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<Text-field style="Heading 3" italic="false" layout="Heading 3"><Font italic="false">Example 2.3.14  Graetz Problem - Finite Difference Solution</Font></Text-field>
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<Text-field style="Text" layout="Normal">The Graetz problem (heat or mass transfer) in cylindrical coordinates with parabolic velocity profile is solved here.  The governing equation for the eigenfunction is<Font superscript="true">[15] [8]</Font> </Text-field>
<Text-field style="Text" layout="Normal">	 	<Equation executable="false" style="2D Math" input-equation="" 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executable="false" style="Text" input-equation="" 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executable="false" style="Text" input-equation="" display="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">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</Equation>                                       (3.2.52)</Text-field>
<Text-field style="Text" layout="Normal">with the boundary conditions:</Text-field>
<Text-field style="Text" layout="Normal">	 	                                                               <Equation executable="false" style="2D Math" input-equation="" display="LUkmbWZyYWNHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2KC1JJW1yb3dHRiQ2JS1JI21vR0YkNi1RMCZEaWZmZXJlbnRpYWxEO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmdW5zZXRGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZGLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EmZmFsc2VGJ0YyRjItRiw2Jy1GSjYjUSFGJ0YuLUZKNiZRInhGJ0ZNLyUscGxhY2Vob2xkZXJHUSV0cnVlRidGMkZSRjIvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmpuLyUpYmV2ZWxsZWRHRk8=">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</Equation><Equation executable="false" style="Text" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtbkdGJDYkUSIwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjRGNC1JI21vR0YkNi1RIj1GJ0Y0LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkxGMEY0">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtbkdGJDYkUSIwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjRGNC1JI21vR0YkNi1RIj1GJ0Y0LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkxGMEY0</Equation>                                                       (3.2.53)</Text-field>
<Text-field style="Text" layout="Normal">and </Text-field>
<Text-field style="Text" layout="Normal">	</Text-field>
<Text-field style="Text" layout="Normal">	 	                                                                 <Equation executable="false" style="Text" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiQtRiM2JC1JI21uR0YkNiRRIjFGJ0YyRjJGMkYy">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiQtRiM2JC1JI21uR0YkNiRRIjFGJ0YyRjJGMkYy</Equation>=0                                                           (3.2.54)</Text-field>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">Equation (3.2.52) is a second order equation with two boundary conditions (equations (3.2.53) and (3.2.54).  In equation (3.2.52), \316\273 is the eigenvalue.  To solve for the eigenvalue an additional boundary condition has to be used.  For this purpose, y at x = 0 is arbitrarily set to 1:</Font></Text-field>
