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<Text-field style="Heading 3" italic="false" layout="Heading 3"><Font italic="false">Example 3.7a</Font></Text-field>
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<Input>
<Text-field style="Text" layout="Normal">Consider the convective diffusion problem (Finlayson,1980)<Font superscript="true">[11]</Font></Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Text Output" alignment="centred"><Equation executable="false" style="Text" input-equation="" display="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">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</Equation></Text-field>
<Text-field style="Text" layout="Text Output" alignment="centred">                                                                             (3.65)</Text-field>
<Text-field style="Text" layout="Text Output" alignment="centred"><Equation executable="false" style="Text" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiQtRiM2JC1JI21uR0YkNiRRIjBGJ0YyRjJGMi1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUi1GOzYkUSIxRidGMkYy">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiQtRiM2JC1JI21uR0YkNiRRIjBGJ0YyRjJGMi1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUi1GOzYkUSIxRidGMkYy</Equation></Text-field>
<Text-field style="Text" layout="Text Output" alignment="centred"></Text-field>
<Text-field style="Text" layout="Text Output" alignment="centred"><Equation executable="false" style="Text" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiQtRiM2JC1JI21uR0YkNiRRIjFGJ0YyRjJGMi1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUi1GOzYkUSIwRidGMkYy">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiQtRiM2JC1JI21uR0YkNiRRIjFGJ0YyRjJGMi1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUi1GOzYkUSIwRidGMkYy</Equation></Text-field>
<Text-field style="Text" layout="Normal" alignment="centred"></Text-field>
<Text-field style="Text" layout="Normal">An analytical solution can be obtained using the exponential matrix method described in section 3.1.2:</Text-field>
<Text-field style="Text" layout="Normal" alignment="centred"></Text-field>
<Text-field style="Normal" layout="Normal" alignment="centred"><Equation executable="false" style="Normal" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZWLUkjbW9HRiQ2LlEifkYnLyUwZm9udF9zdHlsZV9uYW1lR1EqUGxvdH5UZXh0RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y3LyUpc3RyZXRjaHlHRjcvJSpzeW1tZXRyaWNHRjcvJShsYXJnZW9wR0Y3LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkZGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRistSSNtaUdGJDYmUSJjRicvJSdpdGFsaWNHRjdGL0YyLUYsNi5RIj1GJ0YvRjJGNUY4RjpGPEY+RkBGQi9GRVEsMC4yNzc3Nzc4ZW1GJy9GSEZTLUkmbWZyYWNHRiQ2KC1GIzYqLUZKNiZRImVGJ0ZNRi9GMi1JJW1zdXBHRiQ2JS1GSjYjUSFGJ0Zqbi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictRko2JlEjUGVGJ0ZNRi9GMi1GLDYuUSgmbWludXM7RidGL0YyRjVGOEY6RjxGPkZARkIvRkVRLDAuMjIyMjIyMmVtRicvRkhGZ29GWi1GSjYmUSRQZXhGJ0ZNRi9GMi8lK2ZvcmVncm91bmRHUSxbMCwxMDIsMTAyXUYnRjItRiM2KkZaRmBvRitGKy1GLDYuUSomdW1pbnVzMDtGJ0YvRjJGNUY4RjpGPEY+RkBGQkZmb0Zoby1JI21uR0YkNiVRIjFGJ0YvRjJGXHBGMi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGXXEvJSliZXZlbGxlZEdGN0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGKy1JKG1mZW5jZWRHRiQ2JS1GIzYlLUZlcDYlUSUzLjY2RidGL0YyRlxwRjJGL0YyRlxwRjI=">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</Equation></Text-field>
