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<Text-field style="Heading 3" italic="false" layout="Heading 3"><Font italic="false">Example 3.8  Cylindrical Catalyst Pellet</Font></Text-field>
</Input>
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<Input>
<Text-field style="Text" layout="Normal">In the example discussed above, both the boundary conditions are of the Neumann type.  However, many problems involve derivative boundary conditions.  These problems can be handled by using the three point forward and backward differences at x = 0 and x = 1, respectively.  This is illustrated by solving the cylindrical pellet problem solved in example 3.6 with a different boundary condition at the surface (x = 1):</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal" alignment="centred"><Equation executable="false" style="Text" input-equation="" display="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">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</Equation><Equation executable="false" style="Text" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiQtRiM2JC1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGOEY4Rjg=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiQtRiM2JC1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGOEY4Rjg=</Equation>                                                     (3.67)</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">The Maple program developed for the previous example can be modified to handle the derivative boundary conditions.</Text-field>
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<Group labelreference="L42" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">restart:</Text-field>
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<Group labelreference="L4" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">with(linalg):with(plots):</Text-field>
</Input>
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<Group labelreference="L5" drawlabel="true">
<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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<Group labelreference="L6" drawlabel="true">
<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)+1/x*diff(y(x),x)-phi^2*y(x);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjZXFHNiIsKC1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5R0YkNiNJInhHRiQtSSIkR0YoNiRGLSIiIyIiIiomRi0hIiItRic2JEYqRi1GMkYyKiYpSSRwaGlHRiRGMUYyRipGMkY0">LCgtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUYkNiQtSSJ5RzYiNiNJInhHRitGLUYtIiIiKiZGLSEiIkYnRi5GLiomSSRwaGlHRisiIiNGKUYuRjA=</Equation></Text-field>
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<Group labelreference="L8" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc1:=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="PkkkYmMxRzYiLUklZGlmZkclKnByb3RlY3RlZEc2JC1JInlHRiQ2I0kieEdGJEYs">LUklZGlmZkclKnByb3RlY3RlZEc2JC1JInlHNiI2I0kieEdGKEYq</Equation></Text-field>
