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<Text-field style="Text" bold="true" layout="Normal"><Font bold="true">Example 3.2.8  Diffusion of a Substrate in an Enzyme Catalyzed Reaction - BVPs with Removeable Singularity</Font></Text-field>
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<Text-field style="Text" layout="Normal">Boundary value problems in cylindrical and spherical coordinates have an inherent singularity at x = 0.  These problems can be tackled using Maple's inbuilt midpoint methods.  For example, when obtaining the diffusion of a substrate in an enzyme catalyzed reaction,<Font superscript="true">[6]</Font> the governing equation for the dimensionless  concentration is</Text-field>
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<Text-field style="2D Math" layout="Normal" alignment="centred"><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>
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<Text-field style="Text" layout="Normal">where f<Equation executable="false" style="Text" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYmUSJ5RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUqdW5kZXJsaW5lR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y9Rj1GPQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYmUSJ5RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUqdW5kZXJsaW5lR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y9Rj1GPQ==</Equation> is a dimensionless function which describes the change of diffusion coefficient as a function of concentration:</Text-field>
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<Text-field style="Text" layout="Normal" alignment="centred">                                        f<Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYmUSJ5RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUqdW5kZXJsaW5lR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y9Rj1GPQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYmUSJ5RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUqdW5kZXJsaW5lR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y9Rj1GPQ==</Equation>=1+<Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDRitGK0YrRistSShtZmVuY2VkR0YkNiQtRiM2JS1JI21uR0YkNiRRJDMuMkYnRi8tRkw2JFEkLjIwRidGL0YvRi9GLw==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDRitGK0YrRistSShtZmVuY2VkR0YkNiQtRiM2JS1JI21uR0YkNiRRJDMuMkYnRi8tRkw2JFEkLjIwRidGL0YvRi9GLw==</Equation></Text-field>
<Text-field style="Text" layout="Normal">The boundary conditions are:</Text-field>
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<Text-field style="2D Math" layout="Normal" alignment="centred"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZULUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGKy1JJm1mcmFjR0YkNigtRiM2JS1GLDYtUTAmRGlmZmVyZW50aWFsRDtGJ0YvL0YzUSZ1bnNldEYnL0Y2Rk8vRjhGTy9GOkZPL0Y8Rk8vRj5GTy9GQEZPRkFGRC1JI21pR0YkNiVRInlGJy8lJ2l0YWxpY0dGNEYvRi8tRiM2JUZLLUZXNiVRInhGJ0ZaRi9GLy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGYG8vJSliZXZlbGxlZEdGNEYv">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</Equation><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>
<Text-field style="Text" layout="Normal">and</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal" alignment="centred">                                                  y<Equation executable="false" style="Text" input-equation="" display="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">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZlbi1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y0RjQtSSNtb0dGJDYuUSI9RicvJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRidGNC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQC8lKXN0cmV0Y2h5R0ZALyUqc3ltbWV0cmljR0ZALyUobGFyZ2VvcEdGQC8lLm1vdmFibGVsaW1pdHNHRkAvJSdhY2NlbnRHRkAvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZPLUYxNiVGM0Y7RjQtRjg2LlEifkYnRjtGNEY+RkFGQ0ZFRkdGSUZLL0ZOUSYwLjBlbUYnL0ZRRlhGVEZURlRGVEZURlRGVEZURlRGVEZURlRGVEZURlRGVEZURlRGVEZURlRGVEZURlRGVEZURlRGVEZURlRGVEZURlRGVEZURlRGVEZURlRGVEZURlRGVEZURlRGVEZURlRGVEZURlQtRiw2JS1GIzYlLUYxNiVRJDMuMkYnRjtGNC1GMTYlUSQuMjJGJ0Y7RjRGNEY7RjRGNA==</Equation></Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">This boundary value problem is solved in Maple below for different values of  \316\273:</Font></Text-field>
</Input>
</Group>
<Group labelreference="L4" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">restart:</Text-field>
</Input>
</Group>
<Group labelreference="L5" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">with(plots):</Text-field>
</Input>
</Group>