<Text-field style="Text" layout="Normal">	 	                                                                <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiQtRiM2JC1JI21uR0YkNiRRIjBGJ0YyRjJGMkYy">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiQtRiM2JC1JI21uR0YkNiRRIjBGJ0YyRjJGMkYy</Equation>=1                                                            (3.2.55)</Text-field>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">Next, equation (3.2.52) is discretized using finite differences as in section 3.2.3.  This yields N equations for the interior node points.  The boundary conditions (equations (3.2.53) and (3.2.54) are converted to finite difference form.  This yields two equations.  There are a total of N+2 node points including the boundaries.  There are N+2 dependent variables (yi,i = 0..N+1).  There is an additional variable \316\273.  The additional equation is (3.2.55).  Hence, there are N+3 variables (yi,i = 0..N+1 and \316\273) to be solved from N+3 equations.  There are infinite solutions for the differential equation (3.2.52).  Hence, there are multiple solutions for the system of finite difference equations.  This example is solved in Maple below:</Font></Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">restart:</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">with(plots):</Text-field>
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<Group labelreference="L4" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">N:=30;</Text-field>
</Input>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkiTkc2IiIjSQ==">IiNJ</Equation></Text-field>
</Output>
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<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>
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<Group labelreference="L6" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq:=diff(y(x),x$2)+1/x*diff(y(x),x)+lambda^2*(1-x^2)*y(x);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjZXFHNiIsKC1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5R0YkNiNJInhHRiQtSSIkR0YoNiRGLSIiIyIiIiomRi0hIiItRic2JEYqRi1GMkYyKigpSSdsYW1iZGFHRiRGMUYyLCZGMkYyKiQpRi1GMUYyRjRGMkYqRjJGMg==">LCgtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUYkNiQtSSJ5RzYiNiNJInhHRitGLUYtIiIiKiZGLSEiIkYnRi5GLiooSSdsYW1iZGFHRisiIiMsJkYuRi4qJEYtRjNGMEYuRilGLkYu</Equation></Text-field>
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<Group labelreference="L46" drawlabel="true">
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<Text-field style="Text" layout="Normal"></Text-field>
</Input>
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<Group labelreference="L7" 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="L8" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc2:=y(x)-0;</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMyRzYiLUkieUdGJDYjSSJ4R0Yk">LUkieUc2IjYjSSJ4R0Yk</Equation></Text-field>
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<Input>
<Text-field style="Text" layout="Normal">The additional boundary condition is entered here:</Text-field>
</Input>
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<Group labelreference="L9" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc3:=y(x)-1;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMzRzYiLCYtSSJ5R0YkNiNJInhHRiQiIiJGKiEiIg==">LCYtSSJ5RzYiNiNJInhHRiUiIiIhIiJGKA==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L10" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal">The central difference expressions 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>
</Output>
</Group>
<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>
</Output>
</Group>
<Group labelreference="L13" 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="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>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkmZHlkeGZHNiIsJCooIyIiIiIiI0YoLCgmSSJ5R0YkNiNGKSEiIiomIiIlRigmRiw2I0YoRihGKComIiIkRigmRiw2IyIiIUYoRi5GKEkiaEdGJEYuRig=">LCQqJiwoJkkieUc2IjYjIiIjISIiJkYmNiMiIiIiIiUmRiY2IyIiISEiJEYtSSJoR0YnRiojRi1GKQ==</Equation></Text-field>