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<Group labelreference="L9" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc2:=diff(y(x),x)-1+y(x);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMyRzYiLCgtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkieUdGJDYjSSJ4R0YkRi0iIiJGLiEiIkYqRi4=">LCgtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkieUc2IjYjSSJ4R0YpRisiIiIhIiJGLEYnRiw=</Equation></Text-field>
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<Group labelreference="L10" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal"> The central difference expression for the second and first derivatives are</Text-field>
</Input>
</Group>
<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>
</Input>
<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 three point forward and backward difference expressions for the derivative are:</Text-field>
</Input>
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<Group labelreference="L14" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">dydxf:=(-y[2]+4*y[1]-3*y[0])/(2*h);</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkmZHlkeGZHNiIsJCooIyIiIiIiI0YoLCgmSSJ5R0YkNiNGKSEiIiomIiIlRigmRiw2I0YoRihGKComIiIkRigmRiw2IyIiIUYoRi5GKEkiaEdGJEYuRig=">LCQqJiwoJkkieUc2IjYjIiIjISIiJkYmNiMiIiIiIiUmRiY2IyIiISEiJEYtSSJoR0YnRiojRi1GKQ==</Equation></Text-field>
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<Group labelreference="L15" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">dydxb:=(y[N-1]-4*y[N]+3*y[N+1])/(2*h);</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkmZHlkeGJHNiIsJCooIyIiIiIiI0YoLCgmSSJ5R0YkNiMiIiRGKComIiIlRigmRiw2I0YwRighIiIqJkYuRigmRiw2IyIiJkYoRihGKEkiaEdGJEYzRig=">LCQqJiwoJkkieUc2IjYjIiIkIiIiJkYmNiMiIiUhIiUmRiY2IyIiJkYpRipJImhHRichIiIjRioiIiM=</Equation></Text-field>
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<Group labelreference="L16" 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="L17" 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="PiZJI0VxRzYiNiNJIm1HRiUsKComLCgmSSJ5R0YlNiMsJkYnIiIiRi9GL0YvKiYiIiNGLyZGLEYmRi8hIiImRiw2IywmRidGL0YvRjNGL0YvKUkiaEdGJUYxRjNGLyoqI0YvRjFGL0YnRjNGN0YzLCZGK0YvRjRGM0YvRi8qJilJJHBoaUdGJUYxRi9GMkYvRjM=">LCgqJiwoJkkieUc2IjYjLCZJIm1HRiciIiJGK0YrRismRiY2I0YqISIjJkYmNiMsJkYqRishIiJGK0YrRitJImhHRidGLkYrKihGKkYyRjNGLiwmRiVGK0YvRjJGKyNGKyIiIyomSSRwaGlHRidGN0YsRitGMg==</Equation></Text-field>
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<Group labelreference="L18" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal"> The boundary conditions in finite difference form are:</Text-field>