<Group labelreference="L6" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">f(y):=1+lambda/(y(x)+0.01)^2;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi1JImZHNiI2I0kieUdGJSwmIiIiRikqJkknbGFtYmRhR0YlRikpLCYtRic2I0kieEdGJUYpJEYpISIjRikiIiMhIiJGKQ==">LCYiIiJGIyomSSdsYW1iZGFHNiJGIyksJi1JInlHRiY2I0kieEdGJkYjJEYjISIjRiMiIiMhIiJGIw==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L7" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Eq:=1/x^2*diff((x^2*f(y)*diff(y(x),x)),x)=10*y(x)/(y(x)+0.1);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjRXFHNiIvKiYpSSJ4R0YkIiIjISIiLCgqKkYpIiIiRihGLSwmRi1GLSomSSdsYW1iZGFHRiRGLSksJi1JInlHRiQ2I0YoRi0kRi0hIiNGLUYpRipGLUYtLUklZGlmZkclKnByb3RlY3RlZEc2JEYzRihGLUYtKixGKUYtRidGLUYwRi0pRjIiIiRGKilGOEYpRi1GKiooRidGLUYuRi0tRjk2JEYzLUkiJEdGOjYkRihGKUYtRi1GLSwkKigiIzVGLUYzRi0sJkYzRi0kRi1GKkYtRipGLQ==">LyomKUkieEc2IiIiIyEiIiwoKipGJyIiIkYlRissJkYrRisqJkknbGFtYmRhR0YmRispLCYtSSJ5R0YmNiNGJUYrJEYrISIjRitGJ0YoRitGKy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRGMUYlRitGKyosRidGK0YkRitGLkYrKUYwIiIkRigpRjZGJ0YrRigqKEYkRitGLEYrLUY3NiRGNkYlRitGK0YrLCQqKCIjNUYrRjFGKywmRjFGKyRGK0YoRitGKEYr</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L8" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq:=expand(subs(lambda=0,Eq));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjZXFHNiIvLCYqKCIiIyIiIkkieEdGJCEiIi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5R0YkNiNGKkYqRilGKS1GLTYkRjAtSSIkR0YuNiRGKkYoRiksJCooIiM1RilGMEYpLCZGMEYpJEYpRitGKUYrRik=">LywmKigiIiMiIiJJInhHNiIhIiItSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkieUdGKDYjRidGJ0YmRiYtRis2JEYqRidGJiwkKigiIzVGJkYuRiYsJkYuRiYkRiZGKUYmRilGJg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L9" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">BCs:=(D(y)(0),y(1)-1);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkQkNzRzYiNiQtLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjSSJ5R0YkNiMiIiEsJi1GLTYjIiIiRjNGMyEiIg==">NiQtLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSJ5R0YpNiMiIiEsJi1GKzYjIiIiRjFGMSEiIg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L10" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">sol:=dsolve({eq,BCs},{y(x)},type=numeric);</Text-field>
</Input>
<Output>
<Text-field style="Error" layout="Error">Error, (in dsolve/numeric/bvp) system is singular at left endpoint, use midpoint method instead</Text-field>
</Output>
</Group>
<Group labelreference="L19" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Maple identifies the singularity at x = 0 and suggests the midpoint method:</Text-field>
</Input>
</Group>
<Group labelreference="L11" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">sol:=dsolve({eq,BCs},{y(x)},type=numeric,method=bvp[midrich]);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">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</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L12" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">odeplot(sol,[x,y(x)],0..1,thickness=4,title=&quot;Figure Exp. 3.2.15.&quot;,axes=boxed);</Text-field>
</Input>
<Output>
<Text-field style="Maple Plot" layout="Maple Plot"><Plot height="300" type="two-dimensional" width="400" plot-scale="1.0" plot-xtrans="0.0" plot-ytrans="0.0" gridlinevisibility="1" legendvisibility="false">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</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L13" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">sol(0);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NyUvSSJ4RzYiJCIiIUYnLy1JInlHRiU2I0YkJCIzYSlweiczYXpsKCkhIz0vLUklZGlmZkclKnByb3RlY3RlZEc2JEYpRiRGJg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L14" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">L:=[0,1e-2,1e-1,1,10];</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkiTEc2IjcnIiIhJCIiIiEiIyRGKCEiIkYoIiM1">NyciIiEkIiIiISIjJEYlISIiRiUiIzU=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L15" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">MM:=nops(L);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjTU1HNiIiIiY=">IiIm</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L20" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">clr:=[red,green,gold,blue,magenta];</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkY2xyRzYiNydJJHJlZEdGJEkmZ3JlZW5HRiRJJWdvbGRHRiRJJWJsdWVHRiRJKG1hZ2VudGFHRiQ=">NydJJHJlZEc2IkkmZ3JlZW5HRiRJJWdvbGRHRiRJJWJsdWVHRiRJKG1hZ2VudGFHRiQ=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L16" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">for i to MM do </Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq:=expand(subs(lambda=L[i],Eq));</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">sol[i]:=dsolve({eq,BCs},{y(x)},type=numeric,method=bvp[midrich]);</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">p[i]:=odeplot(sol[i],[x,y(x)],0..1,thickness=4,color=clr[i]);</Text-field>
</Input>
<Input>
<Text-field prompt="&gt; " style="Maple Input" linebreak="space" spaceabove="0" rightmargin="0" bullet="none" firstindent="0" bulletsuffix="" linespacing="0.0" pagebreak-before="false" leftmargin="0" alignment="left" initial="0" spacebelow="0">od:</Text-field>
</Input>
</Group>
<Group labelreference="L21" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">pt:=textplot([seq([0.2,rhs(sol[i](0.2)[2])+0.02,lambda=L[i]],i=1..MM)]):</Text-field>
</Input>
</Group>
<Group labelreference="L17" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">display({seq(p[i],i=1..MM),pt},title=&quot;Figure Exp. 3.2.16.&quot;,axes=boxed);</Text-field>
</Input>
<Output>
<Text-field style="Maple Plot" layout="Maple Plot"><Plot height="400" type="two-dimensional" width="400" plot-scale="1.0" plot-xtrans="0.0" plot-ytrans="0.0" gridlinevisibility="1" legendvisibility="false">[encoded data, 25200 characters]</Plot></Text-field>
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<Group labelreference="L18" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"></Text-field>
</Input>
</Group>
</Worksheet>