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<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">This particular problem was chosen as the finite difference solution for this equation and shows oscillations for high Peclet numbers when the central difference expression is used for the first derivative.  This equation is solved below using the procedure described above.</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">restart:</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">with(linalg):with(plots):</Text-field>
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<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">N:=4;</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkiTkc2IiIiJQ==">IiIl</Equation></Text-field>
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<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">L:=1;</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkiTEc2IiIiIg==">IiIi</Equation></Text-field>
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<Group labelreference="L7" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq:=diff(y(x),x$2)-Pe*diff(y(x),x);</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjZXFHNiIsJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5R0YkNiNJInhHRiQtSSIkR0YoNiRGLSIiIyIiIiomSSNQZUdGJEYyLUYnNiRGKkYtRjIhIiI=">LCYtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUYkNiQtSSJ5RzYiNiNJInhHRitGLUYtIiIiKiZJI1BlR0YrRi5GJ0YuISIi</Equation></Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc1:=y(x)-1;</Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMxRzYiLCYtSSJ5R0YkNiNJInhHRiQiIiJGKiEiIg==">LCYtSSJ5RzYiNiNJInhHRiUiIiIhIiJGKA==</Equation></Text-field>
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<Group labelreference="L9" drawlabel="true">
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc2:=y(x);</Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMyRzYiLUkieUdGJDYjSSJ4R0Yk">LUkieUc2IjYjSSJ4R0Yk</Equation></Text-field>
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<Text-field style="Normal" layout="Normal"> Central difference expressions for the second and first derivatives are</Text-field>
</Input>
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<Group labelreference="L11" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">d2ydx2:=(y[m+1]-2*y[m]+y[m-1])/h^2;</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkknZDJ5ZHgyRzYiKiYsKCZJInlHRiQ2IywmSSJtR0YkIiIiRixGLEYsKiYiIiNGLCZGKDYjRitGLCEiIiZGKDYjLCZGK0YsRixGMUYsRiwpSSJoR0YkRi5GMQ==">KiYsKCZJInlHNiI2IywmSSJtR0YmIiIiRipGKkYqJkYlNiNGKSEiIyZGJTYjLCZGKUYqISIiRipGKkYqSSJoR0YmRi0=</Equation></Text-field>
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<Group labelreference="L12" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">dydx:=(y[m+1]-y[m-1])/2/h;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkklZHlkeEc2IiwkKigjIiIiIiIjRigsJiZJInlHRiQ2IywmSSJtR0YkRihGKEYoRigmRiw2IywmRi9GKEYoISIiRjNGKEkiaEdGJEYzRig=">LCQqJiwmJkkieUc2IjYjLCZJIm1HRiciIiJGK0YrRismRiY2IywmRipGKyEiIkYrRi9GK0kiaEdGJ0YvI0YrIiIj</Equation></Text-field>
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<Group labelreference="L13" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal"> The governing equation in finite difference form is:</Text-field>
</Input>