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<Group labelreference="L19" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Eq[0]:=subs(diff(y(x),x)=dydxf,y(x)=y[0],bc1);</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiEsJCooIyIiIiIiI0YrLCgmSSJ5R0YlNiNGLCEiIiomIiIlRismRi82I0YrRitGKyomIiIkRismRi9GJkYrRjFGK0kiaEdGJUYxRis=">LCQqJiwoJkkieUc2IjYjIiIjISIiJkYmNiMiIiIiIiUmRiY2IyIiISEiJEYtSSJoR0YnRiojRi1GKQ==</Equation></Text-field>
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<Group labelreference="L20" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Eq[N+1]:=subs(diff(y(x),x)=dydxb,y(x)=y[N+1],bc2);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiYsKCooIyIiIiIiI0YrLCgmSSJ5R0YlNiMiIiRGKyomIiIlRismRi82I0YzRishIiIqJkYxRismRi9GJkYrRitGK0kiaEdGJUY2RitGK0Y2RjhGKw==">LCgqJiwoJkkieUc2IjYjIiIkIiIiJkYmNiMiIiUhIiUmRiY2IyIiJkYpRipJImhHRichIiIjRioiIiNGM0YqRi9GKg==</Equation></Text-field>
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<Group labelreference="L21" 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="L22" 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>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiIsKComLCgmSSJ5R0YlNiMiIiNGJyomRi5GJyZGLEYmRichIiImRiw2IyIiIUYnRicpSSJoR0YlRi5GMUYnKigjRidGLkYnRjVGMSwmRitGJ0YyRjFGJ0YnKiYpSSRwaGlHRiVGLkYnRjBGJ0Yx">LCgqJiwoJkkieUc2IjYjIiIjIiIiJkYmNiNGKiEiIyZGJjYjIiIhRipGKkkiaEdGJ0YtRioqJkYxRi0sJkYlRipGLiEiIkYqI0YqRikqJkkkcGhpR0YnRilGK0YqRjQ=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiMsKComLCgmSSJ5R0YlNiMiIiQiIiIqJkYnRi8mRixGJkYvISIiJkYsNiNGL0YvRi8pSSJoR0YlRidGMkYvKigjRi8iIiVGL0Y1RjIsJkYrRi9GM0YyRi9GLyomKUkkcGhpR0YlRidGL0YxRi9GMg==">LCgqJiwoJkkieUc2IjYjIiIkIiIiJkYmNiMiIiMhIiMmRiY2I0YqRipGKkkiaEdGJ0YuRioqJkYxRi4sJkYlRipGLyEiIkYqI0YqIiIlKiZJJHBoaUdGJ0YtRitGKkY0</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiQsKComLCgmSSJ5R0YlNiMiIiUiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjRjFGL0YvKUkiaEdGJUYxRjNGLyooI0YvIiInRi9GNkYzLCZGK0YvRjRGM0YvRi8qJilJJHBoaUdGJUYxRi9GMkYvRjM=">LCgqJiwoJkkieUc2IjYjIiIlIiIiJkYmNiMiIiQhIiMmRiY2IyIiI0YqRipJImhHRidGLkYqKiZGMkYuLCZGJUYqRi8hIiJGKiNGKiIiJyomSSRwaGlHRidGMUYrRipGNQ==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiUsKComLCgmSSJ5R0YlNiMiIiYiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiIkRi9GLylJImhHRiVGMUYzRi8qKCNGLyIiKUYvRjdGMywmRitGL0Y0RjNGL0YvKiYpSSRwaGlHRiVGMUYvRjJGL0Yz">LCgqJiwoJkkieUc2IjYjIiImIiIiJkYmNiMiIiUhIiMmRiY2IyIiJEYqRipJImhHRidGLkYqKiZGMkYuLCZGJUYqRi8hIiJGKiNGKiIiKSomSSRwaGlHRiciIiNGK0YqRjU=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L23" 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="PiZJInlHNiI2IyIiISwmKiYjIiIiIiIkRismRiQ2IyIiI0YrISIiKiYjIiIlRixGKyZGJDYjRitGK0Yr">LCYmSSJ5RzYiNiMiIiMjISIiIiIkJkYkNiMiIiIjIiIlRio=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L24" 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="PiZJInlHNiI2IyIiJiomLCgmRiQ2IyIiJCEiIiomIiIlIiIiJkYkNiNGL0YwRjAqJiIiI0YwSSJoR0YlRjBGMEYwLCZGLEYwRjNGMEYt">KiYsKCZJInlHNiI2IyIiJCEiIiZGJTYjIiIlRixJImhHRiYiIiMiIiIsJkYoRi9GLUYuRik=</Equation></Text-field>
