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<Text-field style="Heading 3" layout="Heading 3"><Font italic="false">Example 4.1  Heat Conduction in a Rectangular Slab</Font></Text-field>
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<Text-field style="Text" layout="Normal">Consider the following transient heat conduction problem in a slab.<Font superscript="true">[1-3]</Font>  The governing equation is:</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="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">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</Equation><Font encoding="UTF-8">=\316\261</Font><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZHLUkmbWZyYWNHRiQ2KC1GIzYmLUkjbW9HRiQ2LVErJlBhcnRpYWxEO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOS8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZILUkjbW5HRiQ2JlEiMkYnLyUlc2l6ZUdRIzEwRicvJSxzdXBlcnNjcmlwdEdRJXRydWVGJ0Y0LUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR0Y5RjRGNC1GIzYmRjAtRlY2JVEieEYnRllGNC1GVjYnRk5GT0ZZRlJGNEY0LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zhby8lKWJldmVsbGVkR0Y5LUYxNi5RIn5GJ0ZPRjRGN0Y6RjxGPkZARkJGREZGRklGZm9GZm9GZm9GZm9GZm9GZm9GZm9GZm9GZm9GZm9GZm9GZm9GZm9GZm9GZm9GZm9GZm9GZm9GZm9GZm9GZm9GZm9GZm9GZm9GZm9GZm9GZm9GZm9GZm9GZm9GZm9GZm9GZm9GZm9GNA==">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</Equation></Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">                                                                             u(x,0) = 1</Text-field>
<Text-field style="Text" layout="Normal">                                                                              </Text-field>
<Text-field style="Text" layout="Normal">                                                   </Text-field>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">                                                                             u(0,t) = 0 and u(\342\210\236,t) is defined                                   (4.1.1)</Font></Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">where \316\261 is the thermal diffusivity (m</Font><Font superscript="true" size="10">2</Font>/s).  Equation (4.1.1) is solved below using Maple:</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal256">restart:with(linalg):with(inttrans):with(plots):</Text-field>
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<Text-field style="Text" layout="Normal">The governing equation is stored in the equation:</Text-field>
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<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal256">eq:=diff(u(x,t),t)=alpha*diff(u(x,t),x$2);</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjZXFHNiIvLUklZGlmZkclKnByb3RlY3RlZEc2JC1JInVHRiQ2JEkieEdGJEkidEdGJEYuKiZJJmFscGhhR0YkIiIiLUYnNiRGKi1JIiRHRig2JEYtIiIjRjE=">Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ1RzYiNiRJInhHRilJInRHRilGLComSSZhbHBoYUdGKSIiIi1GJDYkLUYkNiRGJ0YrRitGLw==</Equation></Text-field>
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<Text-field style="Text" layout="Normal">Enter the initial condition here:</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal256">u(x,0):=1;</Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi1JInVHNiI2JEkieEdGJSIiISIiIg==">IiIi</Equation></Text-field>
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<Text-field style="Text" layout="Normal">The boundary condition at x = 0 is entered here:</Text-field>
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<Group labelreference="L5" drawlabel="true">
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc1:=u(0,t)=0;</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMxRzYiLy1JInVHRiQ2JCIiIUkidEdGJEYp">Ly1JInVHNiI2JCIiIUkidEdGJUYn</Equation></Text-field>
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<Text-field style="Text" layout="Normal">Enter the second boundary condition here:</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc2:=u(infinity,t)=defined;</Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMyRzYiLy1JInVHRiQ2JEkpaW5maW5pdHlHJSpwcm90ZWN0ZWRHSSJ0R0YkSShkZWZpbmVkR0Yk">Ly1JInVHNiI2JEkpaW5maW5pdHlHJSpwcm90ZWN0ZWRHSSJ0R0YlSShkZWZpbmVkR0Yl</Equation></Text-field>
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<Text-field style="Text" layout="Normal">The governing equation and the boundary condition at x = 0 are converted to the Laplace domain:</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal256">eqs:=laplace(eq,t,s);</Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXFzRzYiLywmKiZJInNHRiQiIiItSShsYXBsYWNlR0YkNiUtSSJ1R0YkNiRJInhHRiRJInRHRiRGMUYoRilGKUYpISIiKiZJJmFscGhhR0YkRiktSSVkaWZmRyUqcHJvdGVjdGVkRzYkRiotSSIkR0Y3NiRGMCIiI0Yp">LywmKiZJInNHNiIiIiItSShsYXBsYWNlR0YmNiUtSSJ1R0YmNiRJInhHRiZJInRHRiZGL0YlRidGJ0YnISIiKiZJJmFscGhhR0YmRictSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUY0NiRGKEYuRi5GJw==</Equation></Text-field>