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<Group labelreference="L14" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Eq[m]:=subs(diff(y(x),x$2)=d2ydx2,diff(y(x),x)=dydx,y(x)=y[m],x=m*h,eq);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiNJIm1HRiUsJiomLCgmSSJ5R0YlNiMsJkYnIiIiRi9GL0YvKiYiIiNGLyZGLEYmRi8hIiImRiw2IywmRidGL0YvRjNGL0YvKUkiaEdGJUYxRjNGLyoqI0YvRjFGL0kjUGVHRiVGLywmRitGL0Y0RjNGL0Y4RjNGMw==">LCYqJiwoJkkieUc2IjYjLCZJIm1HRiciIiJGK0YrRismRiY2I0YqISIjJkYmNiMsJkYqRishIiJGK0YrRitJImhHRidGLkYrKihJI1BlR0YnRissJkYlRitGL0YyRitGM0YyI0YyIiIj</Equation></Text-field>
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<Group labelreference="L15" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal"> A 'for loop' can be written for the interior node points as</Text-field>
</Input>
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<Group labelreference="L16" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">for i to N do Eq[i]:=subs(m=i,Eq[m]);od;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiIsJiomLCgmSSJ5R0YlNiMiIiNGJyomRi5GJyZGLEYmRichIiImRiw2IyIiIUYnRicpSSJoR0YlRi5GMUYnKiojRidGLkYnSSNQZUdGJUYnLCZGK0YnRjJGMUYnRjZGMUYx">LCYqJiwoJkkieUc2IjYjIiIjIiIiJkYmNiNGKiEiIyZGJjYjIiIhRipGKkkiaEdGJ0YtRioqKEkjUGVHRidGKiwmRiVGKkYuISIiRipGMUY1I0Y1Rik=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiMsJiomLCgmSSJ5R0YlNiMiIiQiIiIqJkYnRi8mRixGJkYvISIiJkYsNiNGL0YvRi8pSSJoR0YlRidGMkYvKiojRi9GJ0YvSSNQZUdGJUYvLCZGK0YvRjNGMkYvRjZGMkYy">LCYqJiwoJkkieUc2IjYjIiIkIiIiJkYmNiMiIiMhIiMmRiY2I0YqRipGKkkiaEdGJ0YuRioqKEkjUGVHRidGKiwmRiVGKkYvISIiRipGMUY1I0Y1Ri0=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiQsJiomLCgmSSJ5R0YlNiMiIiUiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjRjFGL0YvKUkiaEdGJUYxRjNGLyoqI0YvRjFGL0kjUGVHRiVGLywmRitGL0Y0RjNGL0Y3RjNGMw==">LCYqJiwoJkkieUc2IjYjIiIlIiIiJkYmNiMiIiQhIiMmRiY2IyIiI0YqRipJImhHRidGLkYqKihJI1BlR0YnRiosJkYlRipGLyEiIkYqRjJGNiNGNkYx</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiUsJiomLCgmSSJ5R0YlNiMiIiYiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiIkRi9GLylJImhHRiVGMUYzRi8qKiNGL0YxRi9JI1BlR0YlRi8sJkYrRi9GNEYzRi9GOEYzRjM=">LCYqJiwoJkkieUc2IjYjIiImIiIiJkYmNiMiIiUhIiMmRiY2IyIiJEYqRipJImhHRidGLkYqKihJI1BlR0YnRiosJkYlRipGLyEiIkYqRjJGNiNGNiIiIw==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L17" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Eq[0]:=y[0]=1;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiEvJkkieUdGJUYmIiIi">LyZJInlHNiI2IyIiISIiIg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L18" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Eq[N+1]:=y[N+1]=0;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiYvJkkieUdGJUYmIiIh">LyZJInlHNiI2IyIiJiIiIQ==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L19" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">y[0]:=solve(Eq[0],y[0]);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJInlHNiI2IyIiISIiIg==">IiIi</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L20" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">y[N+1]:=solve(Eq[N+1],y[N+1]);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJInlHNiI2IyIiJiIiIQ==">IiIh</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L21" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">h:=L/(N+1);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkiaEc2IiMiIiIiIiY=">IyIiIiIiJg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L22" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">for