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<Text-field style="Heading 3" italic="false" layout="Heading 3"><Font italic="false">Example 3.2.4  Nonlinear Heat Transfer</Font></Text-field>
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<Text-field style="Text" layout="Normal">Consider a modification of the heat transfer discussed in example 3.1.1 with a nonlinear source term.  The governing equation is<Equation executable="false" style="Text" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDRi8=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDRi8=</Equation></Text-field>
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<Text-field style="Text" layout="Normal" alignment="centred"><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>y                                     (3.2.7)</Text-field>
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<Text-field style="Text" layout="Normal">with the boundary conditions:</Text-field>
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<Text-field style="Text" layout="Normal" alignment="centred"><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>
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<Text-field style="Text" layout="Normal">and</Text-field>
<Text-field style="Text" layout="Normal" alignment="centred"><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>
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<Text-field style="Text" layout="Normal">The initial condition, y(0) is taken as</Text-field>
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<Text-field style="Text" layout="Normal" alignment="centred"><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>                                         (3.2.10)</Text-field>
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<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">An initial value for \316\261 is guessed and the governing equation (3.2.7) is solved as an initial value problem.  Then, a new value of \316\261 is obtained using the following relation.</Font><Font superscript="true">[6] [21]</Font></Text-field>
<Text-field style="Text" layout="Normal" alignment="centred"></Text-field>
<Text-field style="Plot Text" size="12" layout="Normal" alignment="centred"><Font encoding="UTF-8" size="12">\316\261</Font><Font subscript="true" size="10">new</Font><Font encoding="UTF-8" size="12">= \316\261</Font><Font subscript="true" size="10">old</Font><Equation executable="false" style="Plot Text" input-equation="" display="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">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</Equation></Text-field>
<Text-field style="Plot Text" size="12" layout="Normal" alignment="centred"></Text-field>
<Text-field style="Plot Text" size="12" layout="Normal" alignment="centred"><Font size="12">=  <Font encoding="UTF-8">\316\261</Font></Font><Font subscript="true" size="10">old</Font><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>
<Text-field style="Plot Text" size="12" layout="Normal" alignment="centred"><Font size="12">                                                                                                           (3.2.11)</Font></Text-field>
<Text-field style="Plot Text" size="12" layout="Normal" alignment="centred"></Text-field>
<Text-field style="Plot Text" size="12" layout="Normal" alignment="centred"><Font size="8">= </Font><Font size="12"> <Font encoding="UTF-8">\316\261</Font></Font><Font subscript="true" size="10">old</Font><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>
<Text-field style="Plot Text" layout="Normal" alignment="centred"></Text-field>
<Text-field style="Text" layout="Normal" alignment="centred">	<Font superscript="true">	</Font></Text-field>
<Text-field style="Text" layout="Normal">In equation (3.2.11),  the Jacobian  <Equation executable="false" style="Text" input-equation="" display="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">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</Equation> is often calculated numerically.  However, the Jacobian can be predicted exactly by differentiating the governing equation (3.2.7) with respect to alpha as</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal" alignment="centred"><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>                   (3.2.12)</Text-field>
