<?xml version="1.0" encoding="UTF-8"?>
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<View-Properties presentation="false"></View-Properties>
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<Font name="Output Labels" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="8" subscript="false" superscript="false" underline="false" placeholder="false"/>
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<Layout name="Heading 4" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Ordered List 5" alignment="left" bullet="ROMAN" firstindent="0" leftmargin="144" rightmargin="0" linespacing="0.0" spaceabove="3" spacebelow="3" linebreak="space" pagebreak-before="false" initial="-1" bulletsuffix=""/>
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<Layout name="Text Output" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Bullet Item" alignment="left" bullet="dot" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="3" spacebelow="3" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Left Justified Maple Output" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.5" spaceabove="0" spacebelow="0" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Help" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Author" alignment="centred" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="8" spacebelow="8" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Diagnostic" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Heading 2" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="8" spacebelow="2" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Ordered List 3" alignment="left" bullet="roman" firstindent="0" leftmargin="72" rightmargin="0" linespacing="0.0" spaceabove="3" spacebelow="3" linebreak="space" pagebreak-before="false" initial="-1" bulletsuffix=""/>
<Layout name="Maple Plot" alignment="centred" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Normal" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Annotation Title" alignment="centred" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="12" spacebelow="12" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Dash Item" alignment="left" bullet="dash" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="3" spacebelow="3" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Heading 3" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Ordered List 4" alignment="left" bullet="ALPHABETIC" firstindent="0" leftmargin="108" rightmargin="0" linespacing="0.0" spaceabove="3" spacebelow="3" linebreak="space" pagebreak-before="false" initial="-1" bulletsuffix=""/>
<Layout name="Maple Output" alignment="centred" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.5" spaceabove="0" spacebelow="0" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="List Item" alignment="left" bullet="indent" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="3" spacebelow="3" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Line Printed Output" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Fixed Width" alignment="centred" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Warning" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Error" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Heading 1" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="8" spacebelow="4" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Title" alignment="centred" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="12" spacebelow="12" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Ordered List 2" alignment="left" bullet="alphabetic" firstindent="0" leftmargin="36" rightmargin="0" linespacing="0.0" spaceabove="3" spacebelow="3" linebreak="space" pagebreak-before="false" initial="-1" bulletsuffix=""/>
<Pencil-style name="Pencil 3" pen-color="[0,0,0]" pen-height="3.0" pen-width="3.0" pen-opacity="1.0"/>
<Pencil-style name="Pencil 1" pen-color="[0,0,0]" pen-height="1.0" pen-width="1.0" pen-opacity="1.0"/>
<Pencil-style name="Pencil 5" pen-color="[255,0,0]" pen-height="5.0" pen-width="5.0" pen-opacity="1.0"/>