i to N do Eq[i]:=eval(Eq[i]);od;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiIsKiomIiNERicmSSJ5R0YlNiMiIiNGJ0YnKiYiI11GJyZGLEYmRichIiJGKkYnKigjIiImRi5GJ0kjUGVHRiVGJywmRitGJ0YnRjJGJ0Yy">LComSSJ5RzYiNiMiIiMiI0QmRiQ2IyIiIiEjXUYoRisqJkkjUGVHRiVGKywmRiNGKyEiIkYrRisjISImRic=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiMsKiomIiNEIiIiJkkieUdGJTYjIiIkRitGKyomIiNdRismRi1GJkYrISIiKiZGKkYrJkYtNiNGK0YrRisqKCMiIiZGJ0YrSSNQZUdGJUYrLCZGLEYrRjVGM0YrRjM=">LComSSJ5RzYiNiMiIiQiI0QmRiQ2IyIiIyEjXSZGJDYjIiIiRigqJkkjUGVHRiVGLywmRiNGL0YtISIiRi8jISImRis=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiQsKiomIiNEIiIiJkkieUdGJTYjIiIlRitGKyomIiNdRismRi1GJkYrISIiKiZGKkYrJkYtNiMiIiNGK0YrKigjIiImRjdGK0kjUGVHRiVGKywmRixGK0Y1RjNGK0Yz">LComSSJ5RzYiNiMiIiUiI0QmRiQ2IyIiJCEjXSZGJDYjIiIjRigqJkkjUGVHRiUiIiIsJkYjRjJGLSEiIkYyIyEiJkYv</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiUsKComIiNdIiIiJkkieUdGJUYmRishIiIqJiIjREYrJkYtNiMiIiRGK0YrKigjIiImIiIjRitJI1BlR0YlRitGMUYrRis=">LCgmSSJ5RzYiNiMiIiUhI10mRiQ2IyIiJCIjRComSSNQZUdGJSIiIkYpRi8jIiImIiIj</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L23" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eqs:=[seq(Eq[i],i=1..N)];</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXFzRzYiNyYsKiomIiNEIiIiJkkieUdGJDYjIiIjRilGKSomIiNdRikmRis2I0YpRikhIiJGKEYpKigjIiImRi1GKUkjUGVHRiRGKSwmRipGKUYpRjJGKUYyLCoqJkYoRikmRis2IyIiJEYpRikqJkYvRilGKkYpRjIqJkYoRilGMEYpRikqKEY0RilGNkYpLCZGOkYpRjBGMkYpRjIsKiomRihGKSZGKzYjIiIlRilGKSomRi9GKUY6RilGMkYnRikqKEY0RilGNkYpLCZGQ0YpRipGMkYpRjIsKComRi9GKUZDRilGMkY5RikqKEY0RilGNkYpRjpGKUYp">NyYsKiZJInlHNiI2IyIiIyIjRCZGJTYjIiIiISNdRilGLComSSNQZUdGJkYsLCZGJEYsISIiRixGLCMhIiZGKCwqJkYlNiMiIiRGKUYkRi1GKkYpKiZGL0YsLCZGNUYsRipGMUYsRjIsKiZGJTYjIiIlRilGNUYtRiRGKSomRi9GLCwmRjtGLEYkRjFGLEYyLChGO0YtRjVGKSomRi9GLEY1RiwjIiImRig=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L24" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">vars:=[seq(y[i],i=1..N)];</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkldmFyc0c2IjcmJkkieUdGJDYjIiIiJkYnNiMiIiMmRic2IyIiJCZGJzYjIiIl">NyYmSSJ5RzYiNiMiIiImRiQ2IyIiIyZGJDYjIiIkJkYkNiMiIiU=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L25" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">A:=genmatrix(eqs,vars,'B1');</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkiQUc2Ii1JJ21hdHJpeEdGJDYjNyY3JiEjXSwmIiNEIiIiKiYjIiImIiIjRi1JI1BlR0YkRi0hIiIiIiFGNDcmLCZGLEYtRi5GLUYqRitGNDcmRjRGNkYqRis3JkY0RjRGNkYq">PTYiNiQ7IiIiIiIlRiVFXFtsMTYkRiZGJyIiITYkRiYiIiRGKjYkRixGJkYqNiRGJ0YmRio2JEYmIiIjLCYiI0RGJkkjUGVHRiMjISImRjA2JEYwRiYsJkYyRiZGMyMiIiZGMDYkRjBGMCEjXTYkRjBGLEYxNiRGLEYnRjE2JEYnRixGNzYkRiZGJkY7NiRGLEYsRjs2JEYnRidGOzYkRidGMEYqNiRGMEYnRio2JEYsRjBGNw==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L26" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">evalm(B1);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="LUkndmVjdG9yRzYiNiM3JiwmIiNEISIiKiYjIiImIiIjIiIiSSNQZUdGJEYuRikiIiFGMEYw">PTYiNiM7IiIiIiIlRVxbbCVGJiwmISNERiZJI1BlR0YjIyEiJiIiI0YuIiIhIiIkRi9GJ0Yv</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L27" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal">Maple generates a row vector, which can be converted to a column vector as:</Text-field>
</Input>
</Group>
