Editing Fractional calculus (section)

===Fractional advection dispersion equation===
This equation{{clarify|date=January 2017}} has been shown useful for modeling contaminant flow in heterogenous porous media.<ref>{{cite journal |last1=Benson |first1=D. |last2=Wheatcraft |first2=S. |last3=Meerschaert |first3=M. |year=2000 |title=Application of a fractional advection-dispersion equation |journal=Water Resources Research |volume=36 |issue=6 |pages=1403–1412 |bibcode=2000WRR....36.1403B |citeseerx=10.1.1.1.4838 |doi=10.1029/2000wr900031|s2cid=7669161 }}</ref><ref>{{cite journal |last1=Benson |first1=D. |last2=Wheatcraft |first2=S. |last3=Meerschaert |first3=M. |s2cid=16579630 |year=2000 |title=The fractional-order governing equation of Lévy motion |journal= Water Resources Research |volume=36 |issue=6 |pages=1413–1423 |bibcode=2000WRR....36.1413B |doi=10.1029/2000wr900032 |doi-access=free}}</ref><ref>{{cite journal |last1=Wheatcraft |first1=Stephen W. |last2=Meerschaert |first2=Mark M. |last3=Schumer |first3=Rina |last4=Benson |first4=David A. |date=2001-01-01 |title=Fractional Dispersion, Lévy Motion, and the MADE Tracer Tests |journal=[[Transport in Porous Media]] |language=en |volume=42 |issue=1–2 |pages=211–240 |citeseerx=10.1.1.58.2062 |doi=10.1023/A:1006733002131 |bibcode=2001TPMed..42..211B |s2cid=189899853 |issn=1573-1634}}</ref>

Atangana and Kilicman extended the fractional advection dispersion equation to a variable order equation. In their work, the hydrodynamic dispersion equation was generalized using the concept of a [[variational order derivative]]. The modified equation was numerically solved via the [[Crank–Nicolson method]]. The stability and convergence in numerical simulations showed that the modified equation is more reliable in predicting the movement of pollution in deformable aquifers than equations with constant fractional and integer derivatives<ref name=Atangana2014a>{{cite journal
|last1=Atangana |first1=Abdon
|last2=Kilicman |first2=Adem
|title=On the Generalized Mass Transport Equation to the Concept of Variable Fractional Derivative
|journal=Mathematical Problems in Engineering
|volume=2014
|year=2014
|page=9
|doi=10.1155/2014/542809
|doi-access=free
}}</ref>