<Pencil-style name="Pencil 2" pen-color="[0,0,255]" pen-height="1.0" pen-width="1.0" pen-opacity="1.0"/>
<Pencil-style name="Pencil 4" pen-color="[0,0,255]" pen-height="3.0" pen-width="3.0" pen-opacity="1.0"/>
<Highlighter-style name="Highlighter 2" pen-color="[255,204,0]" pen-height="14.0" pen-width="14.0" pen-opacity="0.8"/>
<Highlighter-style name="Highlighter 4" pen-color="[0,255,255]" pen-height="32.0" pen-width="32.0" pen-opacity="0.8"/>
<Highlighter-style name="Highlighter 3" pen-color="[51,255,0]" pen-height="24.0" pen-width="24.0" pen-opacity="0.8"/>
<Highlighter-style name="Highlighter 1" pen-color="[255,153,255]" pen-height="12.0" pen-width="8.0" pen-opacity="0.8"/>
<Highlighter-style name="Highlighter 5" pen-color="[255,255,0]" pen-height="48.0" pen-width="48.0" pen-opacity="0.8"/>
</Styles>
<Page-Numbers enabled="false" first-number="1" first-numbered-page="1" horizontal-location="right" style="Page Number" vertical-location="bottom"/>
<Task-table>
    <Task-category name="&lt;default&gt;">
    </Task-category>
</Task-table>
<Annotation-table>
    <Annotation-category name="&lt;default&gt;">
    </Annotation-category>
</Annotation-table>
<Task>
</Task>
<Group labelreference="L48" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Example 7.1  Heat Conduction in a Rectangle</Text-field>
</Input>
</Group>
<Group labelreference="L50" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The Maple program used to solve example 7.1 is given below:</Text-field>
</Input>
</Group>
<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>
</Input>
</Group>
<Group labelreference="L4" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">with(plots):</Text-field>
</Input>
</Group>
<Group labelreference="L51" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The gorging 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)=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>
</Output>
</Group>
<Group labelreference="L53" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The initial condition is entered here:</Text-field>
</Input>
</Group>
<Group labelreference="L6" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">IC:=u(x,0)=1;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjSUNHNiIvLUkidUdGJDYkSSJ4R0YkIiIhIiIi">Ly1JInVHNiI2JEkieEdGJSIiISIiIg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L52" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The boundary conditions are entered here:</Text-field>
</Input>
</Group>
<Group labelreference="L7" 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="L8" 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="L55" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Next, u is separated as u = XT (equation 7.1.5)):</Text-field>
</Input>
</Group>
<Group labelreference="L9" 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>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjRXFHNiIvLUklZGlmZkclKnByb3RlY3RlZEc2JComLUkiWEdGJDYjSSJ4R0YkIiIiLUkiVEdGJDYjSSJ0R0YkRi9GMy1GJzYkRiotSSIkR0YoNiRGLiIiIw==">Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQqJi1JIlhHNiI2I0kieEdGKiIiIi1JIlRHRio2I0kidEdGKkYtRjEtRiQ2JC1GJDYkRidGLEYs</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L10" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Eq:=Eq/X(x)/T(t);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjRXFHNiIvKiYtSSJUR0YkNiNJInRHRiQhIiItSSVkaWZmRyUqcHJvdGVjdGVkRzYkRidGKiIiIiomLUkiWEdGJDYjSSJ4R0YkRistRi02JEYyLUkiJEdGLjYkRjUiIiNGMA==">LyomLUkiVEc2IjYjSSJ0R0YmISIiLUklZGlmZkclKnByb3RlY3RlZEc2JEYkRigiIiIqJi1JIlhHRiY2I0kieEdGJkYpLUYrNiQtRis2JEYwRjNGM0Yu</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L56" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The governing equation for T is:</Text-field>
</Input>
</Group>
