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Fractional Advection Dispersion Equation - Revision history
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imported>Nestor at 23:13, 18 November 2012
2012-11-18T23:13:45Z
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{{</del>See also<del style="font-weight: bold; text-decoration: none;">|</del>Drift-diffusion equations<del style="font-weight: bold; text-decoration: none;">}}</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== </ins>See also <ins style="font-weight: bold; text-decoration: none;">==</ins></div></td></tr>
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<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [[</ins>Drift-diffusion equations<ins style="font-weight: bold; text-decoration: none;">]]</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== References ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== References ==</div></td></tr>
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imported>Nestor
https://web.ma.utexas.edu/mediawiki/index.php?title=Fractional_Advection_Dispersion_Equation&diff=1269&oldid=prev
imported>Nestor at 22:56, 18 November 2012
2012-11-18T22:56:02Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 17:56, 18 November 2012</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\end{equation}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\end{equation}</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>It is important to point out that these fractional derivatives are examples of one-dimensional [[Linear integro-differential operator | linear integro-differential operators]] with non-symmetric kernels (in this case, e.g. $K(y)=\mathbb{1}_{\{y\geq 0\}}|y|^{-1-\alpha}$).</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>It is important to point out that these fractional derivatives are examples of one-dimensional [[Linear integro-differential operator | linear integro-differential operators]] with non-symmetric kernels (in this case, e.g. $K(y)=\mathbb{1}_{\{y\geq 0\}}|y|^{-1-\alpha}$).</div></td></tr>
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<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{See also|Drift-diffusion equations}}</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== References ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== References ==</div></td></tr>
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imported>Nestor
https://web.ma.utexas.edu/mediawiki/index.php?title=Fractional_Advection_Dispersion_Equation&diff=1268&oldid=prev
imported>Nestor at 22:49, 18 November 2012
2012-11-18T22:49:35Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 17:49, 18 November 2012</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The Fractional Advection Dispersion (fADE) was introduced <ref name="MBB1999"/> (and further developed and justified <ref name="BMW"/> <ref name="MBB2001"/> <ref name ="BMW2001"/>) to better account for super-diffusive spreading of tracer particles in aquifers. It has quite a few variations at this point, but a couple of characteristic examples are</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The Fractional Advection Dispersion (fADE) was introduced <ref name="MBB1999"/> (and further developed and justified <ref name="BMW"/> <ref name="MBB2001"/> <ref name ="BMW2001"/>) to better account for super-diffusive spreading of tracer particles in aquifers. It has quite a few variations at this point, but a couple of characteristic examples are</div></td></tr>
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imported>Nestor
https://web.ma.utexas.edu/mediawiki/index.php?title=Fractional_Advection_Dispersion_Equation&diff=1267&oldid=prev
imported>Russell: Created page with "The Fractional Advection Dispersion (fADE) was introduced <ref name="MBB1999"/> (and further developed and justified <ref name="BMW"/> <ref name="MBB2001"/> <ref name ="BMW2001"/..."
2012-11-17T17:56:30Z
<p>Created page with "The Fractional Advection Dispersion (fADE) was introduced <ref name="MBB1999"/> (and further developed and justified <ref name="BMW"/> <ref name="MBB2001"/> <ref name ="BMW2001"/..."</p>
<p><b>New page</b></p><div>The Fractional Advection Dispersion (fADE) was introduced <ref name="MBB1999"/> (and further developed and justified <ref name="BMW"/> <ref name="MBB2001"/> <ref name ="BMW2001"/>) to better account for super-diffusive spreading of tracer particles in aquifers. It has quite a few variations at this point, but a couple of characteristic examples are<br />
<br />
\[<br />
u_t = c\frac{\partial u}{\partial x} + p\frac{\partial^\alpha u}{\partial x^\alpha} + q\frac{\partial^\alpha u}{\partial (-x)^\alpha} ,<br />
\]<br />
where the fractional derivatives are defined as <br />
\begin{equation}<br />
\frac{\partial^{\alpha}}{\partial e^{\alpha}} u(x) = c_{n,\alpha}\int_0^\infty \frac{u(x)-u(x-ye)}{y}y^{-\alpha}dy\ \ \text{and}\ \ \frac{\partial^{\alpha}}{\partial (-e)^{\alpha}} u(x) = c_{n,\alpha}\int_0^\infty \frac{u(x)-u(x+ye)}{y}y^{-\alpha}dy .<br />
\end{equation}<br />
It is important to point out that these fractional derivatives are examples of one-dimensional [[Linear integro-differential operator | linear integro-differential operators]] with non-symmetric kernels (in this case, e.g. $K(y)=\mathbb{1}_{\{y\geq 0\}}|y|^{-1-\alpha}$).<br />
<br />
== References ==<br />
{{reflist|refs=<br />
<ref name="MBB1999">{{Citation | last1=Meerschaert | first1=M.M. | last2=Benson | first2=D. J. | last3=Bäumer | first3=B. | title=Multidimensional advection and fractional dispersion | publisher=APS | year=1999 | journal=[[Physical Review|Physical Review E]] | issn=1539-3755 | volume=59 | issue=5 | pages=5026}}</ref><br />
<ref name="BMW">{{Citation | last1=Benson | first1=D. J. | last2=Wheatcraft | first2=S.W. | last3=Meerschaert | first3=M.M. | title=Application of a fractional advection-dispersion equation | year=2000 | journal=Water Resources Research | volume=36 | issue=6 | pages=1403–1412}}</ref><br />
<ref name="MBB2001">{{Citation | last1=Meerschaert | first1=M.M. | last2=Benson | first2=D. J. | last3=Baeumer | first3=B. | title=Operator Lévy motion and multiscaling anomalous diffusion | publisher=APS | year=2001 | journal=[[Physical Review|Physical Review E]] | issn=1539-3755 | volume=63 | issue=2 | pages=021112}}</ref><br />
<ref name="BMW2001">{{Citation | last1=Benson | first1=D. J. | last2=Schumer | first2=R. | last3=Meerschaert | first3=M.M. | last4=Wheatcraft | first4=S.W. | title=Fractional dispersion, Lévy motion, and the MADE tracer tests | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=2001 | journal=Transport in Porous Media | volume=42 | issue=1 | pages=211–240}}</ref><br />
}}</div>
imported>Russell