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    <Annotation-category name="&lt;default&gt;">
    </Annotation-category>
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<Task>
</Task>
<Group labelreference="L62" drawlabel="true">
<Input>
<Text-field style="Heading 3" italic="false" layout="Heading 3"><Font italic="false">Example 3.2.5  Multiple Steady States in a Catalyst Pellet</Font></Text-field>
</Input>
</Group>
<Group labelreference="L63" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The catalyst pellet problem solved in example 3.2.2 is solved here using the shooting technique.  The Maple program is given below:</Text-field>
</Input>
</Group>
<Group labelreference="L61" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">restart:</Text-field>
</Input>
</Group>
<Group labelreference="L4" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">with(plots):</Text-field>
</Input>
</Group>
<Group labelreference="L64" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The governing equation is entered here (after substituting the parameter values):</Text-field>
</Input>
</Group>
<Group labelreference="L5" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq:=diff(y(x),x$2)-0.04*y(x)*exp(16*(1-y(x))/(1+0.8*(1-y(x))));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjZXFHNiIsJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5R0YkNiNJInhHRiQtSSIkR0YoNiRGLSIiIyIiIiooJCIiJSEiI0YyRipGMi1JJGV4cEc2JEYoSShfc3lzbGliR0YkNiMsJCooIiM7RjIsJkYyRjJGKiEiIkYyLCYkIiM9RkBGMiomJCIiKUZARjJGKkYyRkBGQEYyRjJGQA==">LCYtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUYkNiQtSSJ5RzYiNiNJInhHRitGLUYtIiIiKiZGKUYuLUkkZXhwRzYkRiVJKF9zeXNsaWJHRis2IywkKiYsJkYuRi5GKSEiIkYuLCYkIiM9RjhGLkYpJCEiKUY4RjgiIztGLiQhIiUhIiM=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L6" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eqalpha:=subs(y(x)=Y(x,alpha),eq):</Text-field>
</Input>
</Group>
<Group labelreference="L7" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eqalpha:=diff(eqalpha,alpha):</Text-field>
</Input>
</Group>
<Group labelreference="L8" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eqalpha:=subs(diff(Y(x,alpha),alpha)=y2(x),eqalpha):</Text-field>
</Input>
</Group>
<Group labelreference="L65" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The sensitivity equation is:</Text-field>
</Input>
</Group>
<Group labelreference="L9" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eqalpha:=subs(Y(x,alpha)=y(x),eqalpha);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkoZXFhbHBoYUc2IiwoLUklZGlmZkclKnByb3RlY3RlZEc2JC1JI3kyR0YkNiNJInhHRiQtSSIkR0YoNiRGLSIiIyIiIiooJCIiJSEiI0YyRipGMi1JJGV4cEc2JEYoSShfc3lzbGliR0YkNiMsJCooIiM7RjIsJkYyRjItSSJ5R0YkRiwhIiJGMiwmJCIjPUZCRjIqJiQiIilGQkYyRkBGMkZCRkJGMkYyRkIqKkY0RjJGQEYyLCYqKEY+RjJGKkYyRkNGQkZCKiokIiRHIkZCRjJGP0YyKUZDRjFGQkYqRjJGMkYyRjdGMkZC">LCgtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUYkNiQtSSN5Mkc2IjYjSSJ4R0YrRi1GLSIiIiomRilGLi1JJGV4cEc2JEYlSShfc3lzbGliR0YrNiMsJComLCZGLkYuLUkieUdGK0YsISIiRi4sJiQiIz1GOkYuRjgkISIpRjpGOiIjO0YuJCEiJSEiIyooRjhGLiwmKiZGKUYuRjtGOiEjOyooRjdGLkY7RkNGKUYuJCIkRyJGOkYuRjBGLkZB</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L66" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The variables are stored in vars:</Text-field>
</Input>
</Group>
<Group labelreference="L10" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">vars:=(y(x),y2(x));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkldmFyc0c2IjYkLUkieUdGJDYjSSJ4R0YkLUkjeTJHRiRGKA==">NiQtSSJ5RzYiNiNJInhHRiUtSSN5MkdGJUYm</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L68" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The governing equations are stored in eqs:</Text-field>
</Input>
</Group>
<Group labelreference="L11" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eqs:=(eq,eqalpha);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXFzRzYiNiQsJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5R0YkNiNJInhHRiQtSSIkR0YpNiRGLiIiIyIiIiooJCIiJSEiI0YzRitGMy1JJGV4cEc2JEYpSShfc3lzbGliR0YkNiMsJCooIiM7RjMsJkYzRjNGKyEiIkYzLCYkIiM9RkFGMyomJCIiKUZBRjNGK0YzRkFGQUYzRjNGQSwoLUYoNiQtSSN5MkdGJEYtRi9GMyooRjVGM0ZLRjNGOEYzRkEqKkY1RjNGK0YzLCYqKEY/RjNGS0YzRkJGQUZBKiokIiRHIkZBRjNGQEYzKUZCRjJGQUZLRjNGM0YzRjhGM0ZB">NiQsJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtRiU2JC1JInlHNiI2I0kieEdGLEYuRi4iIiIqJkYqRi8tSSRleHBHNiRGJkkoX3N5c2xpYkdGLDYjLCQqJiwmRi9GL0YqISIiRi8sJiQiIz1GOUYvRiokISIpRjlGOSIjO0YvJCEiJSEiIywoLUYlNiQtRiU2JC1JI3kyR0YsRi1GLkYuRi8qJkZIRi9GMUYvRkAqKEYqRi8sJiomRkhGL0Y6RjkhIzsqKEY4Ri9GOkZCRkhGLyQiJEciRjlGL0YxRi9GQA==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L69" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">The boundary value problem has multiple solutions.  The solution obtained depends on the initial guess provided for \316\261.  An initial guess of 0.9 is given:</Font></Text-field>
</Input>
</Group>
