【作者】 王小东；

【导师】 魏毅强；

【作者基本信息】 太原理工大学 ， 应用数学， 2008， 硕士

【摘要】 作为微积分理论的发展,分数阶微积分的概念人们早已提出,实际应用的介入又为它注入了新的生机。但由于分数阶微积分的定义有多种不同形式,各呈现了不同的运算关系,给应用带来了困难,特别是在非数学领域研究中,运算关系比较混乱,为此本文主要针对Riemann—Liouville分数阶微积分的定义展开讨论。本文的主要工作:第一,针对R-L分数阶微积分的定义,探讨了分数阶微积分中的一些性质,整理并证明了分数阶微分、分数阶积分以及整数阶微积分运算之间的线性性、交换性与结合性,澄清了运算顺序交换的条件与关系。使分数阶微积分与整数阶微积分和谐、统一,进而使实数阶微积分的理论进一步系统化,为工程实际应用提供了依据,便于应用。第二,研究了积分函数D_a^-af（t）连续性问题。给出了积分函数D_a^-af（t）连续的充分条件:如果函数f（t）∈C（[a,b]）且满足H（?）lder条件,则D_a^-af（t）连续。以及D_a^-af（t）绝对连续的充分条件:如果函数f（t）∈L_a^a（[a,b]）,则D_a^-af（t）是在区间[a,b]上绝对连续的函数。第三,研究了函数的分数阶可微性问题,给出了函数分数阶可微的充分条件:如果函数f（t）在区间[a,b]上绝对连续,则f（t）在区间[a,b]上几乎处处a（0＜a≤1）阶可微。更多还原

【Abstract】 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].更多还原