Editing Fractional calculus (section)

==Applications==

===Fractional conservation of mass===
As described by Wheatcraft and Meerschaert (2008),<ref>{{cite journal |last1=Wheatcraft |first1=Stephen W. |last2=Meerschaert |first2=Mark M. |date=October 2008 |title=Fractional conservation of mass |url=https://www.stt.msu.edu/users/mcubed/fCOM.pdf |journal=Advances in Water Resources |language=en |volume=31 |issue=10 |pages=1377–1381 |doi=10.1016/j.advwatres.2008.07.004 |issn=0309-1708 |bibcode=2008AdWR...31.1377W}}</ref> a fractional conservation of mass equation is needed to model fluid flow when the [[control volume]] is not large enough compared to the scale of [[heterogeneity]] and when the flux within the control volume is non-linear. In the referenced paper, the fractional conservation of mass equation for fluid flow is:
<math display="block">-\rho \left(\nabla^\alpha \cdot \vec{u} \right) = \Gamma(\alpha +1)\Delta x^{1-\alpha} \rho \left (\beta_s+\phi \beta_w \right ) \frac{\partial p}{\partial t} </math>

===Electrochemical analysis===
{{See also|Neopolarogram}}
When studying the redox behavior of a substrate in solution, a voltage is applied at an electrode surface to force electron transfer between electrode and substrate. The resulting electron transfer is measured as a current. The current depends upon the concentration of substrate at the electrode surface. As substrate is consumed, fresh substrate diffuses to the electrode as described by [[Fick's laws of diffusion]]. Taking the Laplace transform of Fick's second law yields an ordinary second-order differential equation (here in dimensionless form):
<math display="block">\frac{d^2}{d x^2} C(x,s) = sC(x,s) </math>

whose solution {{math|''C''(''x'',''s'')}} contains a one-half power dependence on {{mvar|s}}. Taking the derivative of {{math|''C''(''x'',''s'')}} and then the inverse Laplace transform yields the following relationship:
<math display="block">\frac{d}{d x} C(x,t) = \frac{d^{\scriptstyle{\frac{1}{2}}}}{d t^{\scriptstyle{\frac{1}{2}}}}C(x,t) </math>

which relates the concentration of substrate at the electrode surface to the current.<ref>Oldham, K. B. ''Analytical Chemistry'' 44(1) '''1972''' 196-198.</ref> This relationship is applied in electrochemical kinetics to elucidate mechanistic behavior. For example, it has been used to study the rate of dimerization of substrates upon electrochemical reduction.<ref>Pospíšil, L. et al. ''Electrochimica Acta'' 300 '''2019''' 284-289.</ref>

===Groundwater flow problem===
In 2013–2014 Atangana et al. described some groundwater flow problems using the concept of a derivative with fractional order.<ref>{{cite journal
|last1=Atangana |first1=Abdon
|last2=Bildik |first2=Necdet
|title=The Use of Fractional Order Derivative to Predict the Groundwater Flow
|year=2013
|journal=Mathematical Problems in Engineering
|volume=2013
|pages=1–9
|doi=10.1155/2013/543026
|doi-access=free
}}</ref><ref>{{cite journal
|last1=Atangana |first1=Abdon
|last2=Vermeulen |first2=P. D.
|title=Analytical Solutions of a Space-Time Fractional Derivative of Groundwater Flow Equation
|year=2014
|journal=Abstract and Applied Analysis
|volume=2014
|pages=1–11
|doi=10.1155/2014/381753
|doi-access=free
}}</ref> In these works, the classical [[Darcy law]] is generalized by regarding the water flow as a function of a non-integer order derivative of the piezometric head. This generalized law and the law of conservation of mass are then used to derive a new equation for groundwater flow.

===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>

===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=1–77 |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=153–192 |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|α}} and {{mvar|β}} are changed into {{math|''α''(''x'', ''t'')}} and {{math|''β''(''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=148–166 |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=2421–2429 |arxiv=1610.06590 |doi=10.1093/mnras/stx261 |doi-access=free |bibcode=2017MNRAS.467.2421C |s2cid=20203131}}</ref>

===Structural damping models===
Fractional derivatives are used to model [[viscoelastic]] [[damping]] in certain types of materials like polymers.<ref name=Mainardi>{{cite book |title=Fractional Calculus and Waves in Linear Viscoelasticity |last=Mainardi |first=Francesco |s2cid=118719247 |date=May 2010 |publisher=[[Imperial College Press]] |isbn=978-1-84816-329-4 |language=en |doi=10.1142/p614}}</ref>

