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<Text-field style="Heading 3" layout="Heading 3"><Font italic="false">Example 3.1 Exponential Method for Linear Boundary Value Problem</Font></Text-field>
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<Text-field style="Text" layout="Normal">Consider the conduciton of heat in a rectangular cooling fin.  The governing differential equation<Font superscript="true">[6]</Font> in dimensionless form is</Text-field>
<Text-field style="Text" layout="Normal" alignment="centred"></Text-field>
<Text-field style="Text" layout="Normal" alignment="centred"><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" alignment="centred"><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" alignment="centred"><Equation executable="false" style="2D Math" input-equation="" display="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">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDRistSSNtaUdGJDYlUScmIzk1MjtGJy8lJ2l0YWxpY0dGNEYvLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtbkdGJDYkUSIwRidGL0YvRi8tRiw2LVEiPUYnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yNzc3Nzc4ZW1GJy9GRUZZLUZSNiRRIjFGJ0YvRistRiw2L1EkYW5kRicvJSVib2xkR0Y0LyUwZm9udF9zdHlsZV9uYW1lR1ElVGV4dEYnRi9GMkY1RjdGOUY7Rj1GP0ZBRkQtSSZtZnJhY0dGJDYoLUYjNiUtRiw2LlEwJkRpZmZlcmVudGlhbEQ7RidGXW9GLy9GM1EmdW5zZXRGJy9GNkZpby9GOEZpby9GOkZpby9GPEZpby9GPkZpby9GQEZpb0ZBRkQtRkc2JkZJRkpGXW9GL0YvLUYjNiVGZW8tRkc2JlEieEYnRkpGXW9GL0YvLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZccS8lKWJldmVsbGVkR0Y0Ri8=</Equation><Equation executable="false" style="Text" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjRGNC1JI21vR0YkNi1RIj1GJ0Y0LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRjE2JFEiMEYnRjRGNA==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjRGNC1JI21vR0YkNi1RIj1GJ0Y0LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRjE2JFEiMEYnRjRGNA==</Equation></Text-field>
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<Text-field style="Text" layout="Normal">This boundary value problem is solved below by following the procedure described earlier.  </Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal262">restart:</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">with(linalg):with(plots):</Text-field>
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<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">N:=2;</Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkiTkc2IiIiIw==">IiIj</Equation></Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq:=diff(theta(x),x$2)-H^2*theta(x);</Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjZXFHNiIsJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSZ0aGV0YUdGJDYjSSJ4R0YkLUkiJEdGKDYkRi0iIiMiIiIqJilJIkhHRiRGMUYyRipGMiEiIg==">LCYtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUYkNiQtSSZ0aGV0YUc2IjYjSSJ4R0YrRi1GLSIiIiomKUkiSEdGKyIiI0YuRilGLiEiIg==</Equation></Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal262">A:=matrix(N,N,[0,1,H^2,0]);</Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkiQUc2Ii1JJ21hdHJpeEdGJDYjNyQ3JCIiISIiIjckKiQpSSJIR0YkIiIjRitGKg==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkqbXZlcmJhdGltR0YkNiNRJSUiQUdGJy1JI21vR0YkNi1RIzo9RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkctSShtZmVuY2VkR0YkNiYtRiM2Iy1JJ210YWJsZUdGJDY2LUkkbXRyR0YkNictSSRtdGRHRiQ2KC1JI21uR0YkNiRRIjBGJ0YzLyUpcm93YWxpZ25HUSFGJy8lLGNvbHVtbmFsaWduR0Zobi8lK2dyb3VwYWxpZ25HRmhuLyUocm93c3BhbkdRIjFGJy8lK2NvbHVtbnNwYW5HRl9vLUZWNigtRlk2JFEiMUYnRjNGZm5GaW5GW29GXW9GYG9GZm5GaW5GW28tRlM2Jy1GVjYoLUYjNiMtRiw2I1FicCklIkhHLUkjbW5HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHNiI2JFEiMjYiLyUsbWF0aHZhcmlhbnRHUSdub3JtYWw2IkYnRmZuRmluRltvRl1vRmBvRlVGZm5GaW5GW28vJSZhbGlnbkdRJWF4aXNGJy9GZ25RKWJhc2VsaW5lRicvRmpuUSdjZW50ZXJGJy9GXG9RJ3xmcmxlZnR8aHJGJy8lL2FsaWdubWVudHNjb3BlR1EldHJ1ZUYnLyUsY29sdW1ud2lkdGhHUSVhdXRvRicvJSZ3aWR0aEdGXnEvJStyb3dzcGFjaW5nR1EmMS4wZXhGJy8lLmNvbHVtbnNwYWNpbmdHUSYwLjhlbUYnLyUpcm93bGluZXNHUSVub25lRicvJSxjb2x1bW5saW5lc0dGaXEvJSZmcmFtZUdGaXEvJS1mcmFtZXNwYWNpbmdHUSwwLjRlbX4wLjVleEYnLyUqZXF1YWxyb3dzR0Y4LyUtZXF1YWxjb2x1bW5zR0Y4LyUtZGlzcGxheXN0eWxlR0Y4LyUlc2lkZUdRJnJpZ2h0RicvJTBtaW5sYWJlbHNwYWNpbmdHRmZxRjMvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRic=</Equation></Text-field>
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<Group labelreference="L9" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">mat:=exponential(A,x);</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkbWF0RzYiLUknbWF0cml4R0YkNiM3JDckLCYqJiMiIiIiIiNGLS1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjKiZJIkhHRiRGLUkieEdGJEYtRi1GLSomRixGLS1GMDYjLCRGNSEiIkYtRi0sJCooRixGLSwmRjlGPEYvRi1GLUY2RjxGLTckLCYqKEYsRi1GNkYtRjlGLUY8KihGLEYtRjZGLUYvRi1GLUYq">[encoded data, 4612 characters]</Equation></Text-field>
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<Text-field style="Normal" layout="Normal"> The exponentials are converted to trigonometric form for convenience.</Text-field>
</Input>
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<Group labelreference="L11" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">mat:=map(convert,mat,trig);</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkbWF0RzYiLUknbWF0cml4R0YkNiM3JDckLUklY29zaEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjKiZJIkhHRiQiIiJJInhHRiRGMiomLUklc2luaEdGLEYvRjJGMSEiIjckLCYqKCNGMiIiI0YyRjFGMiwmRipGMkY1RjdGMkY3KihGO0YyRjFGMiwmRipGMkY1RjJGMkYyRio=">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</Equation></Text-field>
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<Group labelreference="L12" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal262">Y0:=matrix(N,1,[1,c[1]]);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjWTBHNiItSSdtYXRyaXhHRiQ2IzckNyMiIiI3IyZJImNHRiQ2I0Yq">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkqbXZlcmJhdGltR0YkNiNRJiUjWTBHRictSSNtb0dGJDYtUSM6PUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkobWZlbmNlZEdGJDYmLUYjNiMtSSdtdGFibGVHRiQ2Ni1JJG10ckdGJDYmLUkkbXRkR0YkNigtSSNtbkdGJDYkUSIxRidGMy8lKXJvd2FsaWduR1EhRicvJSxjb2x1bW5hbGlnbkdGaG4vJStncm91cGFsaWduR0Zobi8lKHJvd3NwYW5HUSIxRicvJStjb2x1bW5zcGFuR0Zfb0ZmbkZpbkZbby1GUzYmLUZWNigtSSVtc3ViR0YkNiUtRiw2I1ElJSJjR0YnLUYsNiNRX3A2Iy1JI21uRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliRzYiNiRRIjE2Ii8lLG1hdGh2YXJpYW50R1Enbm9ybWFsNiJGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRmZuRmluRltvRl1vRmBvRmZuRmluRltvLyUmYWxpZ25HUSVheGlzRicvRmduUSliYXNlbGluZUYnL0ZqblEnY2VudGVyRicvRlxvUSd8ZnJsZWZ0fGhyRicvJS9hbGlnbm1lbnRzY29wZUdRJXRydWVGJy8lLGNvbHVtbndpZHRoR1ElYXV0b0YnLyUmd2lkdGhHRmBxLyUrcm93c3BhY2luZ0dRJjEuMGV4RicvJS5jb2x1bW5zcGFjaW5nR1EmMC44ZW1GJy8lKXJvd2xpbmVzR1Elbm9uZUYnLyUsY29sdW1ubGluZXNHRltyLyUmZnJhbWVHRltyLyUtZnJhbWVzcGFjaW5nR1EsMC40ZW1+MC41ZXhGJy8lKmVxdWFscm93c0dGOC8lLWVxdWFsY29sdW1uc0dGOC8lLWRpc3BsYXlzdHlsZUdGOC8lJXNpZGVHUSZyaWdodEYnLyUwbWlubGFiZWxzcGFjaW5nR0ZocUYzLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYn</Equation></Text-field>
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<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">sol:=evalm(mat&amp;*Y0);</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pkkkc29sRzYiLUknbWF0cml4R0YkNiM3JDcjLCYtSSVjb3NoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMqJkkiSEdGJCIiIkkieEdGJEYzRjMqKC1JJXNpbmhHRi1GMEYzRjIhIiImSSJjR0YkNiNGM0YzRjM3IywoKigjRjMiIiNGM0YyRjMsJkYrRjNGNkY4RjNGOCooRj9GM0YyRjMsJkYrRjNGNkYzRjNGMyomRitGM0Y5RjNGMw==">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</Equation></Text-field>
