<?xml version="1.0" encoding="UTF-8"?>
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<Annotation-table>
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<Input>
<Text-field style="Text" layout="Normal">Example 7.10 Heat Transfer in a Rectangle</Text-field>
</Input>
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<Group labelreference="L47" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"><Equation executable="true" style="2D Input" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2J1EocmVzdGFydEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjctSSNtb0dGJDYvUSI6RidGL0Y1RjgvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZORi8vJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrZXhlY3V0YWJsZUdGNEY1Rjg=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2J1EocmVzdGFydEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjctSSNtb0dGJDYvUSI6RidGL0Y1RjgvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZO</Equation></Text-field>
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<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">with(plots):</Text-field>
</Input>
</Group>
<Group labelreference="L4" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq:=diff(u(x,t),t)=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="PkkjZXFHNiIvLUklZGlmZkclKnByb3RlY3RlZEc2JC1JInVHRiQ2JEkieEdGJEkidEdGJEYuLUYnNiRGKi1JIiRHRig2JEYtIiIj">Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ1RzYiNiRJInhHRilJInRHRilGLC1GJDYkLUYkNiRGJ0YrRis=</Equation></Text-field>
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<Group labelreference="L5" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">IC:=u(x,0)=1;</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjSUNHNiIvLUkidUdGJDYkSSJ4R0YkIiIhIiIi">Ly1JInVHNiI2JEkieEdGJSIiISIiIg==</Equation></Text-field>
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</Group>
<Group labelreference="L6" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc1:=u(x,t)=0;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMxRzYiLy1JInVHRiQ2JEkieEdGJEkidEdGJCIiIQ==">Ly1JInVHNiI2JEkieEdGJUkidEdGJSIiIQ==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L7" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc2:=u(x,t)=0;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMyRzYiLy1JInVHRiQ2JEkieEdGJEkidEdGJCIiIQ==">Ly1JInVHNiI2JEkieEdGJUkidEdGJSIiIQ==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L8" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Eq:=subs(u(x,t)=X(x)*T(t),eq):</Text-field>
</Input>
</Group>
<Group labelreference="L9" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Eq:=Eq/X(x)/T(t):</Text-field>
</Input>
</Group>
<Group labelreference="L10" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Eq_T:=lhs(Eq)=-lambda^2:</Text-field>
</Input>
</Group>
<Group labelreference="L11" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">T(t):=rhs(dsolve({Eq_T,T(0)=T0},T(t)));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi1JIlRHNiI2I0kidEdGJSomSSNUMEdGJSIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCQqJilJJ2xhbWJkYUdGJSIiI0YqRidGKiEiIkYq">KiZJI1QwRzYiIiIiLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMsJComSSdsYW1iZGFHRiQiIiNJInRHRiRGJSEiIkYl</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L12" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Eq_X:=rhs(Eq)=-lambda^2:</Text-field>
</Input>
</Group>
<Group labelreference="L13" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Eq_X:=expand(Eq_X*X(x));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkklRXFfWEc2Ii8tSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkiWEdGJDYjSSJ4R0YkLUkiJEdGKDYkRi0iIiMsJComRioiIiIpSSdsYW1iZGFHRiRGMUY0ISIi">Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtRiQ2JC1JIlhHNiI2I0kieEdGK0YtRi0sJComRikiIiJJJ2xhbWJkYUdGKyIiIyEiIg==</Equation></Text-field>
