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<Group labelreference="L32" drawlabel="true">
<Input>
<Text-field style="Heading 3" italic="false" layout="Heading 3"><Font italic="false">Example 2.2.3b  Diffusion with a Second Order Reaction</Font></Text-field>
</Input>
</Group>
<Group labelreference="L34" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Example 3.2.1 is solved using finite differences in Maple below.  The program developed for example 3.8 is modified to solve this example.  (Note that y is used as the dependent variable instead of c.)  </Text-field>
</Input>
</Group>
<Group labelreference="L33" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">restart:</Text-field>
</Input>
</Group>
<Group labelreference="L3" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">with(plots):</Text-field>
</Input>
</Group>
<Group labelreference="L35" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The number of node points is entered here:</Text-field>
</Input>
</Group>
<Group labelreference="L4" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">N:=10;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkiTkc2IiIjNQ==">IiM1</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L36" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The length of the domain is entered here:</Text-field>
</Input>
</Group>
<Group labelreference="L5" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">L:=1;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkiTEc2IiIiIg==">IiIi</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L37" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The governing equation is entered below:</Text-field>
</Input>
</Group>
<Group labelreference="L6" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq:=diff(y(x),x$2)-Phi^2*y(x)^2;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjZXFHNiIsJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5R0YkNiNJInhHRiQtSSIkR0YoNiRGLSIiIyIiIiomKUkkUGhpR0YkRjFGMilGKkYxRjIhIiI=">LCYtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUYkNiQtSSJ5RzYiNiNJInhHRitGLUYtIiIiKiYpSSRQaGlHRisiIiNGLilGKUYyRi4hIiI=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L38" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The boundary conditions are entered here:</Text-field>
</Input>
</Group>
<Group labelreference="L7" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc1:=diff(y(x),x);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMxRzYiLUklZGlmZkclKnByb3RlY3RlZEc2JC1JInlHRiQ2I0kieEdGJEYs">LUklZGlmZkclKnByb3RlY3RlZEc2JC1JInlHNiI2I0kieEdGKEYq</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L8" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc2:=y(x)-1;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMyRzYiLCYtSSJ5R0YkNiNJInhHRiQiIiJGKiEiIg==">LCYtSSJ5RzYiNiNJInhHRiUiIiJGKCEiIg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L9" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal">Next, a general program is written to convert the governing equation and the boundary conditions to finite difference form.  The central difference expression for the second and first derivatives are:</Text-field>
</Input>
</Group>
<Group labelreference="L10" 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==">KiYsKCZJInlHNiI2IywmSSJtR0YmIiIiRipGKkYqKiYiIiNGKiZGJTYjRilGKiEiIiZGJTYjLCZGKUYqRipGL0YqRiopSSJoR0YmRixGLw==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L11" 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=">LCQqKCMiIiIiIiNGJSwmJkkieUc2IjYjLCZJIm1HRipGJUYlRiVGJSZGKTYjLCZGLUYlRiUhIiJGMUYlSSJoR0YqRjFGJQ==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L12" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal"> Three point forward and backward difference expressions for the derivative are:</Text-field>
