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<Text-field style="Heading 3" layout="Heading 3"><Font italic="false">Example 4.5  Heat Conduction with a Source Term</Font></Text-field>
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<Input>
<Text-field style="Text" layout="Normal">The technique illustrated in the previous examples can be applied for the case where there is a source term (this source term can be a function of x and t).  Consider the transient heat conduction problem in a slab<Font superscript="true">[5]</Font></Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="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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>
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<Text-field style="Text" layout="Normal">                                                                             u(x,0) =0                                                          (4.1.5)</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) = 0<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">Equation (4.1.5) is solved in Maple below.  The programs given for the previous example can be modified to solve equation (4.1.5). </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>
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<Input>
<Text-field style="Text" layout="Normal">Only the governing equation has to be changed since:</Text-field>
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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)+sin(x)*exp(-t);</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjZXFHNiIvLUklZGlmZkclKnByb3RlY3RlZEc2JC1JInVHRiQ2JEkieEdGJEkidEdGJEYuLCYqJkkmYWxwaGFHRiQiIiItRic2JEYqLUkiJEdGKDYkRi0iIiNGMkYyKiYtSSRzaW5HNiRGKEkoX3N5c2xpYkdGJDYjRi1GMi1JJGV4cEdGPDYjLCRGLiEiIkYyRjI=">Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ1RzYiNiRJInhHRilJInRHRilGLCwmKiZJJmFscGhhR0YpIiIiLUYkNiQtRiQ2JEYnRitGK0YwRjAqJi1JJHNpbkc2JEYlSShfc3lzbGliR0YpNiNGK0YwLUkkZXhwR0Y4NiMsJEYsISIiRjBGMA==</Equation></Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal256">u(x,0):=0;</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi1JInVHNiI2JEkieEdGJSIiIUYo">IiIh</Equation></Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc1:=u(0,t)=0;</Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMxRzYiLy1JInVHRiQ2JCIiIUkidEdGJEYp">Ly1JInVHNiI2JCIiIUkidEdGJUYn</Equation></Text-field>
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<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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<Text-field style="Text" layout="Normal">The following solution and plots are obtained:</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>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXFzRzYiLyomSSJzR0YkIiIiLUkobGFwbGFjZUdGJDYlLUkidUdGJDYkSSJ4R0YkSSJ0R0YkRjBGJ0YoLCYqJkkmYWxwaGFHRiRGKC1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRGKS1JIiRHRjY2JEYvIiIjRihGKComLUkkc2luRzYkRjZJKF9zeXNsaWJHRiQ2I0YvRigsJkYoRihGJ0YoISIiRig=">LyomSSJzRzYiIiIiLUkobGFwbGFjZUdGJTYlLUkidUdGJTYkSSJ4R0YlSSJ0R0YlRi5GJEYmLCYqJkkmYWxwaGFHRiVGJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtRjM2JEYnRi1GLUYmRiYqJi1JJHNpbkc2JEY0SShfc3lzbGliR0YlNiNGLUYmLCZGJkYmRiRGJiEiIkYm</Equation></Text-field>
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<Input>
<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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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXFzRzYiLyomSSJzR0YkIiIiLUkiVUdGJDYjSSJ4R0YkRigsJiomSSZhbHBoYUdGJEYoLUklZGlmZkclKnByb3RlY3RlZEc2JEYpLUkiJEdGMjYkRiwiIiNGKEYoKiYtSSRzaW5HNiRGMkkoX3N5c2xpYkdGJEYrRigsJkYoRihGJ0YoISIiRig=">LyomSSJzRzYiIiIiLUkiVUdGJTYjSSJ4R0YlRiYsJiomSSZhbHBoYUdGJUYmLUklZGlmZkclKnByb3RlY3RlZEc2JC1GLzYkRidGKkYqRiZGJiomLUkkc2luRzYkRjBJKF9zeXNsaWJHRiVGKUYmLCZGJkYmRiRGJiEiIkYm</Equation></Text-field>