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<Input>
<Text-field style="Normal" layout="Normal"> The solution is obtained as a function of x, H and the unknown initial condition c<Font subscript="true">1</Font>.  The first row of sol corresponds to w0 and the second row corresponds to <Equation executable="false" style="Normal" input-equation="" display="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">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</Equation></Text-field>
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<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">theta:=sol[1,1];</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkmdGhldGFHNiIsJi1JJWNvc2hHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyomSSJIR0YkIiIiSSJ4R0YkRi5GLiooLUklc2luaEdGKEYrRi5GLSEiIiZJImNHRiQ2I0YuRi5GLg==">LCYtSSVjb3NoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMqJkkiSEdGKCIiIkkieEdGKEYsRiwqKC1JJXNpbmhHRiVGKUYsRishIiImSSJjR0YoNiNGLEYsRiw=</Equation></Text-field>
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<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">dthetadx:=sol[2,1];</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkpZHRoZXRhZHhHNiIsKCooIyIiIiIiI0YoSSJIR0YkRigsJi1JJWNvc2hHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyomRipGKEkieEdGJEYoRigtSSVzaW5oR0YuRjEhIiJGKEY2KihGJ0YoRipGKCwmRixGKEY0RihGKEYoKiZGLEYoJkkiY0dGJDYjRihGKEYo">LCgqKCMiIiIiIiNGJUkiSEc2IkYlLCYtSSVjb3NoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YoNiMqJkYnRiVJInhHRihGJUYlLUklc2luaEdGLEYvISIiRiVGNCooRiRGJUYnRiUsJkYqRiVGMkYlRiVGJSomRipGJSZJImNHRig2I0YlRiVGJQ==</Equation></Text-field>
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<Group labelreference="L17" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal"> The boundary condition at x = 1 is applied to solve for c<Equation executable="false" style="Normal" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2I1EhRictRiM2JC1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGOC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiLkYnRjgvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGU0Y4">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2I1EhRictRiM2JC1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGOC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiLkYnRjgvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGU0Y4</Equation></Text-field>
</Input>
</Group>
<Group labelreference="L18" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc2:=subs(x=1,dthetadx)=0;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMyRzYiLywoKigjIiIiIiIjRilJIkhHRiRGKSwmLUklY29zaEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRitGKS1JJXNpbmhHRi9GMiEiIkYpRjUqKEYoRilGK0YpLCZGLUYpRjNGKUYpRikqJkYtRikmSSJjR0YkNiNGKUYpRikiIiE=">LywoKigjIiIiIiIjRiZJIkhHNiJGJiwmLUklY29zaEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKTYjRihGJi1JJXNpbmhHRi1GMCEiIkYmRjMqKEYlRiZGKEYmLCZGK0YmRjFGJkYmRiYqJkYrRiYmSSJjR0YpNiNGJkYmRiYiIiE=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L19" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">c[1]:=solve(bc2,c[1]);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJImNHNiI2IyIiIiwkKihJIkhHRiVGJy1JJXNpbmhHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0YqRictSSVjb3NoR0YtRjAhIiJGMw==">LCQqKEkiSEc2IiIiIi1JJXNpbmhHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0YkRiYtSSVjb3NoR0YpRiwhIiJGLw==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L20" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal"><Font style="Text">This value of the constant c</Font><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>