</Input>
</Group>
<Group labelreference="L13" 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>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkmZHlkeGZHNiIsJCooIyIiIiIiI0YoLCgmSSJ5R0YkNiNGKSEiIiomIiIlRigmRiw2I0YoRihGKComIiIkRigmRiw2IyIiIUYoRi5GKEkiaEdGJEYuRig=">LCQqKCMiIiIiIiNGJSwoJkkieUc2IjYjRiYhIiIqJiIiJUYlJkYpNiNGJUYlRiUqJiIiJEYlJkYpNiMiIiFGJUYsRiVJImhHRipGLEYl</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L14" 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>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkmZHlkeGJHNiIsJCooIyIiIiIiI0YoLCgmSSJ5R0YkNiMiIipGKComIiIlRigmRiw2IyIjNUYoISIiKiYiIiRGKCZGLDYjIiM2RihGKEYoSSJoR0YkRjRGKA==">LCQqKCMiIiIiIiNGJSwoJkkieUc2IjYjIiIqRiUqJiIiJUYlJkYpNiMiIzVGJSEiIiomIiIkRiUmRik2IyIjNkYlRiVGJUkiaEdGKkYyRiU=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L15" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal"> The governing equation in finite difference form is:</Text-field>
</Input>
</Group>
<Group labelreference="L16" 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="PiZJI0VxRzYiNiNJIm1HRiUsJiomLCgmSSJ5R0YlNiMsJkYnIiIiRi9GL0YvKiYiIiNGLyZGLEYmRi8hIiImRiw2IywmRidGL0YvRjNGL0YvKUkiaEdGJUYxRjNGLyomKUkkUGhpR0YlRjFGLylGMkYxRi9GMw==">LCYqJiwoJkkieUc2IjYjLCZJIm1HRiciIiJGK0YrRisqJiIiI0YrJkYmNiNGKkYrISIiJkYmNiMsJkYqRitGK0YwRitGKylJImhHRidGLUYwRisqJilJJFBoaUdGJ0YtRispRi5GLUYrRjA=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L17" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal"> The boundary conditions in finite difference form are:</Text-field>
</Input>
</Group>
<Group labelreference="L18" 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>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiEsJCooIyIiIiIiI0YrLCgmSSJ5R0YlNiNGLCEiIiomIiIlRismRi82I0YrRitGKyomIiIkRismRi9GJkYrRjFGK0kiaEdGJUYxRis=">LCQqKCMiIiIiIiNGJSwoJkkieUc2IjYjRiYhIiIqJiIiJUYlJkYpNiNGJUYlRiUqJiIiJEYlJkYpNiMiIiFGJUYsRiVJImhHRipGLEYl</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L19" 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="PiZJI0VxRzYiNiMiIzYsJiZJInlHRiVGJiIiIkYrISIi">LCYmSSJ5RzYiNiMiIzYiIiJGKCEiIg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L20" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal"> A 'for loop' can be written for the interior node points as</Text-field>
</Input>
</Group>
<Group labelreference="L21" 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="PiZJI0VxRzYiNiMiIiIsJiomLCgmSSJ5R0YlNiMiIiNGJyomRi5GJyZGLEYmRichIiImRiw2IyIiIUYnRicpSSJoR0YlRi5GMUYnKiYpSSRQaGlHRiVGLkYnKUYwRi5GJ0Yx">LCYqJiwoJkkieUc2IjYjIiIjIiIiKiZGKUYqJkYmNiNGKkYqISIiJkYmNiMiIiFGKkYqKUkiaEdGJ0YpRi5GKiomKUkkUGhpR0YnRilGKilGLEYpRipGLg==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiMsJiomLCgmSSJ5R0YlNiMiIiQiIiIqJkYnRi8mRixGJkYvISIiJkYsNiNGL0YvRi8pSSJoR0YlRidGMkYvKiYpSSRQaGlHRiVGJ0YvKUYxRidGL0Yy">LCYqJiwoJkkieUc2IjYjIiIkIiIiKiYiIiNGKiZGJjYjRixGKiEiIiZGJjYjRipGKkYqKUkiaEdGJ0YsRi9GKiomKUkkUGhpR0YnRixGKilGLUYsRipGLw==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiQsJiomLCgmSSJ5R0YlNiMiIiUiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjRjFGL0YvKUkiaEdGJUYxRjNGLyomKUkkUGhpR0YlRjFGLylGMkYxRi9GMw==">LCYqJiwoJkkieUc2IjYjIiIlIiIiKiYiIiNGKiZGJjYjIiIkRiohIiImRiY2I0YsRipGKilJImhHRidGLEYwRioqJilJJFBoaUdGJ0YsRiopRi1GLEYqRjA=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiUsJiomLCgmSSJ5R0YlNiMiIiYiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiIkRi9GLylJImhHRiVGMUYzRi8qJilJJFBoaUdGJUYxRi8pRjJGMUYvRjM=">LCYqJiwoJkkieUc2IjYjIiImIiIiKiYiIiNGKiZGJjYjIiIlRiohIiImRiY2IyIiJEYqRiopSSJoR0YnRixGMEYqKiYpSSRQaGlHRidGLEYqKUYtRixGKkYw</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiYsJiomLCgmSSJ5R0YlNiMiIiciIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiIlRi9GLylJImhHRiVGMUYzRi8qJilJJFBoaUdGJUYxRi8pRjJGMUYvRjM=">LCYqJiwoJkkieUc2IjYjIiInIiIiKiYiIiNGKiZGJjYjIiImRiohIiImRiY2IyIiJUYqRiopSSJoR0YnRixGMEYqKiYpSSRQaGlHRidGLEYqKUYtRixGKkYw</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIicsJiomLCgmSSJ5R0YlNiMiIigiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiImRi9GLylJImhHRiVGMUYzRi8qJilJJFBoaUdGJUYxRi8pRjJGMUYvRjM=">LCYqJiwoJkkieUc2IjYjIiIoIiIiKiYiIiNGKiZGJjYjIiInRiohIiImRiY2IyIiJkYqRiopSSJoR0YnRixGMEYqKiYpSSRQaGlHRidGLEYqKUYtRixGKkYw</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIigsJiomLCgmSSJ5R0YlNiMiIikiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiInRi9GLylJImhHRiVGMUYzRi8qJilJJFBoaUdGJUYxRi8pRjJGMUYvRjM=">LCYqJiwoJkkieUc2IjYjIiIpIiIiKiYiIiNGKiZGJjYjIiIoRiohIiImRiY2IyIiJ0YqRiopSSJoR0YnRixGMEYqKiYpSSRQaGlHRidGLEYqKUYtRixGKkYw</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiksJiomLCgmSSJ5R0YlNiMiIioiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiIoRi9GLylJImhHRiVGMUYzRi8qJilJJFBoaUdGJUYxRi8pRjJGMUYvRjM=">LCYqJiwoJkkieUc2IjYjIiIqIiIiKiYiIiNGKiZGJjYjIiIpRiohIiImRiY2IyIiKEYqRiopSSJoR0YnRixGMEYqKiYpSSRQaGlHRidGLEYqKUYtRixGKkYw</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiosJiomLCgmSSJ5R0YlNiMiIzUiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiIpRi9GLylJImhHRiVGMUYzRi8qJilJJFBoaUdGJUYxRi8pRjJGMUYvRjM=">LCYqJiwoJkkieUc2IjYjIiM1IiIiKiYiIiNGKiZGJjYjIiIqRiohIiImRiY2IyIiKUYqRiopSSJoR0YnRixGMEYqKiYpSSRQaGlHRidGLEYqKUYtRixGKkYw</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIzUsJiomLCgmSSJ5R0YlNiMiIzYiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiIqRi9GLylJImhHRiVGMUYzRi8qJilJJFBoaUdGJUYxRi8pRjJGMUYvRjM=">LCYqJiwoJkkieUc2IjYjIiM2IiIiKiYiIiNGKiZGJjYjIiM1RiohIiImRiY2IyIiKkYqRiopSSJoR0YnRixGMEYqKiYpSSRQaGlHRidGLEYqKUYtRixGKkYw</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L39" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The node spacing is given by:</Text-field>
</Input>
</Group>
<Group labelreference="L22" 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="PkkiaEc2IiMiIiIiIzY=">IyIiIiIjNg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L40" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">The value for \316\246 is sustained  in the governing equations.  The governing equations are stored in eqs:</Font></Text-field>
</Input>
</Group>
<Group labelreference="L23" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eqs:=seq(eval(subs(Phi=1,Eq[i])),i=0..N+1);</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="L41" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The variables are stored in vars:</Text-field>
</Input>
</Group>
<Group labelreference="L24" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">vars:=seq(y[i],i=0..N+1);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkldmFyc0c2IjYuJkkieUdGJDYjIiIhJkYnNiMiIiImRic2IyIiIyZGJzYjIiIkJkYnNiMiIiUmRic2IyIiJiZGJzYjIiInJkYnNiMiIigmRic2IyIiKSZGJzYjIiIqJkYnNiMiIzUmRic2IyIjNg==">Ni4mSSJ5RzYiNiMiIiEmRiQ2IyIiIiZGJDYjIiIjJkYkNiMiIiQmRiQ2IyIiJSZGJDYjIiImJkYkNiMiIicmRiQ2IyIiKCZGJDYjIiIpJkYkNiMiIiomRiQ2IyIjNSZGJDYjIiM2</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L42" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The 'fsolve' command sometimes gives negative values when guess values for the dependent variables are not provided.  To avoid this, an initial guess of 1 is provided:</Text-field>