<Group labelreference="L12" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">alpha0:=0.9;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkknYWxwaGEwRzYiJCIiKiEiIg==">JCIiKiEiIg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L13" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">ICs:=(y(0)=alpha0,D(y)(0)=0,y2(0)=1,D(y2)(0)=0);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkSUNzRzYiNiYvLUkieUdGJDYjIiIhJCIiKiEiIi8tLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRihGKUYqLy1JI3kyR0YkRikiIiIvLS1GMTYjRjhGKUYq">NiYvLUkieUc2IjYjIiIhJCIiKiEiIi8tLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjRiVGJ0YoLy1JI3kyR0YmRiciIiIvLS1GLzYjRjZGJ0Yo</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L14" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">sol:=dsolve({eqs,ICs},{vars},type=numeric,abserr=1e-10);</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="L15" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">sol(1);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NycvSSJ4RzYiJCIiIiIiIS8tSSJ5R0YlNiNGJCQiMydcOzBtJ3BFJm8qISM9Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRGKkYkJCIzIz4oZlpOYVEoPiJGLy8tSSN5MkdGJUYsJCIzOipHLjQ6L3VJI0YvLy1GMjYkRjhGJCQhMydlUyxhUzYyPyIhIzw=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L16" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">ypred:=rhs(sol(1)[2]);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkmeXByZWRHNiIkIjMnXDswbSdwRSZvKiEjPQ==">JCIzJ1w7MG0ncEUmbyohIz0=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L17" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">y2pred:=rhs(sol(1)[4]);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkneTJwcmVkRzYiJCIzOipHLjQ6L3VJIyEjPQ==">JCIzOipHLjQ6L3VJIyEjPQ==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L70" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">The new value of \316\261 is obtained as:</Font></Text-field>
</Input>
</Group>
<Group labelreference="L18" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">alpha1:=alpha0+(1-ypred)/y2pred;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkknYWxwaGExRzYiJCIrWThTTzUhIio=">JCIrWThTTzUhIio=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L85" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">For this example, the error is calculated based on the boundary condition at x = 1.</Text-field>
</Input>
</Group>
<Group labelreference="L19" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">err:=abs(1-ypred);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXJyRzYiJCIqTEl0OSQhIzU=">JCIqTEl0OSQhIzU=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L20" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">alpha0:=alpha1;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkknYWxwaGEwRzYiJCIrWThTTzUhIio=">JCIrWThTTzUhIio=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L21" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">k:=1;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pkkia0c2IiIiIg==">IiIi</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L71" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The iteration is performed until the error becomes less than the tolerance limit 1e - 10.</Text-field>
</Input>
</Group>
<Group labelreference="L22" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">tol:=1e-10;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkdG9sRzYiJCIiIiEjNQ==">JCIiIiEjNQ==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L23" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">while err&gt; tol do</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">ICs:=(y(0)=alpha0,D(y)(0)=0,y2(0)=1,D(y2)(0)=0);</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">sol:=dsolve({eqs,ICs},{vars},type=numeric);</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">ypred:=rhs(sol(1)[2]);</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">y2pred:=rhs(sol(1)[4]);</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">alpha1:=alpha0+(1-ypred)/y2pred;</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">err:=abs(1-ypred);</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">alpha0:=alpha1;k:=k+1;</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">end:</Text-field>
</Input>
</Group>
<Group labelreference="L24" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">k;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">IiIn</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L72" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The problem has converged after six iterations.  The concentration at the center of the particle (x = 0) is given by:</Text-field>
</Input>
</Group>
<Group labelreference="L25" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">alpha1;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">JCIrc0xBPCgqISM1</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L73" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The error obtained is:</Text-field>
</Input>
</Group>
<Group labelreference="L26" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">err;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">JCIiIUYj</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L74" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Next, the solution obtained is plotted and stored in p1.</Text-field>
</Input>
</Group>
<Group labelreference="L27" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">p1:=odeplot(sol,[x,y(x)],0..1,axes=boxed,thickness=3,color=blue):</Text-field>
</Input>
</Group>
<Group labelreference="L76" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The same steps are performed for a different initial guess of 0.5.  The solution obtained is stored in p2.</Text-field>
</Input>
</Group>