</Output>
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<Group labelreference="L25" 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="L26" 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="PiZJI0VxRzYiNiMiIiIsKComIyIkKyIiIiRGJyZJInlHRiU2IyIiI0YnRicqJkYqRicmRi5GJkYnISIiKiYpSSRwaGlHRiVGMEYnRjJGJ0Yz">LCgmSSJ5RzYiNiMiIiMjIiQrIiIiJCZGJDYjIiIiIyEkKyJGKiomSSRwaGlHRiVGJ0YrRi0hIiI=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiMsKiomIyIkRCIiIiUiIiImSSJ5R0YlNiMiIiRGLUYtKiYiI11GLSZGL0YmRi0hIiIqJiMiI3ZGLEYtJkYvNiNGLUYtRi0qJilJJHBoaUdGJUYnRi1GNEYtRjU=">LComSSJ5RzYiNiMiIiQjIiREIiIiJSZGJDYjIiIjISNdJkYkNiMiIiIjIiN2RioqJkkkcGhpR0YlRi1GK0YxISIi</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiQsKiomIyIkdiIiIiciIiImSSJ5R0YlNiMiIiVGLUYtKiYiI11GLSZGL0YmRi0hIiIqJiMiJEQiRixGLSZGLzYjIiIjRi1GLSomKUkkcGhpR0YlRjtGLUY0Ri1GNQ==">LComSSJ5RzYiNiMiIiUjIiR2IiIiJyZGJDYjIiIkISNdJkYkNiMiIiMjIiREIkYqKiZJJHBoaUdGJUYxRisiIiIhIiI=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiUsKiomIyIkRCoiI28iIiImSSJ5R0YlNiMiIiRGLUYtKiYjIiR2JiIjTUYtJkYvRiZGLSEiIiMiJEQjRixGLSomKUkkcGhpR0YlIiIjRi1GNkYtRjc=">LComSSJ5RzYiNiMiIiQjIiREKiIjbyZGJDYjIiIlIyEkdiYiI00jIiREI0YqIiIiKiZJJHBoaUdGJSIiI0YrRjMhIiI=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L27" 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="PkkkZXFzRzYiNyYsKComIyIkKyIiIiQiIiImSSJ5R0YkNiMiIiNGK0YrKiZGKEYrJkYtNiNGK0YrISIiKiYpSSRwaGlHRiRGL0YrRjFGK0YzLCoqJiMiJEQiIiIlRismRi02I0YqRitGKyomIiNdRitGLEYrRjMqJiMiI3ZGO0YrRjFGK0YrKiZGNUYrRixGK0YzLCoqJiMiJHYiIiInRismRi02I0Y7RitGKyomRj9GK0Y8RitGMyomI0Y6RkhGK0YsRitGKyomRjVGK0Y8RitGMywqKiYjIiREKiIjb0YrRjxGK0YrKiYjIiR2JiIjTUYrRklGK0YzIyIkRCNGU0YrKiZGNUYrRklGK0Yz">NyYsKCZJInlHNiI2IyIiIyMiJCsiIiIkJkYlNiMiIiIjISQrIkYrKiZJJHBoaUdGJkYoRixGLiEiIiwqJkYlNiNGKyMiJEQiIiIlRiQhI11GLCMiI3ZGOSomRjJGKEYkRi5GMywqJkYlNiNGOSMiJHYiIiInRjVGOkYkI0Y4RkMqJkYyRihGNUYuRjMsKkY1IyIkRCoiI29GPyMhJHYmIiNNIyIkRCNGSUYuKiZGMkYoRj9GLkYz</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L28" 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="L29" 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="PkkiQUc2Ii1JJ21hdHJpeEdGJDYjNyY3JiwmIyIkKyIiIiQhIiIqJClJJHBoaUdGJCIiIyIiIkYuRisiIiFGNDcmIyIjdiIiJSwmIiNdRi5GL0YuIyIkRCJGOEY0NyZGNCNGPCIiJ0Y5IyIkdiJGPzcmRjRGNCMiJEQqIiNvLCYjIiR2JiIjTUYuRi9GLg==">PTYiNiQ7IiIiIiIlRiVFXFtsMTYkIiIkIiIjIyIkRCIiIic2JEYrRiojRi1GJzYkRitGJyIiITYkRipGKiwmISNdRiYqJEkkcGhpR0YjRishIiI2JEYmRiYsJiMhJCsiRipGJkY2Rjg2JEYrRiYjIiN2Ric2JEYqRiZGMjYkRidGJywmIyEkdiYiI01GJkY2Rjg2JEYmRisjIiQrIkYqNiRGJ0YqIyIkRCoiI282JEYqRicjIiR2IkYuNiRGJkYqRjI2JEYnRitGMjYkRidGJkYyNiRGJkYnRjI2JEYrRitGNA==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L30" 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="LUkndmVjdG9yRzYiNiM3JiIiIUYnRicjISREIyIjbw==">PTYiNiM7IiIiIiIlRVxbbCVGJiIiISIiI0YpIiIkRilGJyMhJEQjIiNv</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L31" 