Riesz potential operators and inverses via fractional centred derivatives

By Manuel Duarte Ortigueira. International Journal of Mathematics and Mathematical Sciences, 2006

319 citations recorded by the source corpus.

Fractional centred differences and derivatives definitions are proposed, generalizing to real orders the existing ones valid for even and odd positive integer orders. For each one, suitable integral formulations are obtained. The computations of the involved integrals lead to new generalizations of the Cauchy integral derivative. To compute this integral, a special two-straight-line path was used. With this the referred integrals lead to the well-known Riesz potential operators and their inverses that emerge as true fractional centred derivatives, but that can be computed through summations similar to the well-known Grunwald-Letnikov derivatives.

Subfields: Applied Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/f99d50cc31d306f7c4f55b0219c83f67dc1f7441

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In Applied Mathematics

In Modeling and Simulation

Citations in the corpus

A class of second order difference approximations for solving space fractional diffusion equations (2012, methodology): “Ortigueira [17] gave the “fractional centred derivative” to approximate t h Riesz fractional derivative with second order accuracy, and this method was used by Çelik and Duman in [2] to approximate fractional diffusion equation with the Riesz fractional derivative in a fin ite domain.”
Matrix approach to discrete fractional calculus II: Partial fractional differential equations (2008, methodology): “This leads to the matrix R(β)m = h−α 2 [ −1Um + +1Um ] (28) In the second case (Ortigueira’s definition [48]) we have the following symmetric matix: R(β)m = h −β ω (β) 0 ω (β) 1 ω (β) 2 ω (β) 3 · · · ω(β)m ω (β) 1 ω (β) 0 ω (β) 1 ω (β) 2 · · · ω(β)m−1 ω (β) 2 ω (β) 1 ω (β) 0 ω (β) 1 · · · ω(β)m−2 ”
High-Order Algorithms for Riesz Derivative and Their Applications $(I)$ (2014, background): “From [27], we know that the generating function with coefficients g(α) k satisfies 󵄨󵄨󵄨󵄨󵄨󵄨 2 sin( 2 ) 󵄨󵄨󵄨󵄨󵄨󵄨 α = ∞ ∑”