<Group labelreference="L28" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">alpha0:=0.5;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkknYWxwaGEwRzYiJCIiJiEiIg==">JCIiJiEiIg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L29" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">ICs:=(y(0)=alpha0,D(y)(0)=0,y2(0)=1,D(y2)(0)=0);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkSUNzRzYiNiYvLUkieUdGJDYjIiIhJCIiJiEiIi8tLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRihGKUYqLy1JI3kyR0YkRikiIiIvLS1GMTYjRjhGKUYq">NiYvLUkieUc2IjYjIiIhJCIiJiEiIi8tLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjRiVGJ0YoLy1JI3kyR0YmRiciIiIvLS1GLzYjRjZGJ0Yo</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L30" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">sol:=dsolve({eqs,ICs},{vars},type=numeric,abserr=1e-10);</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="L31" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">sol(1);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NycvSSJ4RzYiJCIiIiIiIS8tSSJ5R0YlNiNGJCQiM3kwKzh5Z1MnZSIhIzwvLUklZGlmZkclKnByb3RlY3RlZEc2JEYqRiQkIjNdTWhqKzooPkQiRi8vLUkjeTJHRiVGLCQhMz0sdF0rPjwvT0YvLy1GMjYkRjhGJCQhM3MueWwwTC9WW0Yv</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L32" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">ypred:=rhs(sol(1)[2]);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkmeXByZWRHNiIkIjN5MCs4eWdTJ2UiISM8">JCIzeTArOHlnUydlIiEjPA==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L33" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">y2pred:=rhs(sol(1)[4]);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkneTJwcmVkRzYiJCEzPSx0XSs+PC9PISM8">JCEzPSx0XSs+PC9PISM8</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L34" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">alpha1:=alpha0+(1-ypred)/y2pred;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkknYWxwaGExRzYiJCIrIkc/cWknISM1">JCIrIkc/cWknISM1</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L35" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">err:=abs(1-ypred);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXJyRzYiJCIqeWdTJ2UhIio=">JCIqeWdTJ2UhIio=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L36" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">alpha0:=alpha1;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkknYWxwaGEwRzYiJCIrIkc/cWknISM1">JCIrIkc/cWknISM1</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L37" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">k:=1;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pkkia0c2IiIiIg==">IiIi</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L38" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">while err&gt; tol do</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">ICs:=(y(0)=alpha0,D(y)(0)=0,y2(0)=1,D(y2)(0)=0);</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">sol:=dsolve({eqs,ICs},{vars},type=numeric);</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">ypred:=rhs(sol(1)[2]);</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">y2pred:=rhs(sol(1)[4]);</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">alpha1:=alpha0+(1-ypred)/y2pred;</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">err:=abs(1-ypred);</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">alpha0:=alpha1;k:=k+1;</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">end:</Text-field>
</Input>
</Group>
<Group labelreference="L39" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">k;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">IiIp</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L77" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The problem has converged after eight iterations.  The concentration at the center of the particle (x = 0) is given by:</Text-field>
</Input>
</Group>
<Group labelreference="L40" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">alpha1;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">JCIrRnUoUnQoISM1</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L41" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">err;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">JCIiIUYj</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L42" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">p2:=odeplot(sol,[x,y(x)],0..1,axes=boxed,thickness=3,color=green):</Text-field>
</Input>
</Group>
<Group labelreference="L78" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">Next, an initial guess of 1e - 4 is used.  For this case the updated \316\261 becomes a negative.  Hence, a scaling factor of \317\201=0.2 is used: </Font></Text-field>
</Input>
</Group>
<Group labelreference="L43" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">alpha0:=1e-4;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkknYWxwaGEwRzYiJCIiIiEiJQ==">JCIiIiEiJQ==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L44" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">ICs:=(y(0)=alpha0,D(y)(0)=0,y2(0)=1,D(y2)(0)=0);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkSUNzRzYiNiYvLUkieUdGJDYjIiIhJCIiIiEiJS8tLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRihGKUYqLy1JI3kyR0YkRilGLC8tLUYxNiNGOEYpRio=">NiYvLUkieUc2IjYjIiIhJCIiIiEiJS8tLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjRiVGJ0YoLy1JI3kyR0YmRidGKi8tLUYvNiNGNkYnRig=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L45" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">sol:=dsolve({eqs,ICs},{vars},type=numeric,abserr=1e-10);</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="L46" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">sol(1);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NycvSSJ4RzYiJCIiIiIiIS8tSSJ5R0YlNiNGJCQiMz1wbz0uc3NePyEjPC8tSSVkaWZmRyUqcHJvdGVjdGVkRzYkRipGJCQiMyhIJylcai04dyJSRi8vLUkjeTJHRiVGLCQiM1U+dztCbicqKkgjISM5Ly1GMjYkRjhGJCQhM3ckPS9ES3VcTychIz8=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L47" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">ypred:=rhs(sol(1)[2]);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkmeXByZWRHNiIkIjM9cG89LnNzXj8hIzw=">JCIzPXBvPS5zc14/ISM8</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L48" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">y2pred:=rhs(sol(1)[4]);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkneTJwcmVkRzYiJCIzVT53O0JuJyoqSCMhIzk=">JCIzVT53O0JuJyoqSCMhIzk=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L49" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">alpha1:=alpha0+(1-ypred)/y2pred;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkknYWxwaGExRzYiJCErSUp6c04hIzg=">JCErSUp6c04hIzg=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L50" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">rho:=0.2;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkcmhvRzYiJCIiIyEiIg==">JCIiIyEiIg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L51" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">alpha1:=alpha0+rho*(1-ypred)/y2pred;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkknYWxwaGExRzYiJCIqU1BUYSkhIzk=">JCIqU1BUYSkhIzk=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L52" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">err:=abs(1-ypred);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXJyRzYiJCIrLnNzXjUhIio=">JCIrLnNzXjUhIio=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L53" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">alpha0:=alpha1;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkknYWxwaGEwRzYiJCIqU1BUYSkhIzk=">JCIqU1BUYSkhIzk=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L54" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">k:=1;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pkkia0c2IiIiIg==">IiIi</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L55" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">while err&gt; tol do</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">ICs:=(y(0)=alpha0,D(y)(0)=0,y2(0)=1,D(y2)(0)=0);</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">sol:=dsolve({eqs,ICs},{vars},type=numeric);</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">ypred:=rhs(sol(1)[2]);</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">y2pred:=rhs(sol(1)[4]);</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">alpha1:=alpha0+rho*(1-ypred)/y2pred;</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">err:=abs(1-ypred);</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">alpha0:=alpha1;k:=k+1;</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">end:</Text-field>
</Input>
</Group>
<Group labelreference="L79" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The problem has converged after 93 iterations.  The concentration at the center particle (x = 0) is given by:</Text-field>
</Input>
</Group>
<Group labelreference="L56" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">k;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">IiMkKg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L57" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">alpha1;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">JCIrJ3onW0w1ISM6</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L58" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">err;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">JCIiIUYj</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L59" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">p3:=odeplot(sol,[x,y(x)],0..1,title=&quot;Figure Exp. 3.2.9&quot;,axes=boxed,thickness=4,color=brown):</Text-field>
</Input>
</Group>
<Group labelreference="L86" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">pt:=textplot([[0.2,0.95,&quot;Steady State 1&quot;],[0.2,0.75,&quot;Steady State 2&quot;],[0.2,0.02,&quot;Steady State 3&quot;]]);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjcHRHNiItSS9JTlRFUkZBQ0VfUExPVEdGJDYlLUklVEVYVEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYkNyQkIiIjISIiJCIjJiohIiNRL1N0ZWFkeX5TdGF0ZX4xRiQtRik2JDckRi8kIiN2RjRRL1N0ZWFkeX5TdGF0ZX4yRiQtRik2JDckRi8kRjBGNFEvU3RlYWR5flN0YXRlfjNGJA==">LSUlUExPVEc2JS1JJVRFWFRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JDckJCIiIyEiIiQiIyYqISIjUS9TdGVhZHl+U3RhdGV+MUYqLUYmNiQ3JEYtJCIjdkYyUS9TdGVhZHl+U3RhdGV+MkYqLUYmNiQ3JEYtJEYuRjJRL1N0ZWFkeX5TdGF0ZX4zRio=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L60" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">display({p1,p2,p3,pt});</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, 9844 characters]</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L81" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">Hence, we observe that the shooting technique can predict three multiple states in a catalyst pellet.  The number of iterations required to obtain a converged solution depends on the initial guess and the scaling factor \317\201.</Font></Text-field>
</Input>
</Group>
<Group labelreference="L67" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"></Text-field>
</Input>
</Group>
</Worksheet>