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="L32" 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="LUknbWF0cml4RzYiNiM3JjcjIiIhRidGJzcjIyEkRCMiI28=">PTYiNiQ7IiIiIiIlO0YmRiZFXFtsJTYkRiZGJiIiITYkIiIjRiZGKzYkIiIkRiZGKzYkRidGJiMhJEQjIiNv</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L33" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal"> The solution is obtained as</Text-field>
</Input>
</Group>
<Group labelreference="L34" 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="PkkiWEc2Ii1JJ21hdHJpeEdGJDYjNyY3IywkKiYiKV03LiMpIiIiLCxGLEYtKiYiKURKcWdGLSlJJHBoaUdGJCIiI0YtRi0qJiIoK11CJkYtKUYyIiIlRi1GLSomIicrRTdGLSlGMiIiJ0YtRi0qJiIkOylGLSlGMiIiKUYtRi0hIiJGLTcjLCQqKCMiKEQxayJGM0YtLCYiJCsiRi0qJiIiJEYtRjFGLUYtRi1GLkZARi03IywkKigiJl1pI0YtLCgiJURKRi0qJiIkXSNGLUYxRi1GLSomRklGLUY2Ri1GLUYtRi5GQEYtNyMsJCooIyIjdkYzRi0sKiIoK3Y9I0YtKiYiJ0QiRyRGLUYxRi1GLSomIiUrJypGLUY2Ri1GLSomIiNzRi1GOkYtRi1GLUYuRkBGLQ==">PTYiNiQ7IiIiIiIlO0YmRiZFXFtsJTYkRiZGJiwkKiQsLCIpXTcuIylGJiokSSRwaGlHRiMiIiMiKURKcWcqJEYwRiciKCtdQiYqJEYwIiInIicrRTcqJEYwIiIpIiQ7KSEiIkYuNiRGMUYmLCQqJiwmIiQrIkYmRi8iIiRGJkYtRjsjIihEMWsiRjE2JEZBRiYsJComLCgiJURKRiZGLyIkXSNGM0ZBRiZGLUY7IiZdaSM2JEYnRiYsJComLCoiKCt2PSNGJkYvIidEIkckRjMiJSsnKkY1IiNzRiZGLUY7IyIjdkYx</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L35" 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="PiZJInlHNiI2IyIiIiwkKiYiKV03LiMpRicsLEYqRicqJiIpREpxZ0YnKUkkcGhpR0YlIiIjRidGJyomIigrXUImRicpRi8iIiVGJ0YnKiYiJytFN0YnKUYvIiInRidGJyomIiQ7KUYnKUYvIiIpRidGJyEiIkYn">LCQqJCwsIildNy4jKSIiIiokSSRwaGlHNiIiIiMiKURKcWcqJEYoIiIlIigrXUImKiRGKCIiJyInK0U3KiRGKCIiKSIkOykhIiJGJQ==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJInlHNiI2IyIiIywkKigjIihEMWsiRiciIiIsJiIkKyJGLComIiIkRiwpSSRwaGlHRiVGJ0YsRixGLCwsIildNy4jKUYsKiYiKURKcWdGLEYxRixGLComIigrXUImRiwpRjIiIiVGLEYsKiYiJytFN0YsKUYyIiInRixGLComIiQ7KUYsKUYyIiIpRixGLCEiIkYs">LCQqJiwmIiQrIiIiIiokSSRwaGlHNiIiIiMiIiRGJiwsIildNy4jKUYmRiciKURKcWcqJEYoIiIlIigrXUImKiRGKCIiJyInK0U3KiRGKCIiKSIkOykhIiIjIihEMWsiRio=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJInlHNiI2IyIiJCwkKigiJl1pIyIiIiwoIiVESkYrKiYiJF0jRispSSRwaGlHRiUiIiNGK0YrKiZGJ0YrKUYxIiIlRitGK0YrLCwiKV03LiMpRisqJiIpREpxZ0YrRjBGK0YrKiYiKCtdQiZGK0Y0RitGKyomIicrRTdGKylGMSIiJ0YrRisqJiIkOylGKylGMSIiKUYrRishIiJGKw==">LCQqJiwoIiVESiIiIiokSSRwaGlHNiIiIiMiJF0jKiRGKCIiJSIiJEYmLCwiKV03LiMpRiZGJyIpREpxZ0YsIigrXUImKiRGKCIiJyInK0U3KiRGKCIiKSIkOykhIiIiJl1pIw==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJInlHNiI2IyIiJSwkKigjIiN2IiIjIiIiLCoiKCt2PSNGLSomIidEIkckRi0pSSRwaGlHRiVGLEYtRi0qJiIlKycqRi0pRjNGJ0YtRi0qJiIjc0YtKUYzIiInRi1GLUYtLCwiKV03LiMpRi0qJiIpREpxZ0YtRjJGLUYtKiYiKCtdQiZGLUY2Ri1GLSomIicrRTdGLUY5Ri1GLSomIiQ7KUYtKUYzIiIpRi1GLSEiIkYt">LCQqJiwqIigrdj0jIiIiKiRJJHBoaUc2IiIiIyInRCJHJCokRigiIiUiJSsnKiokRigiIiciI3NGJiwsIildNy4jKUYmRiciKURKcWdGLCIoK11CJkYvIicrRTcqJEYoIiIpIiQ7KSEiIiMiI3ZGKg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L36" 