<Text-field style="Text" layout="Normal" alignment="centred"></Text-field>
<Text-field style="Text" layout="Normal">Next, the Jacobian is treated as a variable y2.</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal" alignment="centred"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY8LUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGKy1JJm1mcmFjR0YkNigtRiM2Jy1GLDYtUTAmRGlmZmVyZW50aWFsRDtGJ0YvL0YzUSZ1bnNldEYnL0Y2Rk8vRjhGTy9GOkZPL0Y8Rk8vRj5GTy9GQEZPRkFGRC1JJW1zdXBHRiQ2JS1JI21pR0YkNiNRIUYnLUYjNiQtSSNtbkdGJDYkUSIyRidGL0YvLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GVzYlLUZaNiVRInlGJy8lJ2l0YWxpY0dGNEYvLUYjNiRGWUYvRl1vRmluRi8tRiM2J0ZLRistRlc2JS1GWjYlUSJ4RidGZW9GL0ZnbkZdb0ZZRi8vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmVwLyUpYmV2ZWxsZWRHRjRGKy1GLDYtUSI9RidGL0YyRjVGN0Y5RjtGPUY/L0ZCUSwwLjI3Nzc3NzhlbUYnL0ZFRl5xLUkobWZlbmNlZEdGJDYkLUYjNigtRmpuNiRRIjFGJ0YvLUYsNi1RIitGJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjIyMjIyMmVtRicvRkVGXHItRmpuNiRRJDAuMUYnRi9GKy1GWjYlUSN5MkYnRmVvRi9GL0YvRi8=">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</Equation>y  +  <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>             (3.2.13)</Text-field>
<Text-field style="Text" layout="Normal" alignment="centred"></Text-field>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">The initial conditions for y2 are obtained by differentiating equations (3.2.10) and (3.2.8) with respect to \316\261:</Font></Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">	 	                                                     <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYuLUkjbWlHRiQ2JVEjeTJGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtbkdGJDYkUSIwRidGMkYyRjItSSNtb0dGJDYtUSI9RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlItRjs2JFEiMUYnRjItRj82LVEifkYnRjJGQkZERkZGSEZKRkxGTi9GUVEmMC4wZW1GJy9GVEZmbkZYRlhGWEZYRlhGWEYy">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</Equation>                                    (3.2.14)</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">and</Text-field>
<Text-field style="Text" layout="Normal">	 	                                                   <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">This boundary value problem is solved in Maple below:</Text-field>
</Input>
</Group>
<Group labelreference="L26" 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="L29" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Enter the governing equation:</Text-field>
</Input>
</Group>
<Group labelreference="L4" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq:=diff(y(x),x$2)-(1+0.1*y(x))*y(x);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjZXFHNiIsJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5R0YkNiNJInhHRiQtSSIkR0YoNiRGLSIiIyIiIiomLCZGMkYyKiYkRjIhIiJGMkYqRjJGMkYyRipGMkY3">LCYtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUYkNiQtSSJ5RzYiNiNJInhHRitGLUYtIiIiKiYsJkYuRi5GKSRGLiEiIkYuRilGLkYy</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L30" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">The sensitivity equation is developed by treating y as a function of x and \316\261:</Font></Text-field>
</Input>
</Group>
<Group labelreference="L5" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eqalpha:=subs(y(x)=Y(x,alpha),eq);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkoZXFhbHBoYUc2IiwmLUklZGlmZkclKnByb3RlY3RlZEc2JC1JIllHRiQ2JEkieEdGJEkmYWxwaGFHRiQtSSIkR0YoNiRGLSIiIyIiIiomLCZGM0YzKiYkRjMhIiJGM0YqRjNGM0YzRipGM0Y4">LCYtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUYkNiQtSSJZRzYiNiRJInhHRitJJmFscGhhR0YrRi1GLSIiIiomLCZGL0YvRikkRi8hIiJGL0YpRi9GMw==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L31" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The governing equation for the Jacobian <Equation executable="false" style="2D