===PID controllers===
Generalizing [[PID controller]]s to use fractional orders can increase their degree of freedom. The new equation relating the ''control variable'' {{math|''u''(''t'')}} in terms of a measured ''error value'' {{math|''e''(''t'')}} can be written as
<math display="block">u(t) = K_\mathrm{p} e(t) + K_\mathrm{i} D_t^{-\alpha} e(t) + K_\mathrm{d} D_t^{\beta} e(t)</math>

where {{mvar|α}} and {{math|β}} are positive fractional orders and {{math|''K''<sub>p</sub>}}, {{math|''K''<sub>i</sub>}}, and {{math|''K''<sub>d</sub>}}, all non-negative, denote the coefficients for the [[proportional control|proportional]], [[integral]], and [[derivative]] terms, respectively (sometimes denoted {{mvar|P}}, {{mvar|I}}, and {{mvar|D}}).<ref>{{cite journal |last1=Tenreiro Machado |first1=J. A. |last2=Silva |first2=Manuel F. |last3=Barbosa |first3=Ramiro S. |last4=Jesus |first4=Isabel S. |last5=Reis |first5=Cecília M. |last6=Marcos |first6=Maria G. |last7=Galhano |first7=Alexandra F. |date=2010 |title=Some Applications of Fractional Calculus in Engineering |journal=[[Mathematical Problems in Engineering]] |language=en |volume=2010 |pages=1–34 |doi=10.1155/2010/639801 |doi-access=free|hdl=10400.22/13143 |hdl-access=free }}</ref>

===Acoustic wave equations for complex media===
The propagation of acoustical waves in complex media, such as in biological tissue, commonly implies attenuation obeying a frequency power-law. This kind of phenomenon may be described using a causal wave equation which incorporates fractional time derivatives:
<math display="block">\nabla^2 u -\dfrac 1{c_0^2} \frac{\partial^2 u}{\partial t^2} + \tau_\sigma^\alpha \dfrac{\partial^\alpha}{\partial t^\alpha}\nabla^2 u - \dfrac {\tau_\epsilon^\beta}{c_0^2} \dfrac{\partial^{\beta+2} u}{\partial t^{\beta+2}} = 0\,.</math>

See also Holm & Näsholm (2011)<ref>{{cite journal |last1=Holm |first1=S. |last2=Näsholm |first2=S. P. |s2cid=7804006 |year=2011 |title=A causal and fractional all-frequency wave equation for lossy media |journal=Journal of the Acoustical Society of America |volume=130 |issue=4 |pages=2195–2201 |bibcode=2011ASAJ..130.2195H |doi=10.1121/1.3631626 |pmid=21973374|hdl=10852/103311 |hdl-access=free }}</ref> and the references therein. Such models are linked to the commonly recognized hypothesis that multiple relaxation phenomena give rise to the attenuation measured in complex media. This link is further described in Näsholm & Holm (2011b)<ref>{{cite journal |last1=Näsholm |first1=S. P. |last2=Holm |first2=S. |s2cid=10376751 |year=2011 |title=Linking multiple relaxation, power-law attenuation, and fractional wave equations |journal=Journal of the Acoustical Society of America |volume=130 |issue=5 |pages=3038–3045 |bibcode=2011ASAJ..130.3038N |doi=10.1121/1.3641457 |pmid=22087931|hdl=10852/103312 |hdl-access=free }}</ref> and in the survey paper,<ref name=Nasholm2>{{cite journal |last1=Näsholm |first1=S. P. |last2=Holm |first2=S. |year=2012 |title=On a Fractional Zener Elastic Wave Equation |journal=Fract. Calc. Appl. Anal. |volume=16 |pages=26–50 |arxiv=1212.4024 |doi=10.2478/s13540-013-0003-1 |s2cid=120348311}}</ref> as well as the ''[[Acoustic attenuation]]'' article. See Holm & Nasholm (2013)<ref name=HolmNasholm2014>{{cite journal |last1=Holm |first1=S. |last2=Näsholm |first2=S. P. |year=2013 |title=Comparison of fractional wave equations for power law attenuation in ultrasound and elastography |journal=Ultrasound in Medicine & Biology |volume=40 |issue=4 |pages=695–703 |arxiv=1306.6507 |citeseerx=10.1.1.765.120 |doi=10.1016/j.ultrasmedbio.2013.09.033 |pmid=24433745  |s2cid=11983716}}</ref> for a paper which compares fractional wave equations which model power-law attenuation. This book on power-law attenuation also covers the topic in more detail.<ref name=Holm2019>{{cite book |last=Holm |first=S. |url=https://link.springer.com/book/10.1007/978-3-030-14927-7 |title=Waves with Power-Law Attenuation |publisher=Springer and Acoustical Society of America Press |year=2019 |doi=10.1007/978-3-030-14927-7 |bibcode=2019wpla.book.....H |isbn=978-3-030-14926-0|s2cid=145880744 }}</ref>

