On Multivariate Fractional Taylor’s and Cauchy’ Mean Value Theorem

Abstract

In this paper, a generalized multivariate fractional Taylor’s and Cauchy’s mean value theorem of the kind f (x,y)= n ∑ j=0 Djα f (x0,y0) Γ(jα+1) +Rn(ξ,η), f (x,y)− n ∑ j=0 Djα f (x0,y0) Γ(jα+1) g(x,y)− n ∑ j=0 Dg(x0,y0) Γ(jα+1) = Rn(ξ,η) Tα n (ξ,η) , where 0< α≤ 1, is established. Such expression is precisely the classical Taylor’s and Cauchy’s mean value theorem in the particular case α=1. In addition, detailed expressions for Rn(ξ,η) and Tα n (ξ,η) involving the sequential Caputo fractional derivative are also given. AMS subject classifications: 65M70, 65L60, 41A10, 60H35