Math" input-equation="" display="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">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYkLUYjNiQtSSZtZnJhY0dGJDYoLUYjNiUtSSNtb0dGJDYtUTAmRGlmZmVyZW50aWFsRDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJnVuc2V0RicvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGPi8lKnN5bW1ldHJpY0dGPi8lKGxhcmdlb3BHRj4vJS5tb3ZhYmxlbGltaXRzR0Y+LyUnYWNjZW50R0Y+LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTS1JI21pR0YkNiVRInlGJy8lJ2l0YWxpY0dRJmZhbHNlRidGOUY5LUYjNiVGNS1GUTYlUSgmYWxwaGE7RidGVEY5RjkvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRltvLyUpYmV2ZWxsZWRHRlZGOUY5LUY2Ni1RIn5GJ0Y5L0Y9RlYvRkBGVi9GQkZWL0ZERlYvRkZGVi9GSEZWL0ZKRlZGS0ZORjk=</Equation><Font encoding="UTF-8">is obtained by differentiating the governing equation with respect to \316\261:</Font></Text-field>
</Input>
</Group>
<Group labelreference="L6" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eqalpha:=diff(eqalpha,alpha);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkoZXFhbHBoYUc2IiwoLUklZGlmZkclKnByb3RlY3RlZEc2JS1JIllHRiQ2JEkieEdGJEkmYWxwaGFHRiRGLi1JIiRHRig2JEYtIiIjIiIiKigkRjMhIiJGMy1GJzYkRipGLkYzRipGM0Y2KiYsJkYzRjMqJkY1RjNGKkYzRjNGM0Y3RjNGNg==">LCgtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUYkNiQtRiQ2JC1JIllHNiI2JEkieEdGLUkmYWxwaGFHRi1GMEYvRi8iIiIqJkYpRjFGK0YxJCEiIkY0KiYsJkYxRjFGKyRGMUY0RjFGKUYxRjQ=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L32" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">A new variable, y2(x) is used to present the Jacobian <Equation executable="false" style="2D Math" input-equation="" display="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">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobWZlbmNlZEdGJDYkLUYjNiQtSSZtZnJhY0dGJDYoLUYjNiUtSSNtb0dGJDYtUTAmRGlmZmVyZW50aWFsRDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJnVuc2V0RicvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGPi8lKnN5bW1ldHJpY0dGPi8lKGxhcmdlb3BHRj4vJS5tb3ZhYmxlbGltaXRzR0Y+LyUnYWNjZW50R0Y+LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTS1JI21pR0YkNiVRInlGJy8lJ2l0YWxpY0dRJmZhbHNlRidGOUY5LUYjNiVGNS1GUTYlUSgmYWxwaGE7RidGVEY5RjkvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRltvLyUpYmV2ZWxsZWRHRlZGOUY5LUY2Ni1RIn5GJ0Y5L0Y9RlYvRkBGVi9GQkZWL0ZERlYvRkZGVi9GSEZWL0ZKRlZGS0ZOLUY2Ni1RIjpGJ0Y5RmNvRmRvRmVvRmZvRmdvRmhvRmlvL0ZMUSwwLjI3Nzc3NzhlbUYnL0ZPRl5wRjk=</Equation></Text-field>
</Input>
</Group>
<Group labelreference="L7" 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>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkoZXFhbHBoYUc2IiwoLUklZGlmZkclKnByb3RlY3RlZEc2JC1JI3kyR0YkNiNJInhHRiQtSSIkR0YoNiRGLSIiIyIiIiooJEYyISIiRjJGKkYyLUkiWUdGJDYkRi1JJmFscGhhR0YkRjJGNSomLCZGMkYyKiZGNEYyRjZGMkYyRjJGKkYyRjU=">LCgtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUYkNiQtSSN5Mkc2IjYjSSJ4R0YrRi1GLSIiIiomRilGLi1JIllHRis2JEYtSSZhbHBoYUdGK0YuJCEiIkY1KiYsJkYuRi5GMCRGLkY1Ri5GKUYuRjU=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L8" 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="PkkoZXFhbHBoYUc2IiwoLUklZGlmZkclKnByb3RlY3RlZEc2JC1JI3kyR0YkNiNJInhHRiQtSSIkR0YoNiRGLSIiIyIiIiooJEYyISIiRjJGKkYyLUkieUdGJEYsRjJGNSomLCZGMkYyKiZGNEYyRjZGMkYyRjJGKkYyRjU=">LCgtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUYkNiQtSSN5Mkc2IjYjSSJ4R0YrRi1GLSIiIiomRilGLi1JInlHRitGLEYuJCEiIkYzKiYsJkYuRi5GMCRGLkYzRi5GKUYuRjM=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L33" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The variables are stored in vars:</Text-field>
</Input>
</Group>
<Group labelreference="L9" 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="L34" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The original governing equation and the sensitivity equation are stored in eqs:</Text-field>
</Input>
</Group>
<Group labelreference="L10" 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="PkkkZXFzRzYiNiQsJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5R0YkNiNJInhHRiQtSSIkR0YpNiRGLiIiIyIiIiomLCZGM0YzKiYkRjMhIiJGM0YrRjNGM0YzRitGM0Y4LCgtRig2JC1JI3kyR0YkRi1GL0YzKihGN0YzRjxGM0YrRjNGOComRjVGM0Y8RjNGOA==">NiQsJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtRiU2JC1JInlHNiI2I0kieEdGLEYuRi4iIiIqJiwmRi9GL0YqJEYvISIiRi9GKkYvRjMsKC1GJTYkLUYlNiQtSSN5MkdGLEYtRi5GLkYvKiZGOUYvRipGLyRGM0YzKiZGMUYvRjlGL0Yz</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L35" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">The initial value for \316\261 is given here:</Font></Text-field>