</Output>
</Group>
<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>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkmZHlkeGJHNiIsJCooIyIiIiIiI0YoLCgmSSJ5R0YkNiMiI0hGKComIiIlRigmRiw2IyIjSUYoISIiKiYiIiRGKCZGLDYjIiNKRihGKEYoSSJoR0YkRjRGKA==">LCQqJiwoJkkieUc2IjYjIiNIIiIiJkYmNiMiI0khIiUmRiY2IyIjSiIiJEYqSSJoR0YnISIiI0YqIiIj</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L16" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal"> The governing equation in finite difference form is:</Text-field>
</Input>
</Group>
<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="PiZJI0VxRzYiNiNJIm1HRiUsKComLCgmSSJ5R0YlNiMsJkYnIiIiRi9GL0YvKiYiIiNGLyZGLEYmRi8hIiImRiw2IywmRidGL0YvRjNGL0YvKUkiaEdGJUYxRjNGLyoqI0YvRjFGL0YnRjNGN0YzLCZGK0YvRjRGM0YvRi8qKClJJ2xhbWJkYUdGJUYxRi8sJkYvRi8qJilGJ0YxRi9GN0YvRjNGL0YyRi9GLw==">LCgqJiwoJkkieUc2IjYjLCZJIm1HRiciIiJGK0YrRismRiY2I0YqISIjJkYmNiMsJkYqRishIiJGK0YrRitJImhHRidGLkYrKihGKkYyRjNGLiwmRiVGK0YvRjJGKyNGKyIiIyooSSdsYW1iZGFHRidGNywmRitGKyomRipGN0YzRjdGMkYrRixGK0Yr</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L18" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal"> The boundary conditions in finite difference form are:</Text-field>
</Input>
</Group>
<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>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiEsJCooIyIiIiIiI0YrLCgmSSJ5R0YlNiNGLCEiIiomIiIlRismRi82I0YrRitGKyomIiIkRismRi9GJkYrRjFGK0kiaEdGJUYxRis=">LCQqJiwoJkkieUc2IjYjIiIjISIiJkYmNiMiIiIiIiUmRiY2IyIiISEiJEYtSSJoR0YnRiojRi1GKQ==</Equation></Text-field>
</Output>
</Group>
<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="PiZJI0VxRzYiNiMiI0omSSJ5R0YlRiY=">JkkieUc2IjYjIiNK</Equation></Text-field>
</Output>
</Group>
<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>
</Group>
<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>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiIsKComLCgmSSJ5R0YlNiMiIiNGJyomRi5GJyZGLEYmRichIiImRiw2IyIiIUYnRicpSSJoR0YlRi5GMUYnKigjRidGLkYnRjVGMSwmRitGJ0YyRjFGJ0YnKigpSSdsYW1iZGFHRiVGLkYnLCZGJ0YnKiRGNUYnRjFGJ0YwRidGJw==">LCgqJiwoJkkieUc2IjYjIiIjIiIiJkYmNiNGKiEiIyZGJjYjIiIhRipGKkkiaEdGJ0YtRioqJkYxRi0sJkYlRipGLiEiIkYqI0YqRikqKEknbGFtYmRhR0YnRiksJkYqRioqJEYxRilGNEYqRitGKkYq</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiMsKComLCgmSSJ5R0YlNiMiIiQiIiIqJkYnRi8mRixGJkYvISIiJkYsNiNGL0YvRi8pSSJoR0YlRidGMkYvKigjRi8iIiVGL0Y1RjIsJkYrRi9GM0YyRi9GLyooKUknbGFtYmRhR0YlRidGLywmRi9GLyomRjlGL0Y1Ri9GMkYvRjFGL0Yv">LCgqJiwoJkkieUc2IjYjIiIkIiIiJkYmNiMiIiMhIiMmRiY2I0YqRipGKkkiaEdGJ0YuRioqJkYxRi4sJkYlRipGLyEiIkYqI0YqIiIlKihJJ2xhbWJkYUdGJ0YtLCZGKkYqKiRGMUYtISIlRipGK0YqRio=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiQsKComLCgmSSJ5R0YlNiMiIiUiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjRjFGL0YvKUkiaEdGJUYxRjNGLyooI0YvIiInRi9GNkYzLCZGK0YvRjRGM0YvRi8qKClJJ2xhbWJkYUdGJUYxRi8sJkYvRi8qJiIiKkYvRjZGL0YzRi9GMkYvRi8=">LCgqJiwoJkkieUc2IjYjIiIlIiIiJkYmNiMiIiQhIiMmRiY2IyIiI0YqRipJImhHRidGLkYqKiZGMkYuLCZGJUYqRi8hIiJGKiNGKiIiJyooSSdsYW1iZGFHRidGMSwmRipGKiokRjJGMSEiKkYqRitGKkYq</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiUsKComLCgmSSJ5R0YlNiMiIiYiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiIkRi9GLylJImhHRiVGMUYzRi8qKCNGLyIiKUYvRjdGMywmRitGL0Y0RjNGL0YvKigpSSdsYW1iZGFHRiVGMUYvLCZGL0YvKiYiIztGL0Y3Ri9GM0YvRjJGL0Yv">LCgqJiwoJkkieUc2IjYjIiImIiIiJkYmNiMiIiUhIiMmRiY2IyIiJEYqRipJImhHRidGLkYqKiZGMkYuLCZGJUYqRi8hIiJGKiNGKiIiKSooSSdsYW1iZGFHRiciIiMsJkYqRioqJEYyRjohIztGKkYrRipGKg==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiYsKComLCgmSSJ5R0YlNiMiIiciIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiIlRi9GLylJImhHRiVGMUYzRi8qKCNGLyIjNUYvRjdGMywmRitGL0Y0RjNGL0YvKigpSSdsYW1iZGFHRiVGMUYvLCZGL0YvKiYiI0RGL0Y3Ri9GM0YvRjJGL0Yv">LCgqJiwoJkkieUc2IjYjIiInIiIiJkYmNiMiIiYhIiMmRiY2IyIiJUYqRipJImhHRidGLkYqKiZGMkYuLCZGJUYqRi8hIiJGKiNGKiIjNSooSSdsYW1iZGFHRiciIiMsJkYqRioqJEYyRjohI0RGKkYrRipGKg==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIicsKComLCgmSSJ5R0YlNiMiIigiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiImRi9GLylJImhHRiVGMUYzRi8qKCNGLyIjN0YvRjdGMywmRitGL0Y0RjNGL0YvKigpSSdsYW1iZGFHRiVGMUYvLCZGL0YvKiYiI09GL0Y3Ri9GM0YvRjJGL0Yv">LCgqJiwoJkkieUc2IjYjIiIoIiIiJkYmNiMiIichIiMmRiY2IyIiJkYqRipJImhHRidGLkYqKiZGMkYuLCZGJUYqRi8hIiJGKiNGKiIjNyooSSdsYW1iZGFHRiciIiMsJkYqRioqJEYyRjohI09GKkYrRipGKg==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIigsKComLCgmSSJ5R0YlNiMiIikiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiInRi9GLylJImhHRiVGMUYzRi8qKCNGLyIjOUYvRjdGMywmRitGL0Y0RjNGL0YvKigpSSdsYW1iZGFHRiVGMUYvLCZGL0YvKiYiI1xGL0Y3Ri9GM0YvRjJGL0Yv">LCgqJiwoJkkieUc2IjYjIiIpIiIiJkYmNiMiIighIiMmRiY2IyIiJ0YqRipJImhHRidGLkYqKiZGMkYuLCZGJUYqRi8hIiJGKiNGKiIjOSooSSdsYW1iZGFHRiciIiMsJkYqRioqJEYyRjohI1xGKkYrRipGKg==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiksKComLCgmSSJ5R0YlNiMiIioiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiIoRi9GLylJImhHRiVGMUYzRi8qKCNGLyIjO0YvRjdGMywmRitGL0Y0RjNGL0YvKigpSSdsYW1iZGFHRiVGMUYvLCZGL0YvKiYiI2tGL0Y3Ri9GM0YvRjJGL0Yv">LCgqJiwoJkkieUc2IjYjIiIqIiIiJkYmNiMiIikhIiMmRiY2IyIiKEYqRipJImhHRidGLkYqKiZGMkYuLCZGJUYqRi8hIiJGKiNGKiIjOyooSSdsYW1iZGFHRiciIiMsJkYqRioqJEYyRjohI2tGKkYrRipGKg==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIiosKComLCgmSSJ5R0YlNiMiIzUiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiIpRi9GLylJImhHRiVGMUYzRi8qKCNGLyIjPUYvRjdGMywmRitGL0Y0RjNGL0YvKigpSSdsYW1iZGFHRiVGMUYvLCZGL0YvKiYiIyIpRi9GN0YvRjNGL0YyRi9GLw==">LCgqJiwoJkkieUc2IjYjIiM1IiIiJkYmNiMiIiohIiMmRiY2IyIiKUYqRipJImhHRidGLkYqKiZGMkYuLCZGJUYqRi8hIiJGKiNGKiIjPSooSSdsYW1iZGFHRiciIiMsJkYqRioqJEYyRjohIyIpRipGK0YqRio=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIzUsKComLCgmSSJ5R0YlNiMiIzYiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiIqRi9GLylJImhHRiVGMUYzRi8qKCNGLyIjP0YvRjdGMywmRitGL0Y0RjNGL0YvKigpSSdsYW1iZGFHRiVGMUYvLCZGL0YvKiYiJCsiRi9GN0YvRjNGL0YyRi9GLw==">LCgqJiwoJkkieUc2IjYjIiM2IiIiJkYmNiMiIzUhIiMmRiY2IyIiKkYqRipJImhHRidGLkYqKiZGMkYuLCZGJUYqRi8hIiJGKiNGKiIjPyooSSdsYW1iZGFHRiciIiMsJkYqRioqJEYyRjohJCsiRipGK0YqRio=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIzYsKComLCgmSSJ5R0YlNiMiIzciIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiM1Ri9GLylJImhHRiVGMUYzRi8qKCNGLyIjQUYvRjdGMywmRitGL0Y0RjNGL0YvKigpSSdsYW1iZGFHRiVGMUYvLCZGL0YvKiYiJEAiRi9GN0YvRjNGL0YyRi9GLw==">LCgqJiwoJkkieUc2IjYjIiM3IiIiJkYmNiMiIzYhIiMmRiY2IyIjNUYqRipJImhHRidGLkYqKiZGMkYuLCZGJUYqRi8hIiJGKiNGKiIjQSooSSdsYW1iZGFHRiciIiMsJkYqRioqJEYyRjohJEAiRipGK0YqRio=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIzcsKComLCgmSSJ5R0YlNiMiIzgiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiM2Ri9GLylJImhHRiVGMUYzRi8qKCNGLyIjQ0YvRjdGMywmRitGL0Y0RjNGL0YvKigpSSdsYW1iZGFHRiVGMUYvLCZGL0YvKiYiJFciRi9GN0YvRjNGL0YyRi9GLw==">LCgqJiwoJkkieUc2IjYjIiM4IiIiJkYmNiMiIzchIiMmRiY2IyIjNkYqRipJImhHRidGLkYqKiZGMkYuLCZGJUYqRi8hIiJGKiNGKiIjQyooSSdsYW1iZGFHRiciIiMsJkYqRioqJEYyRjohJFciRipGK0YqRio=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIzgsKComLCgmSSJ5R0YlNiMiIzkiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiM3Ri9GLylJImhHRiVGMUYzRi8qKCNGLyIjRUYvRjdGMywmRitGL0Y0RjNGL0YvKigpSSdsYW1iZGFHRiVGMUYvLCZGL0YvKiYiJHAiRi9GN0YvRjNGL0YyRi9GLw==">LCgqJiwoJkkieUc2IjYjIiM5IiIiJkYmNiMiIzghIiMmRiY2IyIjN0YqRipJImhHRidGLkYqKiZGMkYuLCZGJUYqRi8hIiJGKiNGKiIjRSooSSdsYW1iZGFHRiciIiMsJkYqRioqJEYyRjohJHAiRipGK0YqRio=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIzksKComLCgmSSJ5R0YlNiMiIzoiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiM4Ri9GLylJImhHRiVGMUYzRi8qKCNGLyIjR0YvRjdGMywmRitGL0Y0RjNGL0YvKigpSSdsYW1iZGFHRiVGMUYvLCZGL0YvKiYiJCc+Ri9GN0YvRjNGL0YyRi9GLw==">LCgqJiwoJkkieUc2IjYjIiM6IiIiJkYmNiMiIzkhIiMmRiY2IyIjOEYqRipJImhHRidGLkYqKiZGMkYuLCZGJUYqRi8hIiJGKiNGKiIjRyooSSdsYW1iZGFHRiciIiMsJkYqRioqJEYyRjohJCc+RipGK0YqRio=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIzosKComLCgmSSJ5R0YlNiMiIzsiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiM5Ri9GLylJImhHRiVGMUYzRi8qKCNGLyIjSUYvRjdGMywmRitGL0Y0RjNGL0YvKigpSSdsYW1iZGFHRiVGMUYvLCZGL0YvKiYiJEQjRi9GN0YvRjNGL0YyRi9GLw==">LCgqJiwoJkkieUc2IjYjIiM7IiIiJkYmNiMiIzohIiMmRiY2IyIjOUYqRipJImhHRidGLkYqKiZGMkYuLCZGJUYqRi8hIiJGKiNGKiIjSSooSSdsYW1iZGFHRiciIiMsJkYqRioqJEYyRjohJEQjRipGK0YqRio=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIzssKComLCgmSSJ5R0YlNiMiIzwiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiM6Ri9GLylJImhHRiVGMUYzRi8qKCNGLyIjS0YvRjdGMywmRitGL0Y0RjNGL0YvKigpSSdsYW1iZGFHRiVGMUYvLCZGL0YvKiYiJGMjRi9GN0YvRjNGL0YyRi9GLw==">LCgqJiwoJkkieUc2IjYjIiM8IiIiJkYmNiMiIzshIiMmRiY2IyIjOkYqRipJImhHRidGLkYqKiZGMkYuLCZGJUYqRi8hIiJGKiNGKiIjSyooSSdsYW1iZGFHRiciIiMsJkYqRioqJEYyRjohJGMjRipGK0YqRio=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIzwsKComLCgmSSJ5R0YlNiMiIz0iIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiM7Ri9GLylJImhHRiVGMUYzRi8qKCNGLyIjTUYvRjdGMywmRitGL0Y0RjNGL0YvKigpSSdsYW1iZGFHRiVGMUYvLCZGL0YvKiYiJCpHRi9GN0YvRjNGL0YyRi9GLw==">LCgqJiwoJkkieUc2IjYjIiM9IiIiJkYmNiMiIzwhIiMmRiY2IyIjO0YqRipJImhHRidGLkYqKiZGMkYuLCZGJUYqRi8hIiJGKiNGKiIjTSooSSdsYW1iZGFHRiciIiMsJkYqRioqJEYyRjohJCpHRipGK0YqRio=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIz0sKComLCgmSSJ5R0YlNiMiIz4iIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiM8Ri9GLylJImhHRiVGMUYzRi8qKCNGLyIjT0YvRjdGMywmRitGL0Y0RjNGL0YvKigpSSdsYW1iZGFHRiVGMUYvLCZGL0YvKiYiJEMkRi9GN0YvRjNGL0YyRi9GLw==">LCgqJiwoJkkieUc2IjYjIiM+IiIiJkYmNiMiIz0hIiMmRiY2IyIjPEYqRipJImhHRidGLkYqKiZGMkYuLCZGJUYqRi8hIiJGKiNGKiIjTyooSSdsYW1iZGFHRiciIiMsJkYqRioqJEYyRjohJEMkRipGK0YqRio=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIz4sKComLCgmSSJ5R0YlNiMiIz8iIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiM9Ri9GLylJImhHRiVGMUYzRi8qKCNGLyIjUUYvRjdGMywmRitGL0Y0RjNGL0YvKigpSSdsYW1iZGFHRiVGMUYvLCZGL0YvKiYiJGgkRi9GN0YvRjNGL0YyRi9GLw==">LCgqJiwoJkkieUc2IjYjIiM/IiIiJkYmNiMiIz4hIiMmRiY2IyIjPUYqRipJImhHRidGLkYqKiZGMkYuLCZGJUYqRi8hIiJGKiNGKiIjUSooSSdsYW1iZGFHRiciIiMsJkYqRioqJEYyRjohJGgkRipGK0YqRio=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiIz8sKComLCgmSSJ5R0YlNiMiI0AiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiM+Ri9GLylJImhHRiVGMUYzRi8qKCNGLyIjU0YvRjdGMywmRitGL0Y0RjNGL0YvKigpSSdsYW1iZGFHRiVGMUYvLCZGL0YvKiYiJCslRi9GN0YvRjNGL0YyRi9GLw==">LCgqJiwoJkkieUc2IjYjIiNAIiIiJkYmNiMiIz8hIiMmRiY2IyIjPkYqRipJImhHRidGLkYqKiZGMkYuLCZGJUYqRi8hIiJGKiNGKiIjUyooSSdsYW1iZGFHRiciIiMsJkYqRioqJEYyRjohJCslRipGK0YqRio=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiI0AsKComLCgmSSJ5R0YlNiMiI0EiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiM/Ri9GLylJImhHRiVGMUYzRi8qKCNGLyIjVUYvRjdGMywmRitGL0Y0RjNGL0YvKigpSSdsYW1iZGFHRiVGMUYvLCZGL0YvKiYiJFQlRi9GN0YvRjNGL0YyRi9GLw==">LCgqJiwoJkkieUc2IjYjIiNBIiIiJkYmNiMiI0AhIiMmRiY2IyIjP0YqRipJImhHRidGLkYqKiZGMkYuLCZGJUYqRi8hIiJGKiNGKiIjVSooSSdsYW1iZGFHRiciIiMsJkYqRioqJEYyRjohJFQlRipGK0YqRio=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiI0EsKComLCgmSSJ5R0YlNiMiI0IiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiNARi9GLylJImhHRiVGMUYzRi8qKCNGLyIjV0YvRjdGMywmRitGL0Y0RjNGL0YvKigpSSdsYW1iZGFHRiVGMUYvLCZGL0YvKiYiJCVbRi9GN0YvRjNGL0YyRi9GLw==">LCgqJiwoJkkieUc2IjYjIiNCIiIiJkYmNiMiI0EhIiMmRiY2IyIjQEYqRipJImhHRidGLkYqKiZGMkYuLCZGJUYqRi8hIiJGKiNGKiIjVyooSSdsYW1iZGFHRiciIiMsJkYqRioqJEYyRjohJCVbRipGK0YqRio=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiI0IsKComLCgmSSJ5R0YlNiMiI0MiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiNBRi9GLylJImhHRiVGMUYzRi8qKCNGLyIjWUYvRjdGMywmRitGL0Y0RjNGL0YvKigpSSdsYW1iZGFHRiVGMUYvLCZGL0YvKiYiJEgmRi9GN0YvRjNGL0YyRi9GLw==">LCgqJiwoJkkieUc2IjYjIiNDIiIiJkYmNiMiI0IhIiMmRiY2IyIjQUYqRipJImhHRidGLkYqKiZGMkYuLCZGJUYqRi8hIiJGKiNGKiIjWSooSSdsYW1iZGFHRiciIiMsJkYqRioqJEYyRjohJEgmRipGK0YqRio=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiI0MsKComLCgmSSJ5R0YlNiMiI0QiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiNCRi9GLylJImhHRiVGMUYzRi8qKCNGLyIjW0YvRjdGMywmRitGL0Y0RjNGL0YvKigpSSdsYW1iZGFHRiVGMUYvLCZGL0YvKiYiJHcmRi9GN0YvRjNGL0YyRi9GLw==">LCgqJiwoJkkieUc2IjYjIiNEIiIiJkYmNiMiI0MhIiMmRiY2IyIjQkYqRipJImhHRidGLkYqKiZGMkYuLCZGJUYqRi8hIiJGKiNGKiIjWyooSSdsYW1iZGFHRiciIiMsJkYqRioqJEYyRjohJHcmRipGK0YqRio=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiI0QsKComLCgmSSJ5R0YlNiMiI0UiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiNDRi9GLylJImhHRiVGMUYzRi8qKCNGLyIjXUYvRjdGMywmRitGL0Y0RjNGL0YvKigpSSdsYW1iZGFHRiVGMUYvLCZGL0YvKiYiJEQnRi9GN0YvRjNGL0YyRi9GLw==">LCgqJiwoJkkieUc2IjYjIiNFIiIiJkYmNiMiI0QhIiMmRiY2IyIjQ0YqRipJImhHRidGLkYqKiZGMkYuLCZGJUYqRi8hIiJGKiNGKiIjXSooSSdsYW1iZGFHRiciIiMsJkYqRioqJEYyRjohJEQnRipGK0YqRio=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiI0UsKComLCgmSSJ5R0YlNiMiI0YiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiNERi9GLylJImhHRiVGMUYzRi8qKCNGLyIjX0YvRjdGMywmRitGL0Y0RjNGL0YvKigpSSdsYW1iZGFHRiVGMUYvLCZGL0YvKiYiJHcnRi9GN0YvRjNGL0YyRi9GLw==">LCgqJiwoJkkieUc2IjYjIiNGIiIiJkYmNiMiI0UhIiMmRiY2IyIjREYqRipJImhHRidGLkYqKiZGMkYuLCZGJUYqRi8hIiJGKiNGKiIjXyooSSdsYW1iZGFHRiciIiMsJkYqRioqJEYyRjohJHcnRipGK0YqRio=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiI0YsKComLCgmSSJ5R0YlNiMiI0ciIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiNFRi9GLylJImhHRiVGMUYzRi8qKCNGLyIjYUYvRjdGMywmRitGL0Y0RjNGL0YvKigpSSdsYW1iZGFHRiVGMUYvLCZGL0YvKiYiJEgoRi9GN0YvRjNGL0YyRi9GLw==">LCgqJiwoJkkieUc2IjYjIiNHIiIiJkYmNiMiI0YhIiMmRiY2IyIjRUYqRipJImhHRidGLkYqKiZGMkYuLCZGJUYqRi8hIiJGKiNGKiIjYSooSSdsYW1iZGFHRiciIiMsJkYqRioqJEYyRjohJEgoRipGK0YqRio=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiI0csKComLCgmSSJ5R0YlNiMiI0giIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiNGRi9GLylJImhHRiVGMUYzRi8qKCNGLyIjY0YvRjdGMywmRitGL0Y0RjNGL0YvKigpSSdsYW1iZGFHRiVGMUYvLCZGL0YvKiYiJCV5Ri9GN0YvRjNGL0YyRi9GLw==">LCgqJiwoJkkieUc2IjYjIiNIIiIiJkYmNiMiI0chIiMmRiY2IyIjRkYqRipJImhHRidGLkYqKiZGMkYuLCZGJUYqRi8hIiJGKiNGKiIjYyooSSdsYW1iZGFHRiciIiMsJkYqRioqJEYyRjohJCV5RipGK0YqRio=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiI0gsKComLCgmSSJ5R0YlNiMiI0kiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiNHRi9GLylJImhHRiVGMUYzRi8qKCNGLyIjZUYvRjdGMywmRitGL0Y0RjNGL0YvKigpSSdsYW1iZGFHRiVGMUYvLCZGL0YvKiYiJFQpRi9GN0YvRjNGL0YyRi9GLw==">LCgqJiwoJkkieUc2IjYjIiNJIiIiJkYmNiMiI0ghIiMmRiY2IyIjR0YqRipJImhHRidGLkYqKiZGMkYuLCZGJUYqRi8hIiJGKiNGKiIjZSooSSdsYW1iZGFHRiciIiMsJkYqRioqJEYyRjohJFQpRipGK0YqRio=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI0VxRzYiNiMiI0ksKComLCgmSSJ5R0YlNiMiI0oiIiIqJiIiI0YvJkYsRiZGLyEiIiZGLDYjIiNIRi9GLylJImhHRiVGMUYzRi8qKCNGLyIjZ0YvRjdGMywmRitGL0Y0RjNGL0YvKigpSSdsYW1iZGFHRiVGMUYvLCZGL0YvKiYiJCsqRi9GN0YvRjNGL0YyRi9GLw==">LCgqJiwoJkkieUc2IjYjIiNKIiIiJkYmNiMiI0khIiMmRiY2IyIjSEYqRipJImhHRidGLkYqKiZGMkYuLCZGJUYqRi8hIiJGKiNGKiIjZyooSSdsYW1iZGFHRiciIiMsJkYqRioqJEYyRjohJCsqRipGK0YqRio=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L49" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The additional equation for the eigenvalue is:</Text-field>
</Input>
</Group>