<Group labelreference="L11" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Eq_T:=lhs(Eq)=-lambda^2;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkklRXFfVEc2Ii8qJi1JIlRHRiQ2I0kidEdGJCEiIi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRGJ0YqIiIiLCQqJClJJ2xhbWJkYUdGJCIiI0YwRis=">LyomLUkiVEc2IjYjSSJ0R0YmISIiLUklZGlmZkclKnByb3RlY3RlZEc2JEYkRigiIiIsJCokSSdsYW1iZGFHRiYiIiNGKQ==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L57" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">T can be solved as;</Text-field>
</Input>
</Group>
<Group labelreference="L12" 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="L58" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The gorging equation for X is:</Text-field>
</Input>
</Group>
<Group labelreference="L13" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Eq_X:=rhs(Eq)=-lambda^2;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkklRXFfWEc2Ii8qJi1JIlhHRiQ2I0kieEdGJCEiIi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRGJy1JIiRHRi42JEYqIiIjIiIiLCQqJClJJ2xhbWJkYUdGJEYzRjRGKw==">LyomLUkiWEc2IjYjSSJ4R0YmISIiLUklZGlmZkclKnByb3RlY3RlZEc2JC1GKzYkRiRGKEYoIiIiLCQqJEknbGFtYmRhR0YmIiIjRik=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L59" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">X can be solved as:</Text-field>
</Input>
</Group>
<Group labelreference="L14" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">dsolve({Eq_X},X(x));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">PCMvLUkiWEc2IjYjSSJ4RzYiLCYqJkkkX0MxRzYiIiIiLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMqJkknbGFtYmRhRzYiIiIiSSJ4RzYiIiIiIiIiIiIiKiZJJF9DMkc2IiIiIi1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjKiZJJ2xhbWJkYUc2IiIiIkkieEc2IiIiIiIiIiIiIg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L60" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Maple sometimes attaches _C1 with the sine function and sometimes with the cosine function.  To avoid this, we can specify the initial conditions for X and its derivative:</Text-field>
</Input>
</Group>
<Group labelreference="L15" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">X(x):=rhs(dsolve({Eq_X,D(X)(0)=c[1]*lambda,X(0)=c[2]},X(x)));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi1JIlhHNiI2I0kieEdGJSwmKiYmSSJjR0YlNiMiIiJGLS1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjKiZJJ2xhbWJkYUdGJUYtRidGLUYtRi0qJiZGKzYjIiIjRi0tSSRjb3NHRjBGM0YtRi0=">LCYqJiZJImNHNiI2IyIiIkYoLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiMqJkknbGFtYmRhR0YmRihJInhHRiZGKEYoRigqJiZGJTYjIiIjRigtSSRjb3NHRitGLkYoRig=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L61" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The boundary conditions for X are:</Text-field>
</Input>
</Group>
<Group labelreference="L16" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Bc1:=X(x)=0;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkQmMxRzYiLywmKiYmSSJjR0YkNiMiIiJGKy1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjKiZJJ2xhbWJkYUdGJEYrSSJ4R0YkRitGK0YrKiYmRik2IyIiI0YrLUkkY29zR0YuRjFGK0YrIiIh">LywmKiYmSSJjRzYiNiMiIiJGKS1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJzYjKiZJJ2xhbWJkYUdGJ0YpSSJ4R0YnRilGKUYpKiYmRiY2IyIiI0YpLUkkY29zR0YsRi9GKUYpIiIh</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L17" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Bc2:=X(x)=0;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkQmMyRzYiLywmKiYmSSJjR0YkNiMiIiJGKy1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjKiZJJ2xhbWJkYUdGJEYrSSJ4R0YkRitGK0YrKiYmRik2IyIiI0YrLUkkY29zR0YuRjFGK0YrIiIh">LywmKiYmSSJjRzYiNiMiIiJGKS1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJzYjKiZJJ2xhbWJkYUdGJ0YpSSJ4R0YnRilGKUYpKiYmRiY2IyIiI0YpLUkkY29zR0YsRi9GKUYpIiIh</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L62" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The constant c2 is solved using the boundary condition at X = 0:</Text-field>
</Input>
</Group>
