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

摘要

In this paper,a generalized multivariate fractional Taylor 's and Cauchy's mean value theorem of the kind f(x,y)=n∑j=0 Djαf(x0,y0)Γ(jα+1)+Rαn(ξ,η),f(x,y)? n∑j=0 Djαf(x0,y0)Γ(jα+1)g(x,y)? n∑j=0 Djαg(x0,y0)Γ(jα+1)=Rαn(ξ,η)Tαn(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 caseα=1.In addition,detailed expres-sions for Rαn(ξ,η)and Tαn(ξ,η)involving the sequential Caputo fractional derivative are also given.