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<Text-field style="Heading 3" layout="Heading 3"><Font italic="false">Example 4.6</Font></Text-field>
</Input>
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<Input>
<Text-field style="Text" layout="Normal">Consider the transient heat conduction problem in a slab.<Font superscript="true">[4]</Font>  The governing equation in dimensionless form 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="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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) = 1                                                    (4.1.6)</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="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">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</Equation> (0,t) = h*u(0,t)<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"><Font encoding="UTF-8">where \316\261 is the thermal diffusivity (m2/s) and h is the heat transfer coefficient (m-1).  Carslaw and Jaeger</Font><Font superscript="true">[4]</Font> presented solutions for this problem after transforming the governing equation and boundary conditions to a form convenient for similarity transformation.  Equation (4.1.6) is solved in Maple below using the Laplace transform technique (note that the transformation is not necessary with this approach):              </Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal256">restart:with(linalg):with(inttrans):with(plots):</Text-field>
</Input>
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<Group labelreference="L33" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The governing equation is entered here:</Text-field>
</Input>
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<Group labelreference="L4" drawlabel="true">
<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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<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 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">Enter the boundary condition here:</Text-field>
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<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc1:=diff(u(x,t),x)-h*u(x,t)=0;</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMxRzYiLywmLUklZGlmZkclKnByb3RlY3RlZEc2JC1JInVHRiQ2JEkieEdGJEkidEdGJEYuIiIiKiZJImhHRiRGMEYrRjAhIiIiIiE=">LywmLUklZGlmZkclKnByb3RlY3RlZEc2JC1JInVHNiI2JEkieEdGKkkidEdGKkYsIiIiKiZJImhHRipGLkYoRi4hIiIiIiE=</Equation></Text-field>
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<Group labelreference="L7" drawlabel="true">
<Input>
<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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<Group labelreference="L8" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal256">eqs:=laplace(eq,t,s);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXFzRzYiLywmKiZJInNHRiQiIiItSShsYXBsYWNlR0YkNiUtSSJ1R0YkNiRJInhHRiRJInRHRiRGMUYoRilGKUYpISIiKiZJJmFscGhhR0YkRiktSSVkaWZmRyUqcHJvdGVjdGVkRzYkRiotSSIkR0Y3NiRGMCIiI0Yp">LywmKiZJInNHNiIiIiItSShsYXBsYWNlR0YmNiUtSSJ1R0YmNiRJInhHRiZJInRHRiZGL0YlRidGJ0YnISIiKiZJJmFscGhhR0YmRictSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUY0NiRGKEYuRi5GJw==</Equation></Text-field>
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<Group labelreference="L9" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal256">eqs:=subs(laplace(u(x,t),t,s)=U(x),eqs);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXFzRzYiLywmKiZJInNHRiQiIiItSSJVR0YkNiNJInhHRiRGKUYpRikhIiIqJkkmYWxwaGFHRiRGKS1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRGKi1JIiRHRjM2JEYtIiIjRik=">LywmKiZJInNHNiIiIiItSSJVR0YmNiNJInhHRiZGJ0YnRichIiIqJkkmYWxwaGFHRiZGJy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtRjA2JEYoRitGK0Yn</Equation></Text-field>
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<Group labelreference="L10" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal256">bc1:=laplace(bc1,t,s);</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMxRzYiLywmLUklZGlmZkclKnByb3RlY3RlZEc2JC1JKGxhcGxhY2VHRiQ2JS1JInVHRiQ2JEkieEdGJEkidEdGJEYySSJzR0YkRjEiIiIqJkkiaEdGJEY0RitGNCEiIiIiIQ==">LywmLUklZGlmZkclKnByb3RlY3RlZEc2JC1JKGxhcGxhY2VHNiI2JS1JInVHRio2JEkieEdGKkkidEdGKkYwSSJzR0YqRi8iIiIqJkkiaEdGKkYyRihGMiEiIiIiIQ==</Equation></Text-field>
