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Fractional calculus
(section)
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===Time-space fractional diffusion equation models=== Anomalous diffusion processes in complex media can be well characterized by using fractional-order diffusion equation models.<ref>{{cite journal |last1=Metzler |first1=R. |last2=Klafter |first2=J. |year=2000 |title=The random walk's guide to anomalous diffusion: a fractional dynamics approach |journal=Phys. Rep. |volume=339 |issue=1 |pages=1β77 |doi=10.1016/s0370-1573(00)00070-3 |bibcode=2000PhR...339....1M}}</ref><ref>{{cite journal |last1=Mainardi |first1=F. |author-link2=Yuri Luchko |last2=Luchko |first2=Y. |last3=Pagnini |first3=G. |year=2001 |title=The fundamental solution of the space-time fractional diffusion equation |arxiv=cond-mat/0702419 |journal=Fractional Calculus and Applied Analysis |volume=4 |issue=2 |pages=153β192 |bibcode=2007cond.mat..2419M}}</ref> The time derivative term corresponds to long-time heavy tail decay and the spatial derivative for diffusion nonlocality. The time-space fractional diffusion governing equation can be written as <math display="block"> \frac{\partial^\alpha u}{\partial t^\alpha}=-K (-\Delta)^\beta u.</math> A simple extension of the fractional derivative is the variable-order fractional derivative, {{mvar|Ξ±}} and {{mvar|Ξ²}} are changed into {{math|''Ξ±''(''x'', ''t'')}} and {{math|''Ξ²''(''x'', ''t'')}}. Its applications in anomalous diffusion modeling can be found in the reference.<ref name=Atangana2014a/><ref>{{cite book |last1=Gorenflo |first1=Rudolf |last2=Mainardi |first2=Francesco |title=Processes with Long-Range Correlations |date=2007 |editor-last=Rangarajan |editor-first=G. |series=Lecture Notes in Physics |volume=621 |pages=148β166 |chapter=Fractional Diffusion Processes: Probability Distributions and Continuous Time Random Walk |doi=10.1007/3-540-44832-2_8 |arxiv=0709.3990 |editor-last2=Ding |editor-first2=M. |bibcode=2003LNP...621..148G |isbn=978-3-540-40129-2 |s2cid=14946568}}</ref><ref>{{cite journal |last1=Colbrook |first1=Matthew J. |last2=Ma |first2=Xiangcheng |last3=Hopkins |first3=Philip F. |last4=Squire |first4=Jonathan |year=2017 |title=Scaling laws of passive-scalar diffusion in the interstellar medium |journal=[[Monthly Notices of the Royal Astronomical Society]] |volume=467 |issue=2 |pages=2421β2429 |arxiv=1610.06590 |doi=10.1093/mnras/stx261 |doi-access=free |bibcode=2017MNRAS.467.2421C |s2cid=20203131}}</ref>
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