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<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="PkkkYmMxRzYiLy1JKGxhcGxhY2VHRiQ2JS1JInVHRiQ2JCIiIUkidEdGJEYtSSJzR0YkRiw=">Ly1JKGxhcGxhY2VHNiI2JS1JInVHRiU2JCIiIUkidEdGJUYrSSJzR0YlRio=</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(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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<Input>
<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="Pi1JIlVHNiI2I0kieEdGJSwoKiYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyooKUkmYWxwaGFHRiUjIiIiIiIjISIiKUkic0dGJUYzRjRGJ0Y0RjRJJF9DMkdGJUY0RjQqJkY5RjQtRis2IywkRjBGNkY0RjYqKC1JJHNpbkdGLEYmRjQsJkY0RjRGOEY0RjYsJkY4RjRGMkY0RjZGNA==">LCgqJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjKihJJmFscGhhR0YpIyEiIiIiI0kic0dGKSMiIiJGL0kieEdGKUYyRjJJJF9DMkdGKUYyRjIqJkY0RjItRiU2IywkRitGLkYyRi4qKC1JJHNpbkdGJjYjRjNGMiwmRjJGMkYwRjJGLiwmRjBGMkYsRjJGLkYy</Equation></Text-field>
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<Group labelreference="L13" 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>
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<Group labelreference="L14" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">_C2:=solve(eqc,_C2);</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkX0MyRzYiIiIh">IiIh</Equation></Text-field>
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<Group labelreference="L15" 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="Pi1JIlVHNiI2I0kieEdGJSooLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlRiYiIiIsJkYuRi5JInNHRiVGLiEiIiwmRjBGLkkmYWxwaGFHRiVGLkYx">KigtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kieEdGKCIiIiwmRitGK0kic0dGKEYrISIiLCZGLUYrSSZhbHBoYUdGKEYrRi4=</Equation></Text-field>
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<Group labelreference="L16" 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="PkkidUc2IiooLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNJInhHRiQiIiIsJkYtISIiSSZhbHBoYUdGJEYtRi8sJi1JJGV4cEdGKDYjLCQqJkYwRi1JInRHRiRGLUYvRi8tRjM2IywkRjdGL0YtRi0=">KigtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kieEdGKCIiIiwmISIiRitJJmFscGhhR0YoRitGLSwmLUkkZXhwR0YlNiMsJComRi5GK0kidEdGKEYrRi1GLS1GMTYjLCRGNUYtRitGKw==</Equation></Text-field>
</Output>
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<Group labelreference="L17" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">pars:={alpha=1/5};</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkklcGFyc0c2IjwjL0kmYWxwaGFHRiQjIiIiIiIm">PCMvSSZhbHBoYUc2IiMiIiIiIiY=</Equation></Text-field>
</Output>
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<Group labelreference="L19" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">plot3d(subs(pars,u),t=5..0,x=10..0,axes=boxed,title=&quot;Figure Exp. 4.1.11.&quot;,labels=[x,t,&quot;u&quot;]);</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, 13648 characters]</Plot></Text-field>
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<Group labelreference="L20" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">plot([subs(pars,t=0,u),subs(pars,t=0.5,u),subs(pars,t=1,u),subs(pars,t=2,u)],x=0..10,axes=boxed,title=&quot;Figure Exp. 4.1.12.&quot;,thickness=5,labels=[x,&quot;u&quot;],legend=[&quot;t=0&quot;,&quot;t=0.5&quot;,&quot;t=1&quot;,&quot;t=2&quot;]);</Text-field>
</Input>
<Output>
<Text-field style="Maple Plot" layout="Maple Plot"><Plot height="400" type="two-dimensional" width="400" plot-scale="1.0" plot-xtrans="0.0" plot-ytrans="0.0" gridlinevisibility="1" legendvisibility="true">[encoded data, 13336 characters]</Plot></Text-field>
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<Group labelreference="L25" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"></Text-field>
</Input>
</Group>
</Worksheet>