Fractional Differential Equations: A Emergent Field in Applied and Mathematical Sciences

Abstract: The paper is devoted to some aspects of differential equations of fractional order and their applications. It is explained a fact that the subject of fractional differential equations is an emergent topic as a very useful tool to model many anomalous phenomena in nature and in the theory of complexity systems. Various fractional integral and fractional derivative are presented together with some of their properties. Simple partial differential equations are discussed in connection with anomalous diffusion. A new model, useful in both sub-diffusion and super-fast diffusion processes, is introduced. Such a model, generalizing the clasical problem associated with the heat equation, involves the generalized Liouville fractional derivative over the time variable. The one-dimensional case is studied and explicitely solved, and its generalization to the multi-dimensional case is discussed.

Generalized Stirling Functions of Second Kind and Representations of Fractional Order Differences via Derivatives

Abstract: The paper is devoted to the study of generalized Stirling functions of second kind by using difference and differentiation operators of fractional order. The constructions under considerations give extensions of the classical Stirling numbers of second kind S(n, k) to functions S(n, β), whereby the second parameter k becomes any complex β, as well as to functions S(α, β), whereby also the first parameter n becomes any complex α. The main properties of these new functions, including recurrence relations are established. A chief application is a generalization of a basic formula of combinatorial and numerical analysis, namely expressing higher order differences in terms of derivatives, to the fractional instance: essentially a fractional order difference Δ β f(x) leads to S(n, β) in the above sum. Three concrete examples are presented.

Hankel-Schwartz and Hankel-Clifford transforms on \({\cal L}_{\nu,p}\)-spaces

Abstract: The paper is devoted to study Mapping properties of the Hankel-Schwartz integral transform and the Hankel-Clifford integral transform $$ ({\rm H}_{\eta,1}^{s}f)(x)=x^{-\eta}\int_0^\infty J_{\eta}(xt)t^{n+1}f(t)dt(\eta \in {\rm C,Re}(\eta)>-1,x>0) $$ $$ ({\rm H}_{\eta,1}^{s}f)(x)=x^{\eta /2}\int_0^\infty J_{\eta}\bigg (2(xt)1/2\bigg )t^{n/2}f(t)dt(\eta \in {\rm C,Re}(\eta)>-1,x>0) $$ on certain spaces \({\cal L}_{ u,p}( u \in {\rm R}=(-\infty,\infty );1\leq p\leq \infty )\) of measurable functions. Their representations in the form of integral transforms with H-function kernels are presented.