<Group labelreference="L28" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">B:=matrix(N,1):for i to N do B[i,1]:=B1[i]:od:evalm(B);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="LUknbWF0cml4RzYiNiM3JjcjLCYiI0QhIiIqJiMiIiYiIiMiIiJJI1BlR0YkRi9GKjcjIiIhRjFGMQ==">PTYiNiQ7IiIiIiIlO0YmRiZFXFtsJTYkIiIkRiYiIiE2JEYnRiZGLDYkIiIjRiZGLDYkRiZGJiwmISNERiZJI1BlR0YjIyEiJkYv</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L29" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal"> The solution is obtained as:</Text-field>
</Input>
</Group>
<Group labelreference="L30" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">X:=evalm(inverse(A)&amp;*B);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkiWEc2Ii1JJ21hdHJpeEdGJDYjNyY3IywkKioiIzsiIiIsJiIkKyJGLSokKUkjUGVHRiQiIiNGLUYtRi0sKCImKysmRi0qJiIlKzVGLUYxRi1GLSokKUYyIiIlRi1GLSEiIiwmIiNERjsqJiMiIiZGM0YtRjJGLUY7Ri1GOzcjLCQqLCNGM0ZARi0sJiIjNUYtRjJGLUYtLCYiJCskRi1GMEYtRi1GNEY7RjxGLUY7NyMsJCoqIiIpRi0pRkVGM0YtRjRGO0Y8Ri1GOzcjLCQqKkZERi0pRkUiIiRGLUY0RjtGPEYtRjs=">PTYiNiQ7IiIiIiIlO0YmRiZFXFtsJTYkIiIkRiYsJCooLCYiIzVGJkkjUGVHRiNGJiIiIywoIiYrKyZGJiokRjBGMSIlKzUqJEYwRidGJiEiIiwmISNERiZGMCMhIiZGMUYmISIpNiRGJ0YmLCQqKEYuRitGMkY3RjhGJiMhIiMiIiY2JEYxRiYsJCoqRi5GJiwmIiQrJEYmRjRGJkYmRjJGN0Y4RiZGQDYkRiZGJiwkKigsJiIkKyJGJkY0RiZGJkYyRjdGOEYmISM7</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L31" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">for i to N do y[i]:=X[i,1];od;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJInlHNiI2IyIiIiwkKioiIztGJywmIiQrIkYnKiQpSSNQZUdGJSIiI0YnRidGJywoIiYrKyZGJyomIiUrNUYnRi5GJ0YnKiQpRi8iIiVGJ0YnISIiLCYiI0RGOComIyIiJkYwRidGL0YnRjhGJ0Y4">LCQqKCwmIiQrIiIiIiokSSNQZUc2IiIiI0YmRiYsKCImKysmRiZGJyIlKzUqJEYoIiIlRiYhIiIsJiEjREYmRigjISImRipGJiEjOw==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJInlHNiI2IyIiIywkKiwjRiciIiYiIiIsJiIjNUYsSSNQZUdGJUYsRiwsJiIkKyRGLCokKUYvRidGLEYsRiwsKCImKysmRiwqJiIlKzVGLEYzRixGLCokKUYvIiIlRixGLCEiIiwmIiNERjsqJiNGK0YnRixGL0YsRjtGLEY7">LCQqKiwmIiM1IiIiSSNQZUc2IkYmRiYsJiIkKyRGJiokRiciIiNGJkYmLCgiJisrJkYmRisiJSs1KiRGJyIiJUYmISIiLCYhI0RGJkYnIyEiJkYsRiYjISIjIiIm</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJInlHNiI2IyIiJCwkKioiIikiIiIpLCYiIzVGK0kjUGVHRiVGKyIiI0YrLCgiJisrJkYrKiYiJSs1RispRi9GMEYrRisqJClGLyIiJUYrRishIiIsJiIjREY5KiYjIiImRjBGK0YvRitGOUYrRjk=">LCQqKCwmIiM1IiIiSSNQZUc2IkYmIiIjLCgiJisrJkYmKiRGJ0YpIiUrNSokRiciIiVGJiEiIiwmISNERiZGJyMhIiZGKUYmISIp</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJInlHNiI2IyIiJSwkKiojIiIjIiImIiIiKSwmIiM1Ri1JI1BlR0YlRi0iIiRGLSwoIiYrKyZGLSomIiUrNUYtKUYxRitGLUYtKiQpRjFGJ0YtRi0hIiIsJiIjREY6KiYjRixGK0YtRjFGLUY6Ri1GOg==">LCQqKCwmIiM1IiIiSSNQZUc2IkYmIiIkLCgiJisrJkYmKiRGJyIiIyIlKzUqJEYnIiIlRiYhIiIsJiEjREYmRicjISImRi1GJiMhIiMiIiY=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L32" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">y[0]:=eval(y[0]);y[N+1]:=eval(y[N+1]);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJInlHNiI2IyIiISIiIg==">IiIi</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJInlHNiI2IyIiJiIiIQ==">IiIh</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L33" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal"> Next, the result obtained is compared with the exact analytical solution:</Text-field>
</Input>
</Group>