<Group labelreference="L18" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Eq_Bc1:=eval(subs(x=0,Bc1));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkknRXFfQmMxRzYiLyZJImNHRiQ2IyIiIyIiIQ==">LyZJImNHNiI2IyIiIyIiIQ==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L19" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">c[2]:=solve(Eq_Bc1,c[2]);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJImNHNiI2IyIiIyIiIQ==">IiIh</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L63" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The second boundary condition at x = 1 gives:</Text-field>
</Input>
</Group>
<Group labelreference="L20" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Eq_Bc2:=eval(subs(x=1,Bc2));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkknRXFfQmMyRzYiLyomJkkiY0dGJDYjIiIiRiotSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0knbGFtYmRhR0YkRioiIiE=">LyomJkkiY0c2IjYjIiIiRigtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2I0knbGFtYmRhR0YmRigiIiE=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L64" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">If c1 is zero, then we get the trivial solution X = 0.  Hence, sin(\316\273) should be zero:</Font></Text-field>
</Input>
</Group>
<Group labelreference="L21" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Eq_Eig:=sin(lambda)=0;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkknRXFfRWlnRzYiLy1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjSSdsYW1iZGFHRiQiIiE=">Ly1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdsYW1iZGFHRigiIiE=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L65" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">The eigenvalue equation can be solved to get the eigenvalue, \316\273:</Font></Text-field>
</Input>
</Group>
<Group labelreference="L22" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">solve(Eq_Eig,lambda);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">IiIh</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L66" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">By default, Maple picks only one eigenvalue.  We can ask Maple to find all the eigenvalues using the following command:</Text-field>
</Input>
</Group>
<Group labelreference="L23" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">_EnvAllSolutions := true:</Text-field>
</Input>
</Group>
<Group labelreference="L24" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">solve(Eq_Eig,lambda);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnJiM5NjA7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSS1GLDYlUSVfWjF8aXJGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRidGMg==">KiZJI1BpRyUqcHJvdGVjdGVkRyIiIkklX1oxfGlyRzYiRiU=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L67" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">_Z1 means the integer plane.  Hence, the eigenvalues can be taken as \316\273 = n\317\200, n = 1, 2, 3,...\342\210\236.  The solution for u can be written as:</Font></Text-field>
</Input>
</Group>
<Group labelreference="L25" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">U:=eval(X(x)*T(t));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkiVUc2IioqJkkiY0dGJDYjIiIiRiktSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyomSSdsYW1iZGFHRiRGKUkieEdGJEYpRilJI1QwR0YkRiktSSRleHBHRiw2IywkKiYpRjEiIiNGKUkidEdGJEYpISIiRik=">KiomSSJjRzYiNiMiIiJGJy1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjKiZJJ2xhbWJkYUdGJUYnSSJ4R0YlRidGJ0kjVDBHRiVGJy1JJGV4cEdGKjYjLCQqJkYvIiIjSSJ0R0YlRichIiJGJw==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L68" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The constants c1 and T0 can be combined as a single constant A<Font subscript="true">n</Font>:</Text-field>
</Input>
</Group>
<Group labelreference="L26" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Un:=subs(c[1]=A[n]/T0,lambda=lambda[n],U);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjVW5HNiIqKCZJIkFHRiQ2I0kibkdGJCIiIi1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjKiYmSSdsYW1iZGFHRiRGKEYqSSJ4R0YkRipGKi1JJGV4cEdGLTYjLCQqJilGMiIiI0YqSSJ0R0YkRiohIiJGKg==">KigmSSJBRzYiNiNJIm5HRiUiIiItSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyomJkknbGFtYmRhR0YlRiZGKEkieEdGJUYoRigtSSRleHBHRis2IywkKiZGMCIiI0kidEdGJUYoISIiRig=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L69" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Hence, the solution can be written as an infinite series:</Text-field>