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<Group labelreference="L11" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc1:=subs(laplace(u(x,t),t,s)=U(x),bc1);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMxRzYiLywmLUklZGlmZkclKnByb3RlY3RlZEc2JC1JIlVHRiQ2I0kieEdGJEYuIiIiKiZJImhHRiRGL0YrRi8hIiIiIiE=">LywmLUklZGlmZkclKnByb3RlY3RlZEc2JC1JIlVHNiI2I0kieEdGKkYsIiIiKiZJImhHRipGLUYoRi0hIiIiIiE=</Equation></Text-field>
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<Group labelreference="L12" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc1:=convert(bc1,D);</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMxRzYiLywmLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0kiVUdGJDYjSSJ4R0YkIiIiKiZJImhHRiRGMS1GLkYvRjEhIiIiIiE=">LywmLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kiVUdGKjYjSSJ4R0YqIiIiKiZJImhHRipGLy1GLEYtRi8hIiIiIiE=</Equation></Text-field>
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<Input>
<Text-field style="Text" layout="Normal">The boundary condition in the Laplace domain is:</Text-field>
</Input>
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<Group labelreference="L13" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc1:=subs(x=0,bc1);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMxRzYiLywmLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0kiVUdGJDYjIiIhIiIiKiZJImhHRiRGMS1GLkYvRjEhIiJGMA==">LywmLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kiVUdGKjYjIiIhIiIiKiZJImhHRipGLy1GLEYtRi8hIiJGLg==</Equation></Text-field>
</Output>
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<Group labelreference="L39" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">U is solved as:</Text-field>
</Input>
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<Group labelreference="L14" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">U(x):=rhs(dsolve({eqs,bc1}));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi1JIlVHNiI2I0kieEdGJSwoKiojIiIiIiIjRistSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyooKUkmYWxwaGFHRiVGKiEiIilJInNHRiVGKkYrRidGK0YrLCgqKkY3RitJImhHRiVGK0kkX0MxR0YlRitGNEYrRisqJkY8RitGOEYrRitGK0Y2RitGOEY2RisqKkYqRistRi42IywkRjNGNkYrLChGPUYrRjpGNkYrRjZGK0Y4RjZGKyokRjhGNkYr">LCgqKiMiIiIiIiNGJS1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjKigpSSZhbHBoYUdGLEYkISIiKUkic0dGLEYkRiVJInhHRixGJUYlLCgqKkYyRiVJImhHRixGJUkkX0MxR0YsRiVGL0YlRiUqJkY4RiVGM0YlRiVGJUYxRiVGM0YxRiUqKkYkRiUtRig2IywkRi5GMUYlLChGOUYlRjZGMUYlRjFGJUYzRjFGJSokRjNGMUYl</Equation></Text-field>
</Output>
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<Group labelreference="L15" 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="PkkkZXFjRzYiLCQqKCMiIiIiIiNGKCwoKiopSSJzR0YkRidGKEkiaEdGJEYoSSRfQzFHRiRGKClJJmFscGhhR0YkRidGKEYoKiZGL0YoRi1GKEYoRighIiJGKEYtRjNGKA==">LCQqKCMiIiIiIiNGJSwoKiopSSJzRzYiRiRGJUkiaEdGK0YlSSRfQzFHRitGJSlJJmFscGhhR0YrRiRGJUYlKiZGLUYlRipGJUYlRiUhIiJGJUYqRjFGJQ==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L16" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">coef:=solve(eqc,{_C1});</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkklY29lZkc2IjwjL0kkX0MxR0YkKiQsJiooKUkic0dGJCMiIiIiIiNGLkkiaEdGJEYuKUkmYWxwaGFHRiRGLUYuRi5GLEYuISIi">PCMvSSRfQzFHNiIqJCwmKigpSSJzRzYiIyIiIiIiIyIiIkkiaEc2IiIiIilJJmFscGhhRzYiIyIiIiIiIyIiIiIiIkkic0c2IiIiIiEiIg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L40" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The temperature profile in the Laplace domain is:</Text-field>
</Input>
</Group>
