Editing Fractional calculus (section)

===Atangana–Baleanu fractional derivative===
In 2016, Atangana and Baleanu suggested differential operators based on the generalized [[Mittag-Leffler function]] <math> E_{\alpha}</math>. The aim was to introduce fractional differential operators with non-singular nonlocal kernel. Their fractional differential operators are given below in Riemann–Liouville sense and Caputo sense respectively. For a function <math>f(t)</math> of <math>C^1</math> given by <ref name=Algahtani2016/><ref name="doiserbia.nb.rs">{{cite journal |last1=Atangana |first1=Abdon |last2=Baleanu |first2=Dumitru |date=2016 |title=New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model |url=http://www.doiserbia.nb.rs/Article.aspx?ID=0354-98361600018A |journal=Thermal Science |language=en |volume=20 |issue=2 |pages=763–769 |doi=10.2298/TSCI160111018A |arxiv=1602.03408 |issn=0354-9836 |doi-access=free}}</ref>
<math display="block">\sideset{_{\hphantom{AB}a}^{\text{ABC}}}{_t^\alpha}D f(t)=\frac{\operatorname{AB}(\alpha)}{1-\alpha} \int_a^t f'(\tau)E_{\alpha}\left(-\alpha\frac{(t-\tau)^{\alpha}}{1-\alpha}\right)d\tau,</math>

If the function is continuous, the Atangana–Baleanu derivative in Riemann–Liouville sense is given by:
<math display="block">\sideset{_{\hphantom{AB}a}^{\text{ABC}}}{_t^\alpha}D f(t)=\frac{\operatorname{AB}(\alpha)}{1-\alpha} \frac{d}{dt}\int_a^t f(\tau)E_{\alpha}\left(-\alpha\frac{(t-\tau)^{\alpha}}{1-\alpha}\right)d\tau,</math>

The kernel used in Atangana–Baleanu fractional derivative has some properties of a cumulative distribution function. For example, for all {{nowrap|<math>\alpha \in (0, 1]</math>,}} the function <math>E_\alpha</math> is increasing on the real line, converges to <math>0</math> in {{nowrap|<math>- \infty</math>,}} and {{nowrap|<math>E_\alpha (0) = 1</math>.}} Therefore, we have that, the function <math>x \mapsto 1-E_\alpha (-x^\alpha)</math> is the cumulative distribution function of a probability measure on the positive real numbers. The distribution is therefore defined, and any of its multiples is called a [[Mittag-Leffler distribution]] of order {{nowrap|<math>\alpha</math>.}} It is also very well-known that, all these probability distributions are [[absolute continuity|absolutely continuous]]. In particular, the function Mittag-Leffler has a particular case {{nowrap|<math>E_1</math>,}} which is the exponential function, the Mittag-Leffler distribution of order <math>1</math> is therefore an [[exponential distribution]]. However, for {{nowrap|<math>\alpha \in (0, 1)</math>,}} the Mittag-Leffler distributions are [[heavy-tailed distribution|heavy-tailed]]. Their Laplace transform is given by:
<math display="block">\mathbb{E} (e^{- \lambda X_\alpha}) = \frac{1}{1+\lambda^\alpha},</math>

This directly implies that, for {{nowrap|<math>\alpha \in (0, 1)</math>,}} the expectation is infinite. In addition, these distributions are [[geometric stable distribution]]s.