</Input>
</Group>
<Group labelreference="L11" 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="L36" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The initial conditions are stored in ICs:</Text-field>
</Input>
</Group>
<Group labelreference="L12" 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="L37" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Next the numerical solution is obtained and stored in sol:</Text-field>
</Input>
</Group>
<Group labelreference="L13" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">sol:=dsolve({eqs,ICs},{vars},type=numeric);</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="L38" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The solution is evaluated at x = 1:</Text-field>
</Input>
</Group>
<Group labelreference="L14" 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">NycvSSJ4RzYiJCIiIiIiIS8tSSJ5R0YlNiNGJCQiM015Y0Vmcz94eSEjPS8tSSVkaWZmRyUqcHJvdGVjdGVkRzYkRipGJCQiM1dSMGwiKUgnSEcnRi8vLUkjeTJHRiVGLCQiMyNHUm48Pic0MzshIzwvLUYyNiRGOEYkJCIzNGJwVSU9ZSRSOEY8</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L39" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The predicted value of y is stored in ypred:</Text-field>
</Input>
</Group>
<Group labelreference="L15" 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="PkkmeXByZWRHNiIkIjNNeWNFZnM/eHkhIz0=">JCIzTXljRWZzP3h5ISM9</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L40" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The predicted value for the Jacobian is stored in y2pred:</Text-field>
</Input>
</Group>
<Group labelreference="L16" 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="PkkneTJwcmVkRzYiJCIzI0dSbjw+JzQzOyEjPA==">JCIzI0dSbjw+JzQzOyEjPA==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L41" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">The new value for \316\261 is obtained as:</Font></Text-field>
</Input>
</Group>
<Group labelreference="L17" 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="PkknYWxwaGExRzYiJCIrc2QxP2ohIzU=">JCIrc2QxP2ohIzU=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L42" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">The error is calculated based on the value of \316\261:</Font></Text-field>
</Input>
</Group>
<Group labelreference="L18" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">err:=alpha1-alpha0;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXJyRzYiJCIrc2QxPzghIzU=">JCIrc2QxPzghIzU=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L43" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">The new value of \316\261 is then assigned to \316\2610 for the next iteration.</Font></Text-field>
</Input>
</Group>
<Group labelreference="L19" 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="PkknYWxwaGEwRzYiJCIrc2QxP2ohIzU=">JCIrc2QxP2ohIzU=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L44" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">A program is written to update the values of \316\261 until the error is &gt; 1e - 6.  One can set stricter tolerance limits for higher accuracy.</Font></Text-field>
</Input>
</Group>
<Group labelreference="L20" 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="L21" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">while err&gt;1e-6 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(alpha1-alpha0);</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>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkSUNzRzYiNiYvLUkieUdGJDYjIiIhJCIrc2QxP2ohIzUvLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoRilGKi8tSSN5MkdGJEYpIiIiLy0tRjE2I0Y4RilGKg==">NiYvLUkieUc2IjYjIiIhJCIrc2QxP2ohIzUvLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2I0YlRidGKC8tSSN5MkdGJkYnIiIiLy0tRi82I0Y2RidGKA==</Equation></Text-field>
</Output>