<Group labelreference="L85" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">U(x):=evala(subs(coef,U(x)));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi1JIlVHNiI2I0kieEdGJSwmKiwtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywkKigpSSZhbHBoYUdGJSMiIiIiIiMhIiIpSSJzR0YlRjRGNUYnRjVGN0Y1RjNGNUkiaEdGJUY1LCYqJkY6RjVGM0Y1RjUqJkY4RjVGMkY1RjVGN0Y5RjdGNyokRjlGN0Y1">LCYqLC1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqKClJJmFscGhhR0YpIyIiIiIiIyEiIilJInNHRilGL0YwSSJ4R0YpRjBGMkYwRi5GMEkiaEdGKUYwLCYqJkY2RjBGLkYwRjAqJkYzRjBGLUYwRjBGMkY0RjJGMiokRjRGMkYw</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L41" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Maple cannot find the inverse of the Laplace transform:</Text-field>
</Input>
</Group>
<Group labelreference="L18" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">invlaplace(U(x),s,t);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LCotSStpbnZsYXBsYWNlRzYiNiUqJkkic0dGJSEiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCQqKClJJmFscGhhR0YlIyIiIiIiI0YpKUYoRjRGNUkieEdGJUY1RilGNUYoSSJ0R0YlRikqKEkiaEdGJUYpRjJGKS1GJDYlKiZGN0YpRipGNUYoRjlGNUY1KiZGO0YpLUYkNiUqJiwmKiZGO0Y1RjNGNUY1KiZGN0Y1RjJGNUY1RilGKkY1RihGOUY1RilGNUY1</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L42" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The above expression is split into two terms U1 and U2 as shown below:</Text-field>
</Input>
</Group>
<Group labelreference="L19" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">U1:=1/(s);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjVTFHNiIqJEkic0dGJCEiIg==">KiRJInNHNiIhIiI=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L20" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">u1:=invlaplace(U1,s,t);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjdTFHNiIiIiI=">IiIi</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L21" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">U2:=U(x)-U1;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjVTJHNiIsJCosLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMsJCooKUkmYWxwaGFHRiQjIiIiIiIjISIiKUkic0dGJEYxRjJJInhHRiRGMkY0RjJGMEYySSJoR0YkRjIsJiomRjhGMkYwRjJGMiomRjVGMkYvRjJGMkY0RjZGNEY0">LCQqLC1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqKClJJmFscGhhR0YpIyIiIiIiIyEiIilJInNHRilGL0YwSSJ4R0YpRjBGMkYwRi5GMEkiaEdGKUYwLCYqJkY2RjBGLkYwRjAqJkYzRjBGLUYwRjBGMkY0RjJGMg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L87" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The inverse of U2 is:</Text-field>
</Input>
</Group>
<Group labelreference="L88" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">U2:=subs({x=x1,alpha=alpha1},U2);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjVTJHNiIsJCosLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMsJCooKUknYWxwaGExR0YkIyIiIiIiIyEiIilJInNHRiRGMUYySSN4MUdGJEYyRjRGMkYwRjJJImhHRiRGMiwmKiZGOEYyRjBGMkYyKiZGNUYyRi9GMkYyRjRGNkY0RjQ=">LCQqLC1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqKClJJ2FscGhhMUdGKSMiIiIiIiMhIiIpSSJzR0YpRi9GMEkjeDFHRilGMEYyRjBGLkYwSSJoR0YpRjAsJiomRjZGMEYuRjBGMComRjNGMEYtRjBGMEYyRjRGMkYy</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L86" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">assume(x1&gt;0,alpha1&gt;0);</Text-field>
</Input>
</Group>
<Group labelreference="L22" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">u2:=invlaplace(U2,s,t);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjdTJHNiIsJiomLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMqJkkiaEdGJCIiIiwmKihGLkYvSShhbHBoYTF8aXJHRiRGL0kidEdGJEYvRi9JJHgxfGlyR0YkRi9GL0YvLUklZXJmY0dGKTYjLCQqKCNGLyIiI0YvKSomRjJGL0YzRi9GOiEiIiwmRjRGLyoqRjtGL0YuRi9GMkYvRjNGL0YvRi9GL0YvRi8tRjY2IywkKihGOkYvRjxGPkY0Ri9GL0Y+">LCYqJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjKiZJImhHRikiIiIsJiooRixGLUkoYWxwaGExfGlyR0YpRi1JInRHRilGLUYtSSR4MXxpckdGKUYtRi1GLS1JJWVyZmNHRiY2IywkKigjRi0iIiNGLSkqJkYwRi1GMUYtRjghIiIsJkYyRi0qKkY5Ri1GLEYtRjBGLUYxRi1GLUYtRi1GLUYtLUY0NiMsJCooRjhGLUY6RjxGMkYtRi1GPA==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L89" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">u2:=subs({x1=x,alpha1=alpha},u2);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjdTJHNiIsJiomLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMqJkkiaEdGJCIiIiwmKihGLkYvSSZhbHBoYUdGJEYvSSJ0R0YkRi9GL0kieEdGJEYvRi9GLy1JJWVyZmNHRik2IywkKigjRi8iIiNGLykqJkYyRi9GM0YvRjohIiIsJkY0Ri8qKkY7Ri9GLkYvRjJGL0YzRi9GL0YvRi9GL0YvLUY2NiMsJCooRjpGL0Y8Rj5GNEYvRi9GPg==">LCYqJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjKiZJImhHRikiIiIsJiooRixGLUkmYWxwaGFHRilGLUkidEdGKUYtRi1JInhHRilGLUYtRi0tSSVlcmZjR0YmNiMsJCooI0YtIiIjRi0pKiZGMEYtRjFGLUY4ISIiLCZGMkYtKipGOUYtRixGLUYwRi1GMUYtRi1GLUYtRi1GLS1GNDYjLCQqKEY4Ri1GOkY8RjJGLUYtRjw=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L44" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group>
