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<Text-field style="Heading 3" layout="Heading 3"><Font italic="false">Example 4.8  Heat Conduction in a Rectangular Slab</Font></Text-field>
</Input>
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<Group labelreference="L20" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Example 4.1 is solved here with the boundary and initial conditions switched.<Font superscript="true">[4]</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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">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZpby1JI21vR0YkNi1RIn5GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGQ0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRistSSZtZnJhY0dGJDYoLUYjNiUtRiw2LVErJlBhcnRpYWxEO0YnRi9GMkY1RjdGOUY7Rj1GP0ZBRkQtSSNtaUdGJDYlUSJ1RicvJSdpdGFsaWNHRjRGL0YvLUYjNiVGSy1GTzYlUSJ0RidGUkYvRi8vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmhuLyUpYmV2ZWxsZWRHRjRGLw==</Equation><Font encoding="UTF-8">=\316\261</Font><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">                                                                             u(x,0) = 0                                                    (4.1.10)</Text-field>
<Text-field style="Text" layout="Normal"></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="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg==</Equation>(0,t) = 1<Font encoding="UTF-8"> and u(\342\210\236,t) is defined </Font></Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">The following transformation is used to combined the variable:<Font superscript="true">[7]</Font></Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">                                                                                    \316\267=</Font><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"><Font encoding="UTF-8">The variable n in the new coordinate \316\267 is represented by U.  The governing equation (ordinary differential equation) for U is obtained by converting the time and special derivative in equation (4.1.10)  (partial differential equation) to derivatives in the \316\267 coordinate.  The boundary conditions for U are:</Font></Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">                                                                                     U(0) = 1                                              (4.1.12)</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">                                                                                    <Font encoding="UTF-8">U(\342\210\236) = 0                                              (4.1.13)</Font></Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">The governing equation for U is then solved with the above boundary conditions to obtain the final solution.  Example 4.8 is solved in Maple below: </Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group>
<Group labelreference="L3" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">restart:</Text-field>
</Input>
</Group>
<Group labelreference="L21" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The with(student) package is called to facilitate variable transformations:</Text-field>
</Input>
</Group>
<Group labelreference="L4" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">with(student):</Text-field>
</Input>
</Group>
<Group labelreference="L22" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The governing equation is entered here:</Text-field>
</Input>
</Group>
<Group labelreference="L5" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq:=diff(u(x,t),t)-alpha*diff(u(x,t),x$2);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjZXFHNiIsJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ1R0YkNiRJInhHRiRJInRHRiRGLiIiIiomSSZhbHBoYUdGJEYvLUYnNiRGKi1JIiRHRig2JEYtIiIjRi8hIiI=">LCYtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkidUc2IjYkSSJ4R0YpSSJ0R0YpRiwiIiIqJkkmYWxwaGFHRilGLS1GJDYkLUYkNiRGJ0YrRitGLSEiIg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L23" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">First, u(x,t) is transformed to U(\316\267(x,t)):</Font></Text-field>
</Input>
</Group>
<Group labelreference="L6" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq1:=changevar(u(x,t)=U(eta(x,t)),eq);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXExRzYiLCYqJi0tSSJER0koX3N5c2xpYkdGJDYjSSJVR0YkNiMtSSRldGFHRiQ2JEkieEdGJEkidEdGJCIiIi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRGLkYyRjNGMyomSSZhbHBoYUdGJEYzLCYqJi0tLUkjQEBHNiRGNkYqNiRGKSIiI0YrRi1GMyktRjU2JEYuRjFGQkYzRjMqJkYnRjMtRjU2JEYuLUkiJEdGNjYkRjFGQkYzRjNGMyEiIg==">LCYqJi0tSSJER0koX3N5c2xpYkc2IjYjSSJVR0YoNiMtSSRldGFHRig2JEkieEdGKEkidEdGKCIiIi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRGLEYwRjFGMSomSSZhbHBoYUdGKEYxLCYqJi0tLUkjQEBHNiRGNEYnNiRGJiIiI0YpRitGMS1GMzYkRixGL0ZARjEqJkYkRjEtRjM2JEZBRi9GMUYxRjEhIiI=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L24" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">The transformation for \316\267 is substituted here:</Font></Text-field>
</Input>
</Group>
