论文标题
在Riemann-Liouville Sense中具有一般分数衍生物的分数微分方程
Fractional Differential Equations with the General Fractional Derivatives of Arbitrary Order in the Riemann-Liouville Sense
论文作者
论文摘要
在本文中，我们首先要处理Riemann-Liouville Sense中定义的任意顺序的一般分数衍生物。特别是，我们推断出其空空间的明确形式，并证明了分数演算的第二个基本定理，该定理导致其投影仪运营商的封闭式公式。这些结果使我们能够在Riemann-Liouville Sense中使用任意顺序的一般分数衍生物制定分数微分方程的自然初始条件。在本文的第二部分中，我们为Riemann-Liouville Sense中的一般任意顺序的一般分数衍生物开发了Mikusiński类型的操作计算，并将其应用于与这些派生词的单个线性线性差异方程的显式求解形式。解决方案的形式是由相应一般分数积分的内核产生的卷积序列的形式。
In this paper, we first deal with the general fractional derivatives of arbitrary order defined in the Riemann-Liouville sense. In particular, we deduce an explicit form of their null space and prove the second fundamental theorem of Fractional Calculus that leads to a closed form formula for their projector operator. These results allow us to formulate the natural initial conditions for the fractional differential equations with the general fractional derivatives of arbitrary order in the Riemann-Liouville sense. In the second part of the paper, we develop an operational calculus of the Mikusiński type for the general fractional derivatives of arbitrary order in the Riemann-Liouville sense and apply it for derivation of an explicit form of solutions to the Cauchy problems for the single- and multi-term linear fractional differential equations with these derivatives. The solutions are provided in form of the convolution series generated by the kernels of the corresponding general fractional integrals.
