论文标题
Riemann-Liouville和Caputo分数衍生物及其应用的分数Leibniz积分规则
Fractional Leibniz integral rules for Riemann-Liouville and Caputo fractional derivatives and their applications
论文作者
论文摘要
近年来，莱布尼兹（Leibniz）在分数意义上的积分规则的理论无法获得实质性发展。作为要解决的紧迫问题，我们研究了Riemann-Liouville和Caputo类型分化运算符的Leibniz积分规则，其中一般分数阶，为$ n-1 <α\ leq n $，$ n \ in \ mathbb {n} $。与一般顺序的积分符号下的分数分化规则是必要且适用的工具，用于替代候选候选解决方案，以验证不均匀的多项分数微分方程。我们根据最近定义的双变量Mittag-Leffler类型函数来得出广义Bagley-Torvik方程的显式分析解决方案，这些函数基于分数Green的功能方法，并通过应用分数Leibniz积分规则来替换解决方案。此外，我们将振荡器方程式作为通过Leibniz积分规则进行多端口的微分方程的特殊情况。
In recent years, the theory for Leibniz integral rule in the fractional sense has not been able to get substantial development. As an urgent problem to be solved, we study a Leibniz integral rule for Riemann-Liouville and Caputo type differentiation operators with general fractional-order of $n-1 <α\leq n$, $n \in \mathbb{N}$ . A rule of fractional differentiation under integral sign with general order is necessary and applicable tool for verification by substitution for candidate solutions of inhomogeneous multi-term fractional differential equations. We derive explicit analytical solutions of generalized Bagley-Torvik equations in terms of recently defined bivariate Mittag-Leffler type functions that based on fractional Green's function method and verified solutions by substitution in accordance by applying the fractional Leibniz integral rule. Furthermore, we study an oscillator equation as a special case of differential equations with multi-orders via the Leibniz integral rule.