<Group labelreference="L45" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Finally, the temperature distribution is given by:</Text-field>
</Input>
</Group>
<Group labelreference="L26" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">u:=u1+u2;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkidUc2IiwoIiIiRiYqJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjKiZJImhHRiRGJiwmKihGL0YmSSZhbHBoYUdGJEYmSSJ0R0YkRiZGJkkieEdGJEYmRiZGJi1JJWVyZmNHRio2IywkKigjRiYiIiNGJikqJkYyRiZGM0YmRjohIiIsJkY0RiYqKkY7RiZGL0YmRjJGJkYzRiZGJkYmRiZGJkYmLUY2NiMsJCooRjpGJkY8Rj5GNEYmRiZGPg==">LCgiIiJGIyomLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMqJkkiaEdGKkYjLCYqKEYtRiNJJmFscGhhR0YqRiNJInRHRipGI0YjSSJ4R0YqRiNGI0YjLUklZXJmY0dGJzYjLCQqKCNGIyIiI0YjKSomRjBGI0YxRiNGOCEiIiwmRjJGIyoqRjlGI0YtRiNGMEYjRjFGI0YjRiNGI0YjRiMtRjQ2IywkKihGOEYjRjpGPEYyRiNGI0Y8</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L27" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">pars:={alpha=1e-3,h=0.01};</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkklcGFyc0c2IjwkL0kmYWxwaGFHRiQkIiIiISIkL0kiaEdGJCRGKSEiIw==">PCQvSSZhbHBoYUc2IiQiIiIhIiQvSSJoR0YlJEYnISIj</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L28" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">plot3d(subs(pars,u),x=1..0,t=500..0,axes=boxed,title=&quot;Figure Exp. 4.1.13.&quot;,labels=[x,t,&quot;u&quot;],orientation=[110,60]);</Text-field>
</Input>
<Output>
<Text-field style="Maple Plot" layout="Maple Plot"><Plot height="400" 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="L29" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">pfs:=plot([subs(pars,t=10,u),subs(pars,t=100,u),subs(pars,t=200,u),subs(pars,t=500,u)],x=0..1,axes=boxed,title=&quot;Figure Exp. 4.1.14.&quot;,thickness=5,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, 7256 characters]">[encoded data, 7228 characters]</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L90" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">pts:=textplot([[0.2,evalf(subs(pars,t=10,x=0.1,u)),&quot;t=10&quot;],[0.3,evalf(subs(pars,t=100,x=0.2,u)),&quot;t=100&quot;],[0.4,evalf(subs(pars,t=200,x=0.3,u)),&quot;t=200&quot;],[0.5,evalf(subs(pars,t=500,x=0.4,u)),&quot;t=500&quot;]]);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkcHRzRzYiLUkvSU5URVJGQUNFX1BMT1RHRiQ2Ji1JJVRFWFRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2JDckJCIiIyEiIiQiK24qNGcqKiohIzVRJXQ9MTBGJC1GKTYkNyQkIiIkRjEkIitdUCYzKSoqRjRRJnQ9MTAwRiQtRik2JDckJCIiJUYxJCIrUkAwdSoqRjRRJnQ9MjAwRiQtRik2JDckJCIiJkYxJCIrVE07YSoqRjRRJnQ9NTAwRiQ=">LSUlUExPVEc2Ji1JJVRFWFRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JDckJCIiIyEiIiQiK24qNGcqKiohIzVRJXQ9MTBGKi1GJjYkNyQkIiIkRi8kIitdUCYzKSoqRjJRJnQ9MTAwRiotRiY2JDckJCIiJUYvJCIrUkAwdSoqRjJRJnQ9MjAwRiotRiY2JDckJCIiJkYvJCIrVE07YSoqRjJRJnQ9NTAwRio=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L91" 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, 13952 characters]</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L30" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">us:=eval(subs(x=0,u));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjdXNHNiIqJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjKigpSSJoR0YkIiIjIiIiSSZhbHBoYUdGJEYwSSJ0R0YkRjBGMC1JJWVyZmNHRig2IyoqKSomRjFGMEYyRjAjRjBGLyEiIkYuRjBGMUYwRjJGMEYw">KiYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyooKUkiaEdGKCIiIyIiIkkmYWxwaGFHRihGLkkidEdGKEYuRi4tSSVlcmZjR0YlNiMqKikqJkYvRi5GMEYuI0YuRi0hIiJGLEYuRi9GLkYwRi5GLg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L46" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"></Text-field>
</Input>
</Group>
</Worksheet>