</Input>
</Group>
<Group labelreference="L21" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">theta:=eval(theta);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkmdGhldGFHNiIsJi1JJWNvc2hHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyomSSJIR0YkIiIiSSJ4R0YkRi5GLiooLUklc2luaEdGKEYrRi4tRjI2I0YtRi4tRidGNCEiIkY2">LCYtSSVjb3NoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMqJkkiSEdGKCIiIkkieEdGKEYsRiwqKC1JJXNpbmhHRiVGKUYsLUYwNiNGK0YsLUYkRjIhIiJGNA==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L22" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">The expression for \316\270 can be simplified further by using Maple's 'combine' command:</Font></Text-field>
</Input>
</Group>
<Group labelreference="L23" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">theta:=combine(theta);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkmdGhldGFHNiIqJi1JJWNvc2hHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IywmKiZJIkhHRiQiIiJJInhHRiRGL0YvRi4hIiJGLy1GJzYjRi5GMQ==">KiYtSSVjb3NoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJiomSSJIR0YoIiIiSSJ4R0YoRi1GLUYsISIiRi0tRiQ2I0YsRi8=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L24" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal">You can make plots for different values of the heat transfer coefficient, H, by using a 'pars' array and a' do' loop.</Text-field>
</Input>
</Group>
<Group labelreference="L25" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">pars:=[1,2,5,10];</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkklcGFyc0c2IjcmIiIiIiIjIiImIiM1">NyYiIiIiIiMiIiYiIzU=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L38" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">clr:=[red,green,gold,blue]; </Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkY2xyRzYiNyZJJHJlZEdGJEkmZ3JlZW5HRiRJJWdvbGRHRiRJJWJsdWVHRiQ=">NyZJJHJlZEc2IkkmZ3JlZW5HRiRJJWdvbGRHRiRJJWJsdWVHRiQ=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L26" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">for i from 1 to 4 do p[i]:=plot(subs(H=pars[i],theta),x=0..1,color=clr[i]):od:</Text-field>
</Input>
</Group>
<Group labelreference="L27" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">display(seq(p[i],i=1..4),thickness=4,title=&quot;Figure 3.1.1&quot;,axes=boxed,labels=[&quot;X&quot;,&quot;theta&quot;],labeldirections=[HORIZONTAL,VERTICAL]);</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, 12904 characters]</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L28" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">A three dimensional plot for \316\270 can be made for different values of H as x varies between zero and one.</Font></Text-field>
</Input>
</Group>
<Group labelreference="L29" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">plot3d(theta,x=0..1,H=0..10,axes=boxed,title=&quot;Figure 3.1.2&quot;,orientation=[45,45]);</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, 13668 characters]</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L33" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"></Text-field>
</Input>
</Group>
<Group labelreference="L30" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The above picture shows that as H increases the temperature distribution becomes nonuniform.</Text-field>
</Input>
</Group>
<Group labelreference="L31" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="2D Input" layout="Normal"><Equation executable="true" style="2D Input" input-equation="" display="LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2I1EhRic=">JSFH</Equation></Text-field>
</Input>
</Group>
</Worksheet>