Riemann-Liouville Fractional Calculus And The Proving Of The Properties

With the development of the theorem of differential and integral calculus, the fractional-order differential and integral calculus concept has been put forward for a long time,and its practical application makes it vigorous.But because there exists many different concepts on the fractional-order differential and integral,there are not clear about the operational relations,and it's difficult to practical application,especially,the operational relations are disorder in the study of the non-mathematics field,so this present paper is mainly discussed basis on the Riemann-Liouville fractional-order differential and integral calculus concept.The main research work is described in detail in the following:Firstly,mainly discussed some properties of fractional calculus and also we cleared up and prove the linearity,combine and interchange relations between the fractional differentiations,integrals and some formula operators of the integer-order calculus based on the basis of predecessors against the definitions of the Riemann-Liouville fractional-order calculus,Make it clear in proper order to interchange condition and relation,so that the real numerical-order fractional calculus and the fractional calculus harmony and unity,thereby enabled the theory of the real numerical-order fractional calculus systematic and facilitated the practical problems in the application.Secondly,the continuity question of the R-L fractional integral of the real function D_a^-af(t)was studied.And the sufficient condition of the continuity of the integral function D_a^-af(t)was given:If f(t)∈C([a,b])and f(t)verify the H(?)lder condition,then D_a^-af(t)∈C([a,b]).And absolutely condition of the continuity of the integral function D_a^-af(t)was given:If f(t)∈L_a^-a([a,b]),then D_a^-af(t)is absolute continue on[a,b].Thirdly,we study the question that the functions' fractional differentiability and the sufficient condition of the fractional differential were given:If f(t)is absolute continue on interval[a,b],then f(t)is almost a-differentiable (0＜a≤1)on[a,b].