</Input>
</Group>
<Group labelreference="L27" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">u(x,t):=Sum(Un,n=1..infinity);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi1JInVHNiI2JEkieEdGJUkidEdGJS1JJFN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkKigmSSJBR0YlNiNJIm5HRiUiIiItSSRzaW5HRis2IyomJkknbGFtYmRhR0YlRjJGNEYnRjRGNC1JJGV4cEdGKzYjLCQqJilGOSIiI0Y0RihGNCEiIkY0L0YzO0Y0SSlpbmZpbml0eUdGLA==">LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqKCZJIkFHRic2I0kibkdGJyIiIi1JJHNpbkdGJDYjKiYmSSdsYW1iZGFHRidGLEYuSSJ4R0YnRi5GLi1JJGV4cEdGJDYjLCQqJkYzIiIjSSJ0R0YnRi4hIiJGLi9GLTtGLkkpaW5maW5pdHlHRiU=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L70" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The values of the eigenvalues can be substituted as:</Text-field>
</Input>
</Group>
<Group labelreference="L28" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">u(x,t):=subs(lambda[n]=n*Pi,u(x,t));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi1JInVHNiI2JEkieEdGJUkidEdGJS1JJFN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkKigmSSJBR0YlNiNJIm5HRiUiIiItSSRzaW5HRis2IyooRjNGNEkjUGlHRixGNEYnRjRGNC1JJGV4cEdGKzYjLCQqKClGMyIiI0Y0KUY5RkBGNEYoRjQhIiJGNC9GMztGNEkpaW5maW5pdHlHRiw=">LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqKCZJIkFHRic2I0kibkdGJyIiIi1JJHNpbkdGJDYjKihGLUYuSSNQaUdGJUYuSSJ4R0YnRi5GLi1JJGV4cEdGJDYjLCQqKEYtIiIjRjNGOkkidEdGJ0YuISIiRi4vRi07Ri5JKWluZmluaXR5R0Yl</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L71" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The constant A<Font subscript="true">n</Font><Font mathcolor="[0,0,0]" mathvariant="normal" mathsize="12" fontfamily="Times New Roman" mathbackground="[255,255,255]"> is obtained by applying the initial condition and by using the orthogonal property of the eigenfunction:</Font></Text-field>
</Input>
</Group>
<Group labelreference="L29" 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="PkkmZXFfQW5HNiIvLUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiQqJiZJIkFHRiQ2I0kibkdGJCIiIi1JJHNpbkdGKDYjKihGMEYxSSNQaUdGKUYxSSJ4R0YkRjFGMS9GMDtGMUkpaW5maW5pdHlHRilGMQ==">Ly1JJFN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkKiYmSSJBR0YoNiNJIm5HRigiIiItSSRzaW5HRiU2IyooRi5GL0kjUGlHRiZGL0kieEdGKEYvRi8vRi47Ri9JKWluZmluaXR5R0YmRi8=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L30" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">I1:=int((sin(n*Pi*x))^2,x=0..1);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjSTFHNiItSSpQSUVDRVdJU0VHRiQ2JDckIiIhL0kibkdGJEYpNyQsJCoqIyIiIiIiI0YwLCYqJi1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjKiZGK0YwSSNQaUdGN0YwRjAtSSRzaW5HRjZGOUYwISIiRjpGMEYwRitGPkY7Rj5GMEkqb3RoZXJ3aXNlR0Yk">LUkqcGllY2V3aXNlRyUqcHJvdGVjdGVkRzYlL0kibkc2IiIiIUYpLCQqKCwmKiYtSSRjb3NHNiRGJEkoX3N5c2xpYkdGKDYjKiZGJyIiIkkjUGlHRiRGNEY0LUkkc2luR0YwRjJGNCEiIkYzRjRGNEYnRjhGNUY4I0Y0IiIj</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L31" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">I2:=int(sin(n*Pi*x),x=0..1);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjSTJHNiItSSpQSUVDRVdJU0VHRiQ2JDckIiIhL0kibkdGJEYpNyQsJCooLCYtSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyomRisiIiJJI1BpR0YzRjdGN0Y3ISIiRjdGK0Y5RjhGOUY5SSpvdGhlcndpc2VHRiQ=">LUkqcGllY2V3aXNlRyUqcHJvdGVjdGVkRzYlL0kibkc2IiIiIUYpLCQqKCwmLUkkY29zRzYkRiRJKF9zeXNsaWJHRig2IyomRiciIiJJI1BpR0YkRjNGMyEiIkYzRjNGJ0Y1RjRGNUY1</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L72" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">The integrals can be simplified by substituting sin(n\317\200) = 0:</Font></Text-field>
</Input>
</Group>
