A probabilistic interpretation of the fractional-order differentiation

• Autor(a) principal: Tenreiro Machado, J. A.
• Data de Publicação: 2003
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10400.22/13509
• topic: Fractional derivative; Fractional Calculus; Geometric interpretation; Probabilistic interpretation

The theory of fractional calculus (FC) is a useful mathematical tool for applied sciences. Nevertheless, FC is somehow hard to tackle and only in the last decades researchers were motivated for the application of the associated concepts. There are several reasons for this state of affairs, namely the apparent 'sufficient' of classical differential calculus for real-world applications, the plethora of different definitions for fractional derivatives and integrals and the lack of simple interpretation for such formulae. in what concerns the FC usefulness in the case of chaos and fractals lead to the development of fractional-order models and algorithms. On the other hand, the conceptual analysis of a fractional integral or a fractional derivate has also been addressed, but a simple interpretation is not yet completely established. This paper discusses a probabilistic interpretation of the fractional-order derivate, based on the Grunwald-Letnikov definition, that reduces to the standard geometric interpretation for the limit cases of integer order, namely for the derivatives of order one and zero.