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Commit 4812fb43 authored by Michael Kohlhase's avatar Michael Kohlhase
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......@@ -46,14 +46,15 @@
</mfrac>
</mrow>
<mo>=</mo>
<mi mathvariant="normal">&#x2207;<!-- ∇ --></mi>
<mi mathvariant="normal" data-dref="nabla">&#x2207;<!-- ∇ --></mi>
<mo>&#x22C5;<!-- ⋅ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mo maxsize="1.2em" minsize="1.2em">[</mo>
</mrow>
</mrow>
<mi>D</mi>
<mrow data-dref="Dphir">
<mi data-dref="D">D</mi>
<mo stretchy="false">(</mo>
<mi>&#x03D5;<!-- ϕ --></mi>
<mo>,</mo>
......@@ -61,8 +62,9 @@
<mi mathvariant="bold">r</mi>
</mrow>
<mo stretchy="false">)</mo>
</mrow>
<mtext>&#xA0;</mtext>
<mi mathvariant="normal">&#x2207;<!-- ∇ --></mi>
<mi mathvariant="normal" data-dref="nabla">&#x2207;<!-- ∇ --></mi>
<mrow data-dref="phirt">
<mi data-dref="phi">&#x03D5;<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
......@@ -92,8 +94,11 @@
<p>where <span data-decl="phi"><i>ϕ</i>(<b>r</b>, <i>t</i>) is the <a href="/wiki/Density"
title="Density">density</a> of the diffusing material <span data-decl="r"> at location <b>r</b></span>
and <span data-decl="t"> time <i>t</i></span></span> and
<span data-decl="Dphir"><i>D</i>(<i>ϕ</i>, <b>r</b>) is the collective <a href="/wiki/Diffusion_coefficient" class="mw-redirect" title="Diffusion coefficient">diffusion coefficient</a> for density <i>ϕ</i> at location <b>r</b></span>; and ∇ represents the vector <a href="/wiki/Differential_operator" title="Differential operator">differential operator</a> <a href="/wiki/Del" title="Del">del</a>. If the diffusion coefficient depends on the density then the equation is nonlinear, otherwise it is linear.</p>
<p>More generally, when <i>D</i> is a symmetric <a href="/wiki/Positive_definite_matrix" class="mw-redirect" title="Positive definite matrix">positive definite matrix</a>, the equation describes <a href="/wiki/Anisotropy" title="Anisotropy">anisotropic</a> diffusion, which is written (for three dimensional diffusion) as:</p>
<span data-decl="Dphir"><i>D</i>(<i>ϕ</i>, <b>r</b>) is the collective <a
href="/wiki/Diffusion_coefficient" class="mw-redirect" title="Diffusion
coefficient">diffusion coefficient</a> for density <i>ϕ</i> at location <b>r</b></span>;
and <span data-decl="nabla">∇ represents the vector <a href="/wiki/Differential_operator" title="Differential operator">differential operator</a> <a href="/wiki/Del" title="Del">del</a></span>. If the diffusion coefficient depends on the density then the equation is nonlinear, otherwise it is linear.</p>
<p>More generally, when <span data-decl="D"><i>D</i> is a symmetric <a href="/wiki/Positive_definite_matrix" class="mw-redirect" title="Positive definite matrix">positive definite matrix</a></span>, the equation describes <a href="/wiki/Anisotropy" title="Anisotropy">anisotropic</a> diffusion, which is written (for three dimensional diffusion) as:</p>
<dl>
<dd>
<table cellpadding="5" style="border:2px solid #50C878;background: #ECFCF4; text-align: center;">
......@@ -108,15 +113,17 @@ and <span data-decl="t"> time <i>t</i></span></span> and
<mfrac>
<mrow>
<mi mathvariant="normal">&#x2202;<!-- ∂ --></mi>
<mi>&#x03D5;<!-- ϕ --></mi>
<mrow data-dref="phirt">
<mi data-dref="phi">&#x03D5;<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
<mi mathvariant="bold" data-dref="r">r</mi>
</mrow>
<mo>,</mo>
<mi>t</mi>
<mi data-dref="t">t</mi>
<mo stretchy="false">)</mo>
</mrow>
</mrow>
<mrow>
<mi mathvariant="normal">&#x2202;<!-- ∂ --></mi>
<mi>t</mi>
......@@ -163,7 +170,7 @@ and <span data-decl="t"> time <i>t</i></span></span> and
<mrow>
<mo>[</mo>
<mrow>
<msub>
<msub data-dref="D">
<mi>D</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
......@@ -181,14 +188,16 @@ and <span data-decl="t"> time <i>t</i></span></span> and
<mfrac>
<mrow>
<mi mathvariant="normal">&#x2202;<!-- ∂ --></mi>
<mi>&#x03D5;<!-- ϕ --></mi>
<mrow data-dref="phirt">
<mi data-dref="phi">&#x03D5;<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
<mi mathvariant="bold" data-dref="r">r</mi>
</mrow>
<mo>,</mo>
<mi>t</mi>
<mi data-dref="t">t</mi>
<mo stretchy="false">)</mo>
</mrow>
</mrow>
<mrow>
<mi mathvariant="normal">&#x2202;<!-- ∂ --></mi>
......
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