<Group labelreference="L32" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">vars:={sin(n*Pi)=0};</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkldmFyc0c2IjwjLy1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjKiZJIm5HRiQiIiJJI1BpR0YqRi8iIiE=">PCMvLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMqJkkibkc2IiIiIkkjUGlHJSpwcm90ZWN0ZWRHIiIiIiIh</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L33" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">I1:=subs(vars,I1);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjSTFHNiItSSpQSUVDRVdJU0VHRiQ2JDckIiIhL0kibkdGJEYpNyQjIiIiIiIjSSpvdGhlcndpc2VHRiQ=">LUkqcGllY2V3aXNlRyUqcHJvdGVjdGVkRzYlL0kibkc2IiIiIUYpIyIiIiIiIw==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L34" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">A[n]:=I2/I1;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJIkFHNiI2I0kibkdGJSomLUkqUElFQ0VXSVNFR0YlNiQ3JCIiIS9GJ0YtNyQsJCooLCYtSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyomRiciIiJJI1BpR0Y2RjpGOkY6ISIiRjpGJ0Y8RjtGPEY8SSpvdGhlcndpc2VHRiVGOi1GKjYkRiw3JCNGOiIiI0Y9Rjw=">KiYtSSpwaWVjZXdpc2VHJSpwcm90ZWN0ZWRHNiUvSSJuRzYiIiIhRiosJCooLCYtSSRjb3NHNiRGJUkoX3N5c2xpYkdGKTYjKiZGKCIiIkkjUGlHRiVGNEY0ISIiRjRGNEYoRjZGNUY2RjZGNC1GJDYlRidGKiNGNCIiI0Y2</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L73" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The dimensionless temperature profile is given by:</Text-field>
</Input>
</Group>
<Group labelreference="L35" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">u(x,t):=eval(u(x,t));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi1JInVHNiI2JEkieEdGJUkidEdGJS1JJFN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkKiotSSpQSUVDRVdJU0VHRiU2JDckIiIhL0kibkdGJUY0NyQsJCooLCYtSSRjb3NHRis2IyomRjYiIiJJI1BpR0YsRj9GP0Y/ISIiRj9GNkZBRkBGQUZBSSpvdGhlcndpc2VHRiVGPy1GMTYkRjM3JCNGPyIiI0ZCRkEtSSRzaW5HRis2IyooRjZGP0ZARj9GJ0Y/Rj8tSSRleHBHRis2IywkKigpRjZGR0Y/KUZARkdGP0YoRj9GQUY/L0Y2O0Y/SSlpbmZpbml0eUdGLA==">LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqKi1JKnBpZWNld2lzZUdGJTYlL0kibkdGJyIiIUYvLCQqKCwmLUkkY29zR0YkNiMqJkYuIiIiSSNQaUdGJUY3RjchIiJGN0Y3Ri5GOUY4RjlGOUY3LUYrNiVGLUYvI0Y3IiIjRjktSSRzaW5HRiQ2IyooRi5GN0Y4RjdJInhHRidGN0Y3LUkkZXhwR0YkNiMsJCooRi5GPUY4Rj1JInRHRidGN0Y5RjcvRi47RjdJKWluZmluaXR5R0Yl</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L74" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">For plotting purposes, we replace \342\210\236 by, N, an integer:</Font></Text-field>
</Input>
</Group>
<Group labelreference="L36" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">u(x,t):=subs(infinity=N,u(x,t));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi1JInVHNiI2JEkieEdGJUkidEdGJS1JJFN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkKiotSSpQSUVDRVdJU0VHRiU2JDckIiIhL0kibkdGJUY0NyQsJCooLCYtSSRjb3NHRis2IyomRjYiIiJJI1BpR0YsRj9GP0Y/ISIiRj9GNkZBRkBGQUZBSSpvdGhlcndpc2VHRiVGPy1GMTYkRjM3JCNGPyIiI0ZCRkEtSSRzaW5HRis2IyooRjZGP0ZARj9GJ0Y/Rj8tSSRleHBHRis2IywkKigpRjZGR0Y/KUZARkdGP0YoRj9GQUY/L0Y2O0Y/SSJOR0Yl">LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqKi1JKnBpZWNld2lzZUdGJTYlL0kibkdGJyIiIUYvLCQqKCwmLUkkY29zR0YkNiMqJkYuIiIiSSNQaUdGJUY3RjchIiJGN0Y3Ri5GOUY4RjlGOUY3LUYrNiVGLUYvI0Y3IiIjRjktSSRzaW5HRiQ2IyooRi5GN0Y4RjdJInhHRidGN0Y3LUkkZXhwR0YkNiMsJCooRi5GPUY4Rj1JInRHRidGN0Y5RjcvRi47RjdJIk5HRic=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L75" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">N = 20 is enough for this problem:</Text-field>
</Input>
</Group>
<Group labelreference="L37" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">ua:=subs(N=20,u(x,t));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjdWFHNiItSSRTdW1HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2JCoqLUkqUElFQ0VXSVNFR0YkNiQ3JCIiIS9JIm5HRiRGMDckLCQqKCwmLUkkY29zR0YnNiMqJkYyIiIiSSNQaUdGKEY7RjtGOyEiIkY7RjJGPUY8Rj1GPUkqb3RoZXJ3aXNlR0YkRjstRi02JEYvNyQjRjsiIiNGPkY9LUkkc2luR0YnNiMqKEYyRjtGPEY7SSJ4R0YkRjtGOy1JJGV4cEdGJzYjLCQqKClGMkZDRjspRjxGQ0Y7SSJ0R0YkRjtGPUY7L0YyO0Y7IiM/">LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqKi1JKnBpZWNld2lzZUdGJTYlL0kibkdGJyIiIUYvLCQqKCwmLUkkY29zR0YkNiMqJkYuIiIiSSNQaUdGJUY3RjchIiJGN0Y3Ri5GOUY4RjlGOUY3LUYrNiVGLUYvI0Y3IiIjRjktSSRzaW5HRiQ2IyooRi5GN0Y4RjdJInhHRidGN0Y3LUkkZXhwR0YkNiMsJCooRi5GPUY4Rj1JInRHRidGN0Y5RjcvRi47RjciIz8=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L76" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">The solution obtained is plotted at t = 0.  We observe oscillations about initial condition 1.  This is called Gibb's phenomenon.  Theoretically it will take N = \342\210\236 for this profile to become u = 1 at t = 0.</Font></Text-field>