<Group labelreference="L23" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Eqeig:=subs(diff(y(x),x)=dydxf,y(x)=y[0],bc3);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkmRXFlaWdHNiIsJiZJInlHRiQ2IyIiISIiIkYqISIi">LCYmSSJ5RzYiNiMiIiEiIiIhIiJGKA==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L24" 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="PkkiaEc2IiMiIiIiI0o=">IyIiIiIjSg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L50" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The finite difference equations are stored in eqs:</Text-field>
</Input>
</Group>
<Group labelreference="L25" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eqs:=seq(eval(subs(Phi=1,Eq[i])),i=0..N+1),Eqeig;</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="L51" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The dependent variables are stored in vars:</Text-field>
</Input>
</Group>
<Group labelreference="L26" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">vars:=seq(y[i],i=0..N+1),lambda;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkldmFyc0c2IjZDJkkieUdGJDYjIiIhJkYnNiMiIiImRic2IyIiIyZGJzYjIiIkJkYnNiMiIiUmRic2IyIiJiZGJzYjIiInJkYnNiMiIigmRic2IyIiKSZGJzYjIiIqJkYnNiMiIzUmRic2IyIjNiZGJzYjIiM3JkYnNiMiIzgmRic2IyIjOSZGJzYjIiM6JkYnNiMiIzsmRic2IyIjPCZGJzYjIiM9JkYnNiMiIz4mRic2IyIjPyZGJzYjIiNAJkYnNiMiI0EmRic2IyIjQiZGJzYjIiNDJkYnNiMiI0QmRic2IyIjRSZGJzYjIiNGJkYnNiMiI0cmRic2IyIjSCZGJzYjIiNJJkYnNiMiI0pJJ2xhbWJkYUdGJA==">NkMmSSJ5RzYiNiMiIiEmRiQ2IyIiIiZGJDYjIiIjJkYkNiMiIiQmRiQ2IyIiJSZGJDYjIiImJkYkNiMiIicmRiQ2IyIiKCZGJDYjIiIpJkYkNiMiIiomRiQ2IyIjNSZGJDYjIiM2JkYkNiMiIzcmRiQ2IyIjOCZGJDYjIiM5JkYkNiMiIzomRiQ2IyIjOyZGJDYjIiM8JkYkNiMiIz0mRiQ2IyIjPiZGJDYjIiM/JkYkNiMiI0AmRiQ2IyIjQSZGJDYjIiNCJkYkNiMiI0MmRiQ2IyIjRCZGJDYjIiNFJkYkNiMiI0YmRiQ2IyIjRyZGJDYjIiNIJkYkNiMiI0kmRiQ2IyIjSkknbGFtYmRhR0Yl</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L52" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">These variables are solved as:</Text-field>
</Input>
</Group>
<Group labelreference="L27" 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">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</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L53" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Since there are multiple solutions, one has to provide the range for the dependent variables.  The dependent variable y varies between 0 and 1 and the eigenvalue is solved in the range of 0.4.</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group>
<Group labelreference="L28" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">vars:=seq(y[i]=-1..1,i=0..N+1),lambda=0..4;</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="L54" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The solution obtained in stored in sol[1] and plotted:</Text-field>
</Input>
</Group>
<Group labelreference="L29" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">sol[1]:=fsolve({eqs},{vars});</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="L30" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">assign(sol[1]):</Text-field>
</Input>
</Group>
<Group labelreference="L55" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The first eigenvalue is given by:</Text-field>
</Input>
</Group>
<Group labelreference="L31" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">l[1]:=lambda;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJImxHNiI2IyIiIiQiK3QjPk1xIyEiKg==">JCIrdCM+TXEjISIq</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L56" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The first eigenfunction is plotted here:</Text-field>
</Input>
</Group>