Pandey and Holm gave a physical meaning to fractional differential equations by deriving them from physical principles and interpreting the fractional-order in terms of the parameters of the acoustical media, example in fluid-saturated granular unconsolidated marine sediments.<ref name=Pandey2016>{{cite journal |last1=Pandey |first1=Vikash |last2=Holm |first2=Sverre |date=2016-12-01 |title=Connecting the grain-shearing mechanism of wave propagation in marine sediments to fractional order wave equations |journal=The Journal of the Acoustical Society of America |volume=140 |issue=6 |pages=4225–4236 |doi=10.1121/1.4971289 |pmid=28039990 |issn=0001-4966 |arxiv=1612.05557 |bibcode=2016ASAJ..140.4225P |s2cid=29552742}}</ref> Interestingly, Pandey and Holm derived [[Cinna Lomnitz|Lomnitz's law]] in [[seismology]] and Nutting's law in [[non-Newtonian fluid|non-Newtonian rheology]] using the framework of fractional calculus.<ref>{{cite journal |last1=Pandey |first1=Vikash |last2=Holm |first2=Sverre |date=2016-09-23 |title=Linking the fractional derivative and the Lomnitz creep law to non-Newtonian time-varying viscosity |journal=Physical Review E |volume=94 |issue=3 |pages=032606 |doi=10.1103/PhysRevE.94.032606 |pmid=27739858 |bibcode=2016PhRvE..94c2606P |doi-access=free|hdl=10852/53091 |hdl-access=free }}</ref> Nutting's law was used to model the wave propagation in marine sediments using fractional derivatives.<ref name=Pandey2016/>

===Fractional Schrödinger equation in quantum theory===
The [[fractional Schrödinger equation]], a fundamental equation of [[fractional quantum mechanics]], has the following form:<ref>{{cite journal |last=Laskin |first=N. |year=2002 |title=Fractional Schrodinger equation |journal=Phys. Rev. E |volume=66 |issue=5 |pages=056108 |arxiv=quant-ph/0206098 |citeseerx=10.1.1.252.6732 |doi=10.1103/PhysRevE.66.056108 |pmid=12513557 |bibcode=2002PhRvE..66e6108L |s2cid=7520956}}</ref><ref>{{cite book |doi=10.1142/10541 |title=Fractional Quantum Mechanics |year=2018 |last1=Laskin |first1=Nick |isbn=978-981-322-379-0 |citeseerx=10.1.1.247.5449}}</ref>
<math display="block">i\hbar \frac{\partial \psi (\mathbf{r},t)}{\partial t}=D_{\alpha } \left(-\hbar^2\Delta \right)^{\frac{\alpha}{2}}\psi (\mathbf{r},t)+V(\mathbf{r},t)\psi (\mathbf{r},t)\,.</math>

where the solution of the equation is the [[wavefunction]] {{math|''ψ''('''r''', ''t'')}} – the quantum mechanical [[probability amplitude]] for the particle to have a given [[position vector]] {{math|'''r'''}} at any given time {{mvar|t}}, and {{mvar|ħ}} is the [[reduced Planck constant]]. The [[potential energy]] function {{math|''V''('''r''', ''t'')}} depends on the system.

Further, <math display="inline">\Delta = \frac{\partial^2}{\partial\mathbf{r}^2}</math> is the [[Laplace operator]], and {{mvar|D<sub>α</sub>}} is a scale constant with physical [[dimensional analysis|dimension]] {{math|1=[''D<sub>α</sub>''] = J<sup>1 − ''α''</sup>·m<sup>''α''</sup>·s<sup>−''α''</sup> = kg<sup>1 − ''α''</sup>·m<sup>2 − ''α''</sup>·s<sup>''α'' − 2</sup>}}, (at {{math|1=''α'' = 2}}, <math display="inline">D_2 = \frac{1}{2m}</math> for a particle of mass {{mvar|m}}), and the operator {{math|(−''ħ''<sup>2</sup>Δ)<sup>''α''/2</sup>}} is the 3-dimensional fractional quantum Riesz derivative defined by
<math display="block">(-\hbar^2\Delta)^\frac{\alpha}{2}\psi (\mathbf{r},t) = \frac 1 {(2\pi \hbar)^3} \int d^3 p e^{\frac{i}{\hbar} \mathbf{p}\cdot\mathbf{r}}|\mathbf{p}|^\alpha \varphi (\mathbf{p},t) \,.</math>

The index {{mvar|α}} in the fractional Schrödinger equation is the Lévy index, {{math|1 < ''α'' ≤ 2}}.

====Variable-order fractional Schrödinger equation====
As a natural generalization of the [[fractional Schrödinger equation]], the variable-order fractional Schrödinger equation has been exploited to study fractional quantum phenomena:<ref>{{cite journal |last1=Bhrawy |first1=A.H. |last2=Zaky |first2=M.A. |year=2017 |title=An improved collocation method for multi-dimensional space–time variable-order fractional Schrödinger equations |journal=Applied Numerical Mathematics |volume=111 |pages=197–218 |doi=10.1016/j.apnum.2016.09.009}}</ref>
<math display="block">i\hbar \frac{\partial \psi^{\alpha(\mathbf{r})} (\mathbf{r},t)}{\partial t^{\alpha(\mathbf{r})} } = \left(-\hbar^2\Delta \right)^{\frac{\beta(t)}{2}}\psi (\mathbf{r},t)+V(\mathbf{r},t)\psi (\mathbf{r},t),</math>

where <math display="inline">\Delta = \frac{\partial^2}{\partial\mathbf{r}^2}</math> is the [[Laplace operator]] and the operator {{math|(−''ħ''<sup>2</sup>Δ)<sup>''β''(''t'')/2</sup>}} is the variable-order fractional quantum Riesz derivative.