</Input>
</Group>
<Group labelreference="L38" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">plot(eval(subs(t=0,ua)),x=0..1,axes=boxed,title=&quot;Figure Exp. 7.1.&quot;,thickness=3,labels=[x,&quot;u(x,0)&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, 10784 characters]</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L77" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Next, the initial condition is predicted using N = 100 terms in the series:</Text-field>
</Input>
</Group>
<Group labelreference="L39" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">ua2:=subs(N=100,u(x,t));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkdWEyRzYiLUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiQqKi1JKlBJRUNFV0lTRUdGJDYkNyQiIiEvSSJuR0YkRjA3JCwkKigsJi1JJGNvc0dGJzYjKiZGMiIiIkkjUGlHRihGO0Y7RjshIiJGO0YyRj1GPEY9Rj1JKm90aGVyd2lzZUdGJEY7LUYtNiRGLzckI0Y7IiIjRj5GPS1JJHNpbkdGJzYjKihGMkY7RjxGO0kieEdGJEY7RjstSSRleHBHRic2IywkKigpRjJGQ0Y7KUY8RkNGO0kidEdGJEY7Rj1GOy9GMjtGOyIkKyI=">LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqKi1JKnBpZWNld2lzZUdGJTYlL0kibkdGJyIiIUYvLCQqKCwmLUkkY29zR0YkNiMqJkYuIiIiSSNQaUdGJUY3RjchIiJGN0Y3Ri5GOUY4RjlGOUY3LUYrNiVGLUYvI0Y3IiIjRjktSSRzaW5HRiQ2IyooRi5GN0Y4RjdJInhHRidGN0Y3LUkkZXhwR0YkNiMsJCooRi5GPUY4Rj1JInRHRidGN0Y5RjcvRi47RjciJCsi</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L40" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">plot(eval(subs(t=0,ua2)),x=0..1,axes=boxed,title=&quot;Figure Exp. 7.2.&quot;,thickness=3,labels=[x,&quot;u(x,0)&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, 9528 characters]</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L78" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">We observe better predictions as the number of terms increase.  However, we need more terms at t = 0.  For values of t &gt; 0, N = 20 is enough.  Hence, a piecewise polynomial can be used to  define the initial condition at t = 0 and the separation of variables solution obtained can be used for values of t &gt; 0.</Text-field>
</Input>
</Group>
<Group labelreference="L41" 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>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjdXVHNiItSSpQSUVDRVdJU0VHRiQ2JDckIiIiL0kidEdGJCIiITckLUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiQqKi1GJjYkNyRGLC9JIm5HRiRGLDckLCQqKCwmLUkkY29zR0YwNiMqJkY5RilJI1BpR0YxRilGKUYpISIiRilGOUZDRkJGQ0ZDSSpvdGhlcndpc2VHRiRGKS1GJjYkRjc3JCNGKSIiI0ZERkMtSSRzaW5HRjA2IyooRjlGKUZCRilJInhHRiRGKUYpLUkkZXhwR0YwNiMsJCooKUY5RklGKSlGQkZJRilGK0YpRkNGKS9GOTtGKSIjPzJGLEYr">LUkqcGllY2V3aXNlRyUqcHJvdGVjdGVkRzYmL0kidEc2IiIiISIiIjJGKUYnLUkkU3VtRzYkRiRJKF9zeXNsaWJHRig2JCoqLUYjNiUvSSJuR0YoRilGKSwkKigsJi1JJGNvc0dGLjYjKiZGNUYqSSNQaUdGJEYqRiohIiJGKkYqRjVGPkY9Rj5GPkYqLUYjNiVGNEYpI0YqIiIjRj4tSSRzaW5HRi42IyooRjVGKkY9RipJInhHRihGKkYqLUkkZXhwR0YuNiMsJCooRjVGQkY9RkJGJ0YqRj5GKi9GNTtGKiIjPw==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L79" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">A three dimensional plot can be made as:</Text-field>
</Input>
</Group>
<Group labelreference="L42" 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. 7.3.&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, 13688 characters]</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L80" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The profiles across the slab at different times can be plotted as:</Text-field>
</Input>
</Group>
<Group labelreference="L43" 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. 7.4.&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>
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