<Group labelreference="L60" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">colorlist:=[black,red,blue,green];</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkqY29sb3JsaXN0RzYiNyZJJmJsYWNrR0YkSSRyZWRHRiRJJWJsdWVHRiRJJmdyZWVuR0Yk">NyZJJmJsYWNrRzYiSSRyZWRHRiRJJWJsdWVHRiRJJmdyZWVuR0Yk</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L32" 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,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="L33" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">p[1]:=plot([seq([i*h,y[i]],i=0..N+1)],thickness=4,axes=boxed,labels=[x,y],color=colorlist[1]):</Text-field>
</Input>
</Group>
<Group labelreference="L65" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">pt[1]:=textplot([0.4,0.9,'lambda=l[1]']):</Text-field>
</Input>
</Group>
<Group labelreference="L34" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">for i from 0 to N+1 do unassign('y[i]'):od:</Text-field>
</Input>
</Group>
<Group labelreference="L35" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">unassign('lambda'):</Text-field>
</Input>
</Group>
<Group labelreference="L36" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">M:=4;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkiTUc2IiIiJQ==">IiIl</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L57" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The next three eigenvalues are found by changing the range in increments of 4:</Text-field>
</Input>
</Group>
<Group labelreference="L37" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">for j from 2 to M do </Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">vars:=seq(y[i]=-1..1,i=0..N+1),lambda=l[j-1]+1..l[j-1]+4;</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">sol[j]:=fsolve({eqs},{vars});</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">assign(sol[j]):</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">l[j]:=lambda;</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">p[j]:=plot([seq([i*h,y[i]],i=0..N+1)],thickness=4,axes=boxed,labels=[x,y],color=colorlist[j]):</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">for i from 0 to N+1 do unassign('y[i]'):od:</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">unassign('lambda'):</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">od:</Text-field>
</Input>
</Group>
<Group labelreference="L58" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The same program was run rerun for N= 30 node points.  The results obtained are:</Text-field>
</Input>
</Group>
<Group labelreference="L38" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">print(seq(l[i],i=1..M));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiYkIit0Iz5NcSMhIiokIistNHJvbUYlJCIrKnlfTDEiISIpJCIrUUMqcFgiRio=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L66" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">pt[2]:=textplot([0.4,0.4,lambda=l[2]]):
pt[3]:=textplot([0.87,0.35,lambda=l[3]]):
pt[4]:=textplot([0.8,-0.05,lambda=l[4]]):</Text-field>
</Input>
</Group>
<Group labelreference="L59" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Even after using N=30 interior node points we get only two digit accuracy with the exact solution.  In addition, we observe more errors for  higher eigenvalues than for the lower eigenvalues.</Text-field>
</Input>
</Group>
<Group labelreference="L39" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">display({seq(p[i],i=1..M),seq(pt[i],i=1..M)});</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, 17240 characters]</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L40" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"></Text-field>
</Input>
</Group>
</Worksheet>