<Group labelreference="L7" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq2:=expand(simplify(subs(eta(x,t)=x/2/(alpha*t)^(1/2),eq1)));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXEyRzYiLCYqLCMiIiIiIiVGKEkidEdGJCEiIikqJkkmYWxwaGFHRiRGKEYqRigjRigiIiNGKy0tSSJER0koX3N5c2xpYkdGJDYjSSJVR0YkNiMsJCooRi9GKEkieEdGJEYoRixGK0YoRihGOkYoRisqKEYnRigtLS1JI0BARzYkJSpwcm90ZWN0ZWRHRjQ2JEYzRjBGNUY3RihGKkYrRis=">LCYqKkkidEc2IiEiIiomSSZhbHBoYUdGJSIiIkYkRikjRiYiIiMtLUkiREdJKF9zeXNsaWJHRiU2I0kiVUdGJTYjLCQqJkkieEdGJUYpRidGKiNGKUYrRilGNUYpI0YmIiIlKiYtLS1JI0BARzYkJSpwcm90ZWN0ZWRHRi82JEYuRitGMEYyRilGJEYmRjc=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L25" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The governing equation is further simplified here:</Text-field>
</Input>
</Group>
<Group labelreference="L8" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq2:=expand(eq2*t);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXEyRzYiLCYqKiMiIiIiIiVGKCkqJkkmYWxwaGFHRiRGKEkidEdGJEYoI0YoIiIjISIiLS1JIkRHSShfc3lzbGliR0YkNiNJIlVHRiQ2IywkKihGLkYoSSJ4R0YkRihGKkYwRihGKEY6RihGMComRidGKC0tLUkjQEBHNiQlKnByb3RlY3RlZEdGNDYkRjNGL0Y1RjdGKEYw">LCYqKComSSZhbHBoYUc2IiIiIkkidEdGJkYnIyEiIiIiIy0tSSJER0koX3N5c2xpYkdGJjYjSSJVR0YmNiMsJComSSJ4R0YmRidGJEYpI0YnRitGJ0Y1RicjRioiIiUtLS1JI0BARzYkJSpwcm90ZWN0ZWRHRi82JEYuRitGMEYyRjc=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L9" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq2:=subs(x=eta*2*(alpha*t)^(1/2),eq2);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXEyRzYiLCYqKCMiIiIiIiNGKC0tSSJER0koX3N5c2xpYkdGJDYjSSJVR0YkNiNJJGV0YUdGJEYoRjFGKCEiIiomI0YoIiIlRigtLS1JI0BARzYkJSpwcm90ZWN0ZWRHRi02JEYsRilGLkYwRihGMg==">LCYqJi0tSSJER0koX3N5c2xpYkc2IjYjSSJVR0YoNiNJJGV0YUdGKCIiIkYsRi0jISIiIiIjLS0tSSNAQEc2JCUqcHJvdGVjdGVkR0YnNiRGJkYwRilGKyNGLyIiJQ==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L10" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq2:=convert(eq2,diff);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXEyRzYiLCYqKCMiIiIiIiNGKC1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJVR0YkNiNJJGV0YUdGJEYxRihGMUYoISIiKiYjRigiIiVGKC1GKzYkRi4tSSIkR0YsNiRGMUYpRihGMg==">LCYqJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJVRzYiNiNJJGV0YUdGKkYsIiIiRixGLSMhIiIiIiMtRiU2JEYkRiwjRi8iIiU=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L26" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The final form for the governing equation is:</Text-field>
</Input>
</Group>
<Group labelreference="L11" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq2:=expand(-2*eq2);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXEyRzYiLCYqJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJVR0YkNiNJJGV0YUdGJEYuIiIiRi5GL0YvKiYjRi8iIiNGLy1GKDYkRistSSIkR0YpNiRGLkYyRi9GLw==">LCYqJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJVRzYiNiNJJGV0YUdGKkYsIiIiRixGLUYtLUYlNiRGJEYsI0YtIiIj</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L27" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Enter the boundary condition here:</Text-field>
</Input>
</Group>
<Group labelreference="L12" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc1:=U(0)=1;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMxRzYiLy1JIlVHRiQ2IyIiISIiIg==">Ly1JIlVHNiI2IyIiISIiIg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L13" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc2:=U(infinity)=0;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMyRzYiLy1JIlVHRiQ2I0kpaW5maW5pdHlHJSpwcm90ZWN0ZWRHIiIh">Ly1JIlVHNiI2I0kpaW5maW5pdHlHJSpwcm90ZWN0ZWRHIiIh</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L28" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">U is solved as:</Text-field>
</Input>
</Group>
<Group labelreference="L14" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">U:=rhs(dsolve({eq2,bc1,bc2},U(eta)));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkiVUc2IiwmIiIiRiYtSSRlcmZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0kkZXRhR0YkISIi">LCYiIiJGIy1JJGVyZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSRldGFHRikhIiI=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L15" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">U:=convert(U,erfc);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkiVUc2Ii1JJWVyZmNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0kkZXRhR0Yk">LUklZXJmY0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSRldGFHRic=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L29" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Next, us as a function of x and 1 is obtained as:</Text-field>
</Input>
</Group>
<Group labelreference="L16" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">u:=subs(eta=x/2/(alpha*t)^(1/2),U);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkidUc2Ii1JJWVyZmNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IywkKigjIiIiIiIjRi5JInhHRiRGLikqJkkmYWxwaGFHRiRGLkkidEdGJEYuRi0hIiJGLg==">LUklZXJmY0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJkkieEdGJyIiIiomSSZhbHBoYUdGJ0YsSSJ0R0YnRiwjISIiIiIjI0YsRjI=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L30" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The solution obtained can be plotted:</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. 4.1.17.&quot;,labels=[x,t,&quot;u&quot;],orientation=[-60,60]);</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="L18" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"></Text-field>
</Input>
</Group>
</Worksheet>