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<Group labelreference="L49" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal"><Font encoding="UTF-8">The eigenvalue \316\273 is obtained using the shooting technique.  The third condition at x = 0 is assumed for </Font><Equation executable="false" style="Normal" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY4LUkmbWZyYWNHRiQ2KC1GIzYlLUkjbW9HRiQ2LVEwJkRpZmZlcmVudGlhbEQ7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZ1bnNldEYnLyUqc2VwYXJhdG9yR0Y5LyUpc3RyZXRjaHlHRjkvJSpzeW1tZXRyaWNHRjkvJShsYXJnZW9wR0Y5LyUubW92YWJsZWxpbWl0c0dGOS8lJ2FjY2VudEdGOS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkgtSSNtaUdGJDYlUSJYRicvJSdpdGFsaWNHUSZmYWxzZUYnRjRGNC1GIzYmRjAtRjE2LlEifkYnLyUwZm9udF9zdHlsZV9uYW1lR1ElVGV4dEYnRjQvRjhGUS9GO0ZRL0Y9RlEvRj9GUS9GQUZRL0ZDRlEvRkVGUUZGRkktRkw2JlEieEYnRk9GV0Y0RjQvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmNvLyUpYmV2ZWxsZWRHRlEtRjE2LlEiLkYnRldGNEZaRmVuRmZuRmduRmhuRmluRmpuRkZGSUZURlQtRkw2JlEkVGhlRidGT0ZXRjRGVC1GTDYmUSxzZW5zaXRpdml0eUYnRk9GV0Y0RlQtRkw2JlEpZXF1YXRpb25GJ0ZPRldGNEZULUZMNiZRI2lzRidGT0ZXRjRGVC1GTDYmUSpkZXZlbG9wZWRGJ0ZPRldGNEZULUZMNiZRI2ZvRidGT0ZXRjRGVC1GTDYmUSR0aGVGJ0ZPRldGNEZURl5wRlQtRkw2JlEpdmFyaWFibGVGJ0ZPRldGNEY0">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</Equation></Text-field>
<Text-field style="Text" layout="Normal">X2 = <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>
</Input>
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<Group labelreference="L14" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eqlambda:=subs(X(x)=X(x,lambda),Eq_X):</Text-field>
</Input>
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<Group labelreference="L15" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eqlambda:=diff(eqlambda,lambda):</Text-field>
</Input>
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<Group labelreference="L16" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eqlambda:=subs(diff(X(x,lambda),lambda)=X2(x),eqlambda):</Text-field>
</Input>
</Group>
<Group labelreference="L17" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eqlambda:=subs(X(x,lambda)=X(x),eqlambda);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkpZXFsYW1iZGFHNiIvLUklZGlmZkclKnByb3RlY3RlZEc2JC1JI1gyR0YkNiNJInhHRiQtSSIkR0YoNiRGLSIiIywmKiZGKiIiIilJJ2xhbWJkYUdGJEYxRjQhIiIqKEYxRjQtSSJYR0YkRixGNEY2RjRGNw==">Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtRiQ2JC1JI1gyRzYiNiNJInhHRitGLUYtLCYqJkYpIiIiSSdsYW1iZGFHRisiIiMhIiIqJi1JIlhHRitGLEYwRjFGMCEiIw==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L52" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The dependent variables are:</Text-field>
</Input>
</Group>
<Group labelreference="L18" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">vars:=(X(x),X2(x));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkldmFyc0c2IjYkLUkiWEdGJDYjSSJ4R0YkLUkjWDJHRiRGKA==">NiQtSSJYRzYiNiNJInhHRiUtSSNYMkdGJUYm</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L54" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The initial conditions for X and X2 are:</Text-field>
</Input>
</Group>
<Group labelreference="L19" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">ICs:=(X(0)=0,D(X)(0)=1,X2(0)=0,D(X2)(0)=0);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkSUNzRzYiNiYvLUkiWEdGJDYjIiIhRiovLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoRikiIiIvLUkjWDJHRiRGKUYqLy0tRi42I0Y2RilGKg==">NiYvLUkiWEc2IjYjIiIhRigvLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2I0YlRiciIiIvLUkjWDJHRiZGJ0YoLy0tRiw2I0Y0RidGKA==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L53" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">A tolerance 1e - 9 is set.  A scaling factor of 1/2 is used for obtaining eigenvalues.</Text-field>
</Input>
</Group>
<Group labelreference="L20" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">tol:=1e-9;rho:=1/2;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkdG9sRzYiJCIiIiEiKg==">JCIiIiEiKg==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkcmhvRzYiIyIiIiIiIw==">IyIiIiIiIw==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L55" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Different initial guesses are used to predict the first ten eigenvalues:</Text-field>
</Input>
</Group>
<Group labelreference="L21" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">lambdaguess:=[3,6,9,12,15,18,21,24,27,30];</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkksbGFtYmRhZ3Vlc3NHNiI3LCIiJCIiJyIiKiIjNyIjOiIjPSIjQCIjQyIjRiIjSQ==">NywiIiQiIiciIioiIzciIzoiIz0iI0AiI0MiI0YiI0k=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L22" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">MM:=nops(lambdaguess);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjTU1HNiIiIzU=">IiM1</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L23" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Xexp:=0:</Text-field>
</Input>
</Group>