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<Text-field style="Text" layout="Normal">The given partial differential equation is transformed to an ordinary differential equation in the Laplace domain since</Text-field>
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<Group labelreference="L8" drawlabel="true">
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal256">eqs:=subs(laplace(u(x,t),t,s)=U(x),eqs);</Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXFzRzYiLywmKiZJInNHRiQiIiItSSJVR0YkNiNJInhHRiRGKUYpRikhIiIqJkkmYWxwaGFHRiRGKS1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRGKi1JIiRHRjM2JEYtIiIjRik=">LywmKiZJInNHNiIiIiItSSJVR0YmNiNJInhHRiZGJ0YnRichIiIqJkkmYWxwaGFHRiZGJy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtRjA2JEYoRitGK0Yn</Equation></Text-field>
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<Text-field style="Text" layout="Normal">where U(x) is the dependent variable in the Laplace domain:</Text-field>
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<Group labelreference="L9" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal256">bc1:=laplace(bc1,t,s);</Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMxRzYiLy1JKGxhcGxhY2VHRiQ2JS1JInVHRiQ2JCIiIUkidEdGJEYtSSJzR0YkRiw=">Ly1JKGxhcGxhY2VHNiI2JS1JInVHRiU2JCIiIUkidEdGJUYrSSJzR0YlRio=</Equation></Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc1:=subs(laplace(u(0,t),t,s)=U(0),bc1);</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMxRzYiLy1JIlVHRiQ2IyIiIUYp">Ly1JIlVHNiI2IyIiIUYn</Equation></Text-field>
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<Text-field style="Text" layout="Normal">Next, the dependent variable in the Laplace domain is solved using the 'dsolve' command since (see chapter 3.1.6):</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">U(x):=rhs(dsolve({eqs,bc1},U(x)));</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi1JIlVHNiI2I0kieEdGJSwoKiYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyooKUkmYWxwaGFHRiUjIiIiIiIjISIiKUkic0dGJUYzRjRGJ0Y0RjRJJF9DMkdGJUY0RjQqKC1GKzYjLCRGMEY2RjQsJiomRjlGNEY4RjRGNEY0RjRGNEY4RjZGNiokRjhGNkY0">LCgqJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjKigpSSZhbHBoYUdGKSMiIiIiIiMhIiIpSSJzR0YpRi5GL0kieEdGKUYvRi9JJF9DMkdGKUYvRi8qKC1GJTYjLCRGK0YxRi8sJiomRjVGL0YzRi9GL0YvRi9GL0YzRjFGMSokRjNGMUYv</Equation></Text-field>
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<Group labelreference="L40" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">The constant _C2 is found using the boundary condition at x = \342\210\236.  The dependent variable U(x) is defined at x = \342\210\236.  In the above expression </Font><Equation executable="false" style="Text" input-equation="" display="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">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</Equation><Font encoding="UTF-8"> becomes \342\210\236 as x tends to \342\210\236.  Hence, in the above expression, the coefficient of </Font><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>
</Input>
</Group>
<Group labelreference="L12" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal256">eqc:=coeff(U(x),exp(1/alpha^(1/2)*s^(1/2)*x));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXFjRzYiSSRfQzJHRiQ=">SSRfQzJHNiI=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L13" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">_C2:=solve(eqc,_C2);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkX0MyRzYiIiIh">IiIh</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L42" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">This simplifies the solution since:</Text-field>
</Input>
</Group>
<Group labelreference="L14" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal256">U(x):=eval(U(x));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi1JIlVHNiI2I0kieEdGJSwmKiYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywkKigpSSZhbHBoYUdGJSMiIiIiIiMhIiIpSSJzR0YlRjRGNUYnRjVGN0Y1RjlGN0Y3KiRGOUY3RjU=">LCYqJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqKClJJmFscGhhR0YpIyIiIiIiIyEiIilJInNHRilGL0YwSSJ4R0YpRjBGMkYwRjRGMkYyKiRGNEYyRjA=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L43" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The solution obtained in the Laplace domain is converted to the time domain since:</Text-field>
