论文标题
在分数延迟微分方程的初始条件下
On initial conditions for fractional delay differential equations
论文作者
论文摘要
分数顺序的衍生物以不同的方式引入：作为分数积分的剩余或通过概括定义整数导数的差异商的极限。尽管这两种方法（在标准平滑度假设下）引导到等效运算符，但第一个方法不涉及在初始点左侧的函数，而后者则迫使后者迫使函数假设所选值。使用分数延迟微分方程新问题出现：延迟的存在不仅在初始点，而且在整个间隔内分配解决方案。由于选择初始函数的自由，因此有可能与分数衍生物强制的值不一致，并且运算符可能不再是等效的。在本文中，我们讨论了分数延迟差分方程的初始化，我们还研究了初始条件不仅对溶液的影响，而且还研究了分数操作员的效果，我们研究了通过在记忆分数衍生物中纳入或不是初始功能获得的解决方案之间的差异。线性方程家族的精确解是通过拉普拉斯变换获得的，而数值方法则用于解决非线性问题。因此，显示和评论不同的结果。
Derivatives of fractional order are introduced in different ways: as left-inverse of the fractional integral or by generalizing the limit of the difference quotient defining integer-order derivatives. Although the two approaches lead (under standard smoothness assumptions) to equivalent operators, the first one does not involve the function at the left of the initial point where, instead, the latter forces the function to assume selected values. With fractional delay differential equations new problems arise: the presence of the delay imposes to assign the solution not just at the initial point but on an entire interval. Due to the freedom in the choice of the initial function, some inconsistencies with the values forced by the fractional derivative are possible and the operators may no longer be equivalent. In this paper we discuss the initialization of fractional delay differential equations, we investigate the effects of the initial condition not only on the solution but also on the fractional operator as well and we study the difference between solutions obtained by incorporating or not the initial function in the memory of the fractional derivative. The exact solution of a family of linear equations is obtained by the Laplace transform whilst numerical methods are used to solve nonlinear problems; the different results are therefore shown and commented.