<Group labelreference="L34" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">ya:=(exp(Pe)-exp(Pe*x))/(exp(Pe)-1);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjeWFHNiIqJiwmLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNJI1BlR0YkIiIiLUYoNiMqJkYtRi5JInhHRiRGLiEiIkYuLCZGJ0YuRi5GM0Yz">KiYsJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSNQZUdGKSIiIi1GJTYjKiZGK0YsSSJ4R0YpRiwhIiJGLCwmRiRGLEYxRixGMQ==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L35" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">p1:=plot([seq([i*h,subs(Pe=1,y[i])],i=0..N+1)],thickness=4,color=blue,axes=boxed):</Text-field>
</Input>
</Group>
<Group labelreference="L36" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">p2:=plot(subs(Pe=1,ya),x=0..1,thickness=8,color=brown,axes=boxed,linestyle=2):</Text-field>
</Input>
</Group>
<Group labelreference="L50" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">pt:=textplot([.5,.5,&quot;Pe=1&quot;]);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjcHRHNiItSS9JTlRFUkZBQ0VfUExPVEdGJDYjLUklVEVYVEdGJDYkNyQkIiImISIiRixRJVBlPTFGJA==">LSUlUExPVEc2Iy1JJVRFWFRHNiI2JDckJCIiJiEiIkYqUSVQZT0xRic=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L37" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">display({p1,p2,pt},title=&quot;Figure Exp. 3.1.9.&quot;,labels=[x,&quot;y&quot;]);</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, 4348 characters]</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L49" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">We observe that both the finite difference solution and the analytical solution match exactly when the Peclet number is 1.  New plots can be obtained for different values of the Peclet number as follows:</Text-field>
</Input>
</Group>
<Group labelreference="L38" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">p1:=plot([seq([i*h,subs(Pe=50,y[i])],i=0..N+1)],color=blue,thickness=4,axes=boxed):</Text-field>
</Input>
</Group>
<Group labelreference="L39" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">p2:=plot(subs(Pe=50,ya),x=0..1,thickness=5,color=brown,axes=boxed,linestyle=2):</Text-field>
</Input>
</Group>
<Group labelreference="L51" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">pt:=textplot([.5,.5,&quot;Pe=50&quot;]);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjcHRHNiItSS9JTlRFUkZBQ0VfUExPVEdGJDYjLUklVEVYVEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYkNyQkIiImISIiRi9RJlBlPTUwRiQ=">LSUlUExPVEc2Iy1JJVRFWFRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JDckJCIiJiEiIkYtUSZQZT01MEYq</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L40" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">display({p1,p2,pt},title=&quot;Figure Exp. 3.1.10.&quot;,labels=[x,&quot;y&quot;]);</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">[encoded data, 5000 characters]</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L41" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal"> This shows that for Pe = 50, four interior node points are not enough and we observe oscillations.<Font superscript="true">[11][12]</Font>  This happens usually when central difference approximations are used for the convective term <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>.  Use a forward approximation for the first derivative to solve this problem.  Only dydx in the Maple program needs to be changed:</Text-field>
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