论文标题
Dirac Delta函数的分数衍生物和非理性函数逆向拉普拉斯变换的新结果
The fractional derivative of the Dirac delta function and new results on the inverse Laplace transform of irrational functions
论文作者
论文摘要
从关于异常扩散的研究中，我们表明，内存功能$ m（t）复杂材料的蠕变合规性遵循了权力法，$ j（t）\ sim t^q $带有$ q \ in \ mathbb {r}^+$，是dirac delta功能的分数， $ \ frac {\ mathrm {d}^qδ（t-0）} {\ mathrm {d} t^q} $ at \ in \ mathbb {r}^+$。这导致发现，任何$ s^q $的反向转换\ in \ mathbb {r}^+$是dirac delta函数的分数衍生物，$ \ frac {\ frac {\ mathrm {d}^qΔ（t-nqδ（t-0）}}该结果与卷积定理相关，使$ \ frac {s^q} {s^α\mpλ} $的逆拉动转换的计算是可能$ \ varepsilon_ {α-1}（\pmλ，t）= t^{α-1} e_ {α，α}（\pmλt^α）$。 Rabotnov功能的$ Q \ $ q \的分数导数，$ \ VAREPSILON_ {α-1}（α-1}（\pmλ，t）$产生的奇异性会产生奇异性，这些奇异性是由有限数量的dirac delta函数的有限分数衍生物提取的，依赖于dirac delta功能$ $ $ $ $ q的强度，依赖于$ q Mittag-Leffler功能。
Motivated from studies on anomalous diffusion, we show that the memory function $M(t)$ of complex materials, that their creep compliance follows a power law, $J(t)\sim t^q$ with $q\in \mathbb{R}^+$, is the fractional derivative of the Dirac delta function, $\frac{\mathrm{d}^qδ(t-0)}{\mathrm{d}t^q}$ with $q\in \mathbb{R}^+$. This leads to the finding that the inverse Laplace transform of $s^q$ for any $q\in \mathbb{R}^+$ is the fractional derivative of the Dirac delta function, $\frac{\mathrm{d}^qδ(t-0)}{\mathrm{d}t^q}$. This result, in association with the convolution theorem, makes possible the calculation of the inverse Laplace transform of $\frac{s^q}{s^α\mpλ}$ where $α<q\in\mathbb{R}^+$ which is the fractional derivative of order $q$ of the Rabotnov function $\varepsilon_{α-1}(\pmλ, t)=t^{α-1}E_{α, α}(\pmλt^α)$. The fractional derivative of order $q\in \mathbb{R}^+$ of the Rabotnov function, $\varepsilon_{α-1}(\pmλ, t)$ produces singularities which are extracted with a finite number of fractional derivatives of the Dirac delta function depending on the strength of $q$ in association with the recurrence formula of the two-parameter Mittag-Leffler function.
