Editing Fractional calculus (section)

=== Conformable fractional derivative ===
The conformable fractional derivative of a function <math>f</math> of order <math>\alpha</math> is given by<math display="block"> T_a(f)(t) = \lim_{\epsilon \rightarrow 0}\frac{f\left(t+\epsilon t^{1-\alpha}\right) - f(t)}{\epsilon} </math>Unlike other definitions of the fractional derivative, the conformable fractional derivative obeys the [[Product rule|product]] and [[quotient rule]] has analogs to [[Rolle's theorem]] and the [[mean value theorem]].<ref>{{Cite journal |last1=Khalil |first1=R. |last2=Al Horani |first2=M. |last3=Yousef |first3=A. |last4=Sababheh |first4=M. |date=2014-07-01 |title=A new definition of fractional derivative |url=https://www.sciencedirect.com/science/article/pii/S0377042714000065 |journal=Journal of Computational and Applied Mathematics |volume=264 |pages=65–70 |doi=10.1016/j.cam.2014.01.002 |issn=0377-0427|doi-access=free }}</ref><ref name=":0">{{Cite journal |last1=Gao |first1=Feng |last2=Chi |first2=Chunmei |date=2020 |title=Improvement on Conformable Fractional Derivative and Its Applications in Fractional Differential Equations |journal=Journal of Function Spaces |language=en |volume=2020 |issue=1 |pages=5852414 |doi=10.1155/2020/5852414 |doi-access=free |issn=2314-8888}}</ref> However, this fractional derivative produces significantly different results compared to the Riemann-Liouville and Caputo fractional derivative. In 2020, Feng Gao and Chunmei Chi defined the improved Caputo-type conformable fractional derivative, which more closely approximates the behavior of the Caputo fractional derivative:<ref name=":0" /><math display="block"> ^C_a\widetilde{T}_a(f)(t) = \lim_{\epsilon \rightarrow 0}\left[(1-\alpha)(f(t)-f(a))+\alpha\frac{f\left(t+\epsilon (t-a)^{1-\alpha}\right) - f(t)}{\epsilon}\right] </math>where <math>a</math> and <math>t</math> are [[real numbers]] and <math>a<t</math>. They also defined the improved Riemann-Liouville-type conformable fractional derivative to similarly approximate the Riemann-Liouville fractional derivative:<ref name=":0" />

<math display="block"> ^{RL}_a\widetilde{T}_a(f)(t) = \lim_{\epsilon \rightarrow 0}\left[(1-\alpha)f(t)+\alpha\frac{f\left(t+\epsilon (t-a)^{1-\alpha}\right) - f(t)}{\epsilon}\right] </math>where <math>a</math> and <math>t</math> are [[real numbers]] and <math>a<t</math>. Both improved conformable fractional derivatives have analogs to Rolle's theorem and the [[interior extremum theorem]].<ref>{{Cite journal |last=Hasanah |first=Dahliatul |date=2022-10-31 |title=On continuity properties of the improved conformable fractional derivatives |url=http://mail.fourier.or.id/index.php/FOURIER/article/view/176 |journal=Jurnal Fourier |language=en |volume=11 |issue=2 |pages=88–96 |doi=10.14421/fourier.2022.112.88-96 |issn=2541-5239|doi-access=free }}</ref>