<Group labelreference="L24" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">for i from 1 to MM do </Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">lambda0:=lambdaguess[i];</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">k:=1;err:=1;</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">while err&gt;tol do</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eqs:=subs(lambda=lambda0,Eq_X),subs(lambda=lambda0,eqlambda);</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Sol[i]:=dsolve({eqs,ICs},{vars},type=numeric,output=listprocedure);</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Xpred:=rhs(Sol[i](1)[2]);</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">X2pred:=rhs(Sol[i](1)[4]);</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">lambda1:=lambda0+rho*(Xexp-Xpred)/X2pred;</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">err:=abs(lambda1-lambda0);</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">lambda0:=lambda1;k:=k+1;</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">end;</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">l[i]:=lambda0;</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">kk[i]:=k;</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Err[i]:=err;</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">od:</Text-field>
</Input>
</Group>
<Group labelreference="L56" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The first ten eigenvalues are:</Text-field>
</Input>
</Group>
<Group labelreference="L25" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">seq(l[i],i=1..MM);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiwkIittRWZUSiEiKiQiKyVSJj0kRydGJSQiK2cieVpVKkYlJCIrITRQbUQiISIpJCIraWp6cTpGLCQiK09jJlwpPUYsJCIrM1w2Kj4jRiwkIiskPXVLXiNGLCQiK2ZNVkZHRiwkIitLRmZUSkYs</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L57" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The number of iterations required to get the eigenvalues are:</Text-field>
</Input>
</Group>
<Group labelreference="L26" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">seq(kk[i],i=1..MM);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiwiI0ciI0giI0lGI0YjRiNGIyIjRkYjRiQ=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L58" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The error associated with the eigenvalues is:</Text-field>
</Input>
</Group>
<Group labelreference="L27" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">seq(Err[i],i=1..MM);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiwkIiIiISIqRiNGIyQiIiFGJ0YmRiZGJkYmRiZGJg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L59" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The first ten eigenfunctions are plotted as:</Text-field>
</Input>
</Group>
<Group labelreference="L28" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">for i to MM do XX[i]:=subs(Sol[i],X(x)):od:</Text-field>
</Input>
</Group>
<Group labelreference="L29" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">for i to MM do p[i]:=plot(XX[i](x),x=0..1,thickness=3,title=&quot;Figure Exp. 7.23.&quot;,axes=boxed):od:</Text-field>
</Input>
</Group>
<Group labelreference="L30" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">display({seq(p[i],i=1..MM)},labels=[x,&quot;X&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="false">[encoded data, 47476 characters]</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L60" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The first eigenfunction does not cross the x axis.  The second eigenfunction crosses the x axis once.  Similarly, the n<Font superscript="true">th</Font> eigenfunction crosses the x axis n - 1 times.</Text-field>
</Input>
</Group>
<Group labelreference="L31" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">U:=eval(X(x)*T(t)):</Text-field>
</Input>
</Group>
<Group labelreference="L61" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The solution can be taken as:</Text-field>
</Input>
</Group>
<Group labelreference="L32" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Un:=A[n]*XX[n](x)*exp(-lambda[n]^2*t);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjVW5HNiIqKCZJIkFHRiQ2I0kibkdGJCIiIi0mSSNYWEdGJEYoNiNJInhHRiRGKi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjLCQqJikmSSdsYW1iZGFHRiRGKCIiI0YqSSJ0R0YkRiohIiJGKg==">KigmSSJBRzYiNiNJIm5HRiUiIiItJkkjWFhHRiVGJjYjSSJ4R0YlRigtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywkKiYmSSdsYW1iZGFHRiVGJiIiI0kidEdGJUYoISIiRig=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L33" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">for i to MM do lambda[i]:=l[i];od:</Text-field>
</Input>
</Group>