<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>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkmeXByZWRHNiIkIjMqKj41JVE7XjYrIiEjPA==">JCIzKio+NSVRO142KyIhIzw=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkneTJwcmVkRzYiJCIzaClmXEQnemJEOyEjPA==">JCIzaClmXEQnemJEOyEjPA==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkknYWxwaGExRzYiJCIrPFQpSEonISM1">JCIrPFQpSEonISM1</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXJyRzYiJCIoYjszKCEjNQ==">JCIoYjszKCEjNQ==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkknYWxwaGEwRzYiJCIrPFQpSEonISM1">JCIrPFQpSEonISM1</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pkkia0c2IiIiIw==">IiIj</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkSUNzRzYiNiYvLUkieUdGJDYjIiIhJCIrPFQpSEonISM1Ly0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKEYpRiovLUkjeTJHRiRGKSIiIi8tLUYxNiNGOEYpRio=">NiYvLUkieUc2IjYjIiIhJCIrPFQpSEonISM1Ly0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiNGJUYnRigvLUkjeTJHRiZGJyIiIi8tLUYvNiNGNkYnRig=</Equation></Text-field>
</Output>
<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>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkmeXByZWRHNiIkIjM/aUonSC4rKysiISM8">JCIzP2lKJ0guKysrIiEjPA==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkneTJwcmVkRzYiJCIzIio9eDZeUllEOyEjPA==">JCIzIio9eDZeUllEOyEjPA==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkknYWxwaGExRzYiJCIrOVIpSEonISM1">JCIrOVIpSEonISM1</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXJyRzYiJCIkLiMhIzU=">JCIkLiMhIzU=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkknYWxwaGEwRzYiJCIrOVIpSEonISM1">JCIrOVIpSEonISM1</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pkkia0c2IiIiJA==">IiIk</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L22" 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">IiIk</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L45" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">Three iterations were required for this problem.  The error in \316\261 is found to be:</Font></Text-field>
</Input>
</Group>
<Group labelreference="L23" 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">JCIkLiMhIzU=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L46" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The solution obtained is then plotted:</Text-field>
</Input>
</Group>
<Group labelreference="L24" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">p:=odeplot(sol,[x,y(x)],0..1,color=brown,thickness=4):
pt:=textplot([0.4,0.8,'alpha=alpha1']):
display({p,pt},axes=boxed,title=&quot;Figure Exp. 3.2.8.&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, 6056 characters]</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L47" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">For this boundary value problem, it takes only three iterations for the solution to converge.  Depending on the problem and the initial guess provided, the program might take any number of iterations to converge.  In addition, for updating \316\261 (equation 3.2.11) the Jacobian  in the denominator might approach zero for certain problems.  Sometimes it is necessary to scale the update of \316\261 using the following relations:</Font></Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal" alignment="centred"><Font encoding="UTF-8">                                    \316\261</Font><Font subscript="true" size="10">new</Font><Font size="8">= </Font> <Font encoding="UTF-8">\316\261</Font><Font subscript="true" size="10">old</Font><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>                                             (3.2.16)</Text-field>
<Text-field style="Text" layout="Normal" alignment="centred"></Text-field>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">where \317\201 is the number between 0 and 1.  The lower the value of \317\201 the higher is the number of iterations required for convergence.  The value of \317\201 depends on the problem.  This is illustrated in the next example.</Font></Text-field>
</Input>
</Group>
<Group labelreference="L28" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"></Text-field>
</Input>
</Group>
</Worksheet>