</Input>
</Group>
<Group labelreference="L15" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal256">u:=invlaplace(U(x),s,t);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkidUc2IiwmLUkraW52bGFwbGFjZUdGJDYlKiYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IywkKigpSSZhbHBoYUdGJCMiIiIiIiMhIiIpSSJzR0YkRjRGNUkieEdGJEY1RjdGNUY5RjdGOUkidEdGJEY3RjVGNQ==">LCYtSStpbnZsYXBsYWNlRzYiNiUqJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCQqKClJJmFscGhhR0YlIyIiIiIiIyEiIilJInNHRiVGMkYzSSJ4R0YlRjNGNUYzRjdGNUY3SSJ0R0YlRjVGM0Yz</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L44" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">This solution can be further simplified since:</Text-field>
</Input>
</Group>
<Group labelreference="L16" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">u:=convert(u,erf);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkidUc2IiwmLUkraW52bGFwbGFjZUdGJDYlKiYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IywkKigpSSZhbHBoYUdGJCMiIiIiIiMhIiIpSSJzR0YkRjRGNUkieEdGJEY1RjdGNUY5RjdGOUkidEdGJEY3RjVGNQ==">LCYtSStpbnZsYXBsYWNlRzYiNiUqJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCQqKClJJmFscGhhR0YlIyIiIiIiIyEiIilJInNHRiVGMkYzSSJ4R0YlRjNGNUYzRjdGNUY3SSJ0R0YlRjVGM0Yz</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L45" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">The solution obtained can be plotted for a particular value of the parameter \316\261 since:</Font></Text-field>
</Input>
</Group>
<Group labelreference="L17" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">plot3d(subs(alpha=0.001,u),x=1..0,t=500..0,axes=boxed,title=&quot;Figure Exp. 4.1.1.&quot;,labels=[x,t,&quot;u&quot;],orientation=[120,60]);</Text-field>
</Input>
<Output>
<Text-field style="Maple Plot" layout="Maple Plot"><Plot height="300" type="three-dimensional" width="400" plot-scale="1.0" plot-xtrans="0.0" plot-ytrans="0.0">[encoded data, 13696 characters]</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L47" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">The same plot is made for a different value of \316\261 here:</Font></Text-field>
</Input>
</Group>
<Group labelreference="L18" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">plot3d(subs(alpha=0.1,u),x=1..0,t=50..0,axes=boxed,title=&quot;Figure Exp. 4.1.2.&quot;,labels=[x,t,&quot;u&quot;],orientation=[120,60]);</Text-field>
</Input>
<Output>
<Text-field style="Maple Plot" layout="Maple Plot"><Plot height="300" type="three-dimensional" width="400" plot-scale="1.0" plot-xtrans="0.0" plot-ytrans="0.0">[encoded data, 13692 characters]</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L46" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Next, the dimensionless temperature u is plotted versus x for different values of time as shown below:</Text-field>
</Input>
</Group>
<Group labelreference="L19" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">pfs:=plot([subs(alpha=0.001,t=1,u),subs(alpha=0.001,t=10,u),subs(alpha=0.001,t=100,u),subs(alpha=0.001,t=200,u)],x=0..1,axes=boxed,title=&quot;Figure Exp. 4.1.3.&quot;,thickness=4,labels=[x,&quot;u&quot;]);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="[encoded data, 5284 characters]">[encoded data, 5256 characters]</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L50" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">pts:=textplot([[0.12,evalf(subs(alpha=0.001,t=1,x=0.08,u)),&quot;t=1&quot;],[0.25,evalf(subs(alpha=0.001,t=10,x=0.2,u)),&quot;t=10&quot;],[0.58,evalf(subs(alpha=0.001,t=100,x=0.5,u)),&quot;t=100&quot;],[0.69,evalf(subs(alpha=0.001,t=200,x=0.6,u)),&quot;t=200&quot;]]);</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 prompt="&gt; " style="Maple Input" layout="Normal">display({pfs,pts});</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, 12244 characters]</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L49" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">An animation in time can be made since:</Text-field>
</Input>
</Group>
<Group labelreference="L20" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">animate(subs(alpha=0.001,u),x=0..1,t=1..500,thickness=4,title=&quot;Figure Exp. 4.1.4.&quot;,axes=boxed,labels=[x,&quot;u&quot;]);
 
</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">[encoded data, 27028 characters]</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L29" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"></Text-field>
</Input>
</Group>
</Worksheet>