<Group labelreference="L34" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">u(x,t):=Sum(Un,n=1..MM);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi1JInVHNiI2JEkieEdGJUkidEdGJS1JJFN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkKigmSSJBR0YlNiNJIm5HRiUiIiItJkkjWFhHRiVGMjYjRidGNC1JJGV4cEdGKzYjLCQqJikmSSdsYW1iZGFHRiVGMiIiI0Y0RihGNCEiIkY0L0YzO0Y0IiM1">LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqKCZJIkFHRic2I0kibkdGJyIiIi0mSSNYWEdGJ0YsNiNJInhHRidGLi1JJGV4cEdGJDYjLCQqJiZJJ2xhbWJkYUdGJ0YsIiIjSSJ0R0YnRi4hIiJGLi9GLTtGLiIjNQ==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L62" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The constant A<Font subscript="true">n</Font> is obtained as:</Text-field>
</Input>
</Group>
<Group labelreference="L35" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq_An:=eval(subs(t=0,u(x,t)))=rhs(IC);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkmZXFfQW5HNiIvLUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiQqJiZJIkFHRiQ2I0kibkdGJCIiIi0mSSNYWEdGJEYvNiNJInhHRiRGMS9GMDtGMSIjNUYx">Ly1JJFN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkKiYmSSJBR0YoNiNJIm5HRigiIiItJkkjWFhHRihGLTYjSSJ4R0YoRi8vRi47Ri8iIzVGLw==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L36" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">I1:=int(X(x)^2,x=0..1):</Text-field>
</Input>
</Group>
<Group labelreference="L37" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">I2:=int(X(x),x=0..1):</Text-field>
</Input>
</Group>
<Group labelreference="L38" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">An:=I2/I1;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjQW5HNiIqJi1JJGludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYkLUkiWEdGJDYjSSJ4R0YkL0YvOyIiISIiIkYzLUYnNiQqJClGLCIiI0YzRjAhIiI=">KiYtSSRpbnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JC1JIlhHRig2I0kieEdGKC9GLTsiIiEiIiJGMS1GJDYkKiRGKiIiI0YuISIi</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L63" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The coefficient  A<Font subscript="true">n</Font><Font mathcolor="[0,0,0]" mathvariant="normal" mathsize="12" fontfamily="Times New Roman" mathbackground="[255,255,255]"> , n = 1.. .10 are numerically obtained as:</Font></Text-field>
</Input>
</Group>
<Group labelreference="L39" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">for j to MM do </Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">N:=200:I1:=0:I2:=0:</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">for i from 1 to N-1 do I1:=I1+XX[j](i/N)^2*2/(2*N);I2:=I2+XX[j](i/N)*2/(2*N)od:</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">I1:=I1+(XX[j](0)^2+XX[j](1)^2)/(2*N):</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">I2:=I2+(XX[j](0)+XX[j](1))/(2*N):</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">A[j]:=I2/I1;od:</Text-field>
</Input>
</Group>
<Group labelreference="L40" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">seq(A[j],j=1..MM);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiwkIismUjwqKipSISIqJCErQDQ7MzwhIzokIishKW9EKipSRiUkISteZmN5UkYoJCIrZnckeipSRiUkIStsZFozaEYoJCIrTCwnZipSRiUkISs5OnItJClGKCQiK0NQSyQqUkYlJCEraFQnPi0iISM5</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L41" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">u(x,t):=eval(u(x,t)):</Text-field>
</Input>
</Group>
<Group labelreference="L42" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">ua:=evalf(u(x,t));</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="L43" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">uu:=piecewise(t=0,rhs(IC),t&gt;0,ua):</Text-field>
</Input>
</Group>
<Group labelreference="L44" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">plot3d(uu,x=1..0,t=0.3..0,axes=boxed,title=&quot;Figure Exp. 7.24.&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, 13788 characters]</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L45" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">plot([subs(t=0,uu),subs(t=0.01,uu),subs(t=0.05,uu),subs(t=0.1,uu),subs(t=0.2,uu)],x=0..1,axes=boxed,title=&quot;Figure Exp. 7.25.&quot;,thickness=5,labels=[x,&quot;u&quot;],legend=[&quot;t=0&quot;,&quot;t=0.01&quot;,&quot;t=0.05&quot;,&quot;t=0.1&quot;,&quot;t=0.2&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="true">[encoded data, 13752 characters]</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L46" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"></Text-field>
</Input>
</Group>
</Worksheet>