</Input>
</Group>
<Group labelreference="L25" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">fsolve({eqs},{vars});</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">PC4vJkkieUc2IjYjIiM2JCIrKysrKzUhIiovJkYlNiMiIiMkIit3Qy4zcyEjNS8mRiU2IyIiISQiKzVsbEJyRjIvJkYlNiMiIiIkIissMHZXckYyLyZGJTYjIiIqJCIrZzUjeScqKUYyLyZGJTYjIiM1JCIrWz0uWiUqRjIvJkYlNiMiIiQkIit0SUQ5dEYyLyZGJTYjIiIlJCIrUnJva3VGMi8mRiU2IyIiJiQiK3c/PGh3RjIvJkYlNiMiIickIismM2toIXpGMi8mRiU2IyIiKCQiK0deIkc/KUYyLyZGJTYjIiIpJCIrImV1XWIpRjI=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L26" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">vars:=seq(y[i]=1,i=0..N+1);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkldmFyc0c2IjYuLyZJInlHRiQ2IyIiISIiIi8mRig2I0YrRisvJkYoNiMiIiNGKy8mRig2IyIiJEYrLyZGKDYjIiIlRisvJkYoNiMiIiZGKy8mRig2IyIiJ0YrLyZGKDYjIiIoRisvJkYoNiMiIilGKy8mRig2IyIiKkYrLyZGKDYjIiM1RisvJkYoNiMiIzZGKw==">Ni4vJkkieUc2IjYjIiIhIiIiLyZGJTYjRilGKS8mRiU2IyIiI0YpLyZGJTYjIiIkRikvJkYlNiMiIiVGKS8mRiU2IyIiJkYpLyZGJTYjIiInRikvJkYlNiMiIihGKS8mRiU2IyIiKUYpLyZGJTYjIiIqRikvJkYlNiMiIzVGKS8mRiU2IyIjNkYp</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L27" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">sol:=fsolve({eqs},{vars});</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pkkkc29sRzYiPC4vJkkieUdGJDYjIiIjJCIrd0MuM3MhIzUvJkYoNiMiIiEkIis1bGxCckYtLyZGKDYjIiIiJCIrLDB2V3JGLS8mRig2IyIiKiQiK2c1I3knKilGLS8mRig2IyIjNSQiK1s9LlolKkYtLyZGKDYjIiIkJCIrdElEOXRGLS8mRig2IyIiJSQiK1Jyb2t1Ri0vJkYoNiMiIiYkIit3Pzxod0YtLyZGKDYjIiInJCIrJjNraCF6Ri0vJkYoNiMiIigkIitHXiJHPylGLS8mRig2IyIiKSQiKyJldV1iKUYtLyZGKDYjIiM2JEY3RjE=">PC4vJkkieUc2IjYjIiIjJCIrd0MuM3MhIzUvJkYlNiMiIiEkIis1bGxCckYrLyZGJTYjIiIiJCIrLDB2V3JGKy8mRiU2IyIiKiQiK2c1I3knKilGKy8mRiU2IyIjNSQiK1s9LlolKkYrLyZGJTYjIiIkJCIrdElEOXRGKy8mRiU2IyIiJSQiK1Jyb2t1RisvJkYlNiMiIiYkIit3Pzxod0YrLyZGJTYjIiInJCIrJjNraCF6RisvJkYlNiMiIigkIitHXiJHPylGKy8mRiU2IyIiKSQiKyJldV1iKUYrLyZGJTYjIiM2JEY1Ri8=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L43" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The solution obtained is assigned and plotted:</Text-field>
</Input>
</Group>
<Group labelreference="L28" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">assign(sol):</Text-field>
</Input>
</Group>
<Group labelreference="L29" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">plot([seq([i*h,y[i]],i=0..N+1)],thickness=4,axes=boxed,title=&quot;Figure Exp. 3.2.6.&quot;,labels=[x,y]);</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="L44" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The value for y[0] obtained with N=10 interior node points:</Text-field>
</Input>
</Group>
<Group labelreference="L30" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">y[0];</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">JCIrNWxsQnIhIzU=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L31" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"></Text-field>
</Input>
</Group>
<Group labelreference="L46" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Hence, we conclude that the solution obtained has converged.  The finite difference solution is an easy technique to apply.  However, the number of node points required might increase drastically for stiff boundary value problems.</Text-field>
</Input>
</Group>
<Group labelreference="L45" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group>
</Worksheet>