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="PiZJInlHNiI2IyIiISwmKigjIid2b2EiIiMiIiIsJiIkKyJGLSomIiIkRi0pSSRwaGlHRiVGLEYtRi1GLSwsIildNy4jKUYtKiYiKURKcWdGLUYyRi1GLSomIigrXUImRi0pRjMiIiVGLUYtKiYiJytFN0YtKUYzIiInRi1GLSomIiQ7KUYtKUYzIiIpRi1GLSEiIkZEKiYiKitdUDQiRi1GNEZERi0=">LCYqJiwmIiQrIiIiIiokSSRwaGlHNiIiIiMiIiRGJiwsIildNy4jKUYmRiciKURKcWcqJEYoIiIlIigrXUImKiRGKCIiJyInK0U3KiRGKCIiKSIkOykhIiIjISd2b2FGKiokRixGOCIqK11QNCI=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJInlHNiI2IyIiJiwoKigjIiddNzgiIzwiIiIsKCIlREpGLSomIiRdI0YtKUkkcGhpR0YlIiIjRi1GLSomIiIkRi0pRjMiIiVGLUYtRi0sLCIpXTcuIylGLSomIilESnFnRi1GMkYtRi0qJiIoK11CJkYtRjdGLUYtKiYiJytFN0YtKUYzIiInRi1GLSomIiQ7KUYtKUYzIiIpRi1GLSEiIkZHKigjIiRdKEYsRi0sKiIoK3Y9I0YtKiYiJ0QiRyRGLUYyRi1GLSomIiUrJypGLUY3Ri1GLSomIiNzRi1GQUYtRi1GLUY5RkdGLSNGNEYsRi0=">LCgqJiwoIiVESiIiIiokSSRwaGlHNiIiIiMiJF0jKiRGKCIiJSIiJEYmLCwiKV03LiMpRiZGJyIpREpxZ0YsIigrXUImKiRGKCIiJyInK0U3KiRGKCIiKSIkOykhIiIjISddNzgiIzwqJiwqIigrdj0jRiZGJyInRCJHJEYsIiUrJypGMyIjc0YmRi9GOSMiJF0oRjwjRipGPEYm</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L37" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal"><Font encoding="UTF-8"> Now the result obtained is plotted for different values of Thiele modulus \316\246:</Font></Text-field>
</Input>
</Group>
<Group labelreference="L49" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">pars:=[0.1,0.5,1,2,5];</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkklcGFyc0c2IjcnJCIiIiEiIiQiIiZGKEYnIiIjRio=">NyckIiIiISIiJCIiJkYlRiQiIiNGJw==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L48" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">clr:=[black,red,green,gold,blue];</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkY2xyRzYiNydJJmJsYWNrR0YkSSRyZWRHRiRJJmdyZWVuR0YkSSVnb2xkR0YkSSVibHVlR0Yk">NydJJmJsYWNrRzYiSSRyZWRHRiRJJmdyZWVuR0YkSSVnb2xkR0YkSSVibHVlR0Yk</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L50" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">for j from 1 to 5 do p[j]:=plot([seq([i*h,subs(phi=pars[j],y[i])],i=0..N+1)],thickness=3,color=clr[j]):od:
</Text-field>
</Input>
</Group>
<Group labelreference="L51" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">pt:=textplot([seq([3*h,evalf(subs(phi=pars[j],y[3])-0.02),phi=pars[j]],j=1..5)]):</Text-field>
</Input>
</Group>
<Group labelreference="L47" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">display({seq(p[i],i=1..5),pt},title=&quot;Figure Exp 3.1.13.&quot;,axes=boxed,labels=[x,y]);</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, 17084 characters]</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L46" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"></Text-field>
</Input>
</Group>
</Worksheet>