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<Text-field style="Text" layout="Normal">Example 7.3  Mass Transfer in a Spherical Pellet</Text-field>
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</Group>
<Group labelreference="L48" 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="L3" drawlabel="true">
<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)+2/x*diff(u(x,t),x);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjZXFHNiIvLUklZGlmZkclKnByb3RlY3RlZEc2JC1JInVHRiQ2JEkieEdGJEkidEdGJEYuLCYtRic2JEYqLUkiJEdGKDYkRi0iIiMiIiIqKEY1RjZGLSEiIi1GJzYkRipGLUY2RjY=">Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ1RzYiNiRJInhHRilJInRHRilGLCwmLUYkNiQtRiQ2JEYnRitGKyIiIiomRishIiJGMEYyIiIj</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L5" 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="L6" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc1:=diff(u(x,t),x)=0;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMxRzYiLy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ1R0YkNiRJInhHRiRJInRHRiRGLSIiIQ==">Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ1RzYiNiRJInhHRilJInRHRilGKyIiIQ==</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:=expand(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>
<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="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="PkklRXFfWEc2Ii8sJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJYR0YkNiNJInhHRiQtSSIkR0YpNiRGLiIiIyIiIiooRjJGM0YuISIiLUYoNiRGK0YuRjNGMywkKiZGK0YzKUknbGFtYmRhR0YkRjJGM0Y1">LywmLUklZGlmZkclKnByb3RlY3RlZEc2JC1GJTYkLUkiWEc2IjYjSSJ4R0YsRi5GLiIiIiomRi4hIiJGKEYvIiIjLCQqJkYqRi9JJ2xhbWJkYUdGLEYyRjE=</Equation></Text-field>
</Output>
</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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">PCMvLUkiWEc2IjYjSSJ4RzYiLCYqKEkkX0MxRzYiIiIiSSJ4RzYiISIiLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMqJkknbGFtYmRhRzYiIiIiSSJ4RzYiIiIiIiIiIiIiKihJJF9DMkc2IiIiIkkieEc2IiEiIi1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjKiZJJ2xhbWJkYUc2IiIiIkkieEc2IiIiIiIiIiIiIg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L50" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The solution for X(x) can be taken as:</Text-field>
</Input>
</Group>
<Group labelreference="L15" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">X(x):=c[1]*sin(lambda*x)/x+c[2]*cos(lambda*x)/x;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi1JIlhHNiI2I0kieEdGJSwmKigmSSJjR0YlNiMiIiJGLS1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjKiZJJ2xhbWJkYUdGJUYtRidGLUYtRichIiJGLSooJkYrNiMiIiNGLS1JJGNvc0dGMEYzRi1GJ0Y2Ri0=">LCYqKCZJImNHNiI2IyIiIkYoLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiMqJkknbGFtYmRhR0YmRihJInhHRiZGKEYoRjEhIiJGKCooJkYlNiMiIiNGKC1JJGNvc0dGK0YuRihGMUYyRig=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L16" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Bc1:=diff(X(x),x)=0;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkQmMxRzYiLywqKiomSSJjR0YkNiMiIiJGKy1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjKiZJJ2xhbWJkYUdGJEYrSSJ4R0YkRitGK0YzRitGNCEiIkYrKihGKEYrLUkkc2luR0YuRjFGKylGNCIiI0Y1RjUqKiZGKTYjRjpGK0Y3RitGM0YrRjRGNUY1KihGPEYrRixGK0Y5RjVGNSIiIQ==">LywqKiomSSJjRzYiNiMiIiJGKS1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJzYjKiZJJ2xhbWJkYUdGJ0YpSSJ4R0YnRilGKUYxRilGMiEiIkYpKihGJUYpLUkkc2luR0YsRi9GKUYyISIjRjMqKiZGJjYjIiIjRilGNUYpRjFGKUYyRjNGMyooRjlGKUYqRilGMkY3RjMiIiE=</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="PkkkQmMyRzYiLywmKigmSSJjR0YkNiMiIiJGKy1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjKiZJJ2xhbWJkYUdGJEYrSSJ4R0YkRitGK0Y0ISIiRisqKCZGKTYjIiIjRistSSRjb3NHRi5GMUYrRjRGNUYrIiIh">LywmKigmSSJjRzYiNiMiIiJGKS1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJzYjKiZJJ2xhbWJkYUdGJ0YpSSJ4R0YnRilGKUYyISIiRikqKCZGJjYjIiIjRiktSSRjb3NHRixGL0YpRjJGM0YpIiIh</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L51" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">x = 0 cannot be substituted.  Hence, the limit at x = 0 is obtained:</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="Error" layout="Error">Error, numeric exception: division by zero</Text-field>
</Output>
</Group>
<Group labelreference="L19" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">limit(Bc1,x=0);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LywkKiYtSSdzaWdudW1HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyZJImNHRio2IyIiIyIiIkkpaW5maW5pdHlHRihGMCEiIiIiIQ==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L52" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">So the constant c2 should be zero.</Text-field>
</Input>
</Group>
<Group labelreference="L20" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">c[2]:=0;</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="L21" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Eq_Bc2:=eval(subs(x=1,Bc2)):</Text-field>
</Input>
</Group>
<Group labelreference="L22" 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="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="L25" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">U:=eval(X(x)*T(t)):</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>
</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>
</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="Pi1JInVHNiI2JEkieEdGJUkidEdGJS1JJFN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkKiomSSJBR0YlNiNJIm5HRiUiIiItSSRzaW5HRis2IyooRjNGNEkjUGlHRixGNEYnRjRGNEYnISIiLUkkZXhwR0YrNiMsJCooKUYzIiIjRjQpRjlGQUY0RihGNEY6RjQvRjM7RjRJKWluZmluaXR5R0Ys">LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqKiZJIkFHRic2I0kibkdGJyIiIi1JJHNpbkdGJDYjKihGLUYuSSNQaUdGJUYuSSJ4R0YnRi5GLkY0ISIiLUkkZXhwR0YkNiMsJCooRi0iIiNGM0Y7SSJ0R0YnRi5GNUYuL0YtO0YuSSlpbmZpbml0eUdGJQ==</Equation></Text-field>
</Output>
</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="PkkmZXFfQW5HNiIvLUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiQqKCZJIkFHRiQ2I0kibkdGJCIiIi1JJHNpbkdGKDYjKihGMEYxSSNQaUdGKUYxSSJ4R0YkRjFGMUY3ISIiL0YwO0YxSSlpbmZpbml0eUdGKUYx">Ly1JJFN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkKigmSSJBR0YoNiNJIm5HRigiIiItSSRzaW5HRiU2IyooRi5GL0kjUGlHRiZGL0kieEdGKEYvRi9GNSEiIi9GLjtGL0kpaW5maW5pdHlHRiZGLw==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L53" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The eigenfunction is:</Text-field>
</Input>
</Group>
<Group labelreference="L30" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">phi[n]:=sin(n*Pi*x)/x;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJJHBoaUc2IjYjSSJuR0YlKiYtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyooRiciIiJJI1BpR0YsRjBJInhHRiVGMEYwRjIhIiI=">KiYtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyooSSJuR0YoIiIiSSNQaUdGJkYsSSJ4R0YoRixGLEYuISIi</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L54" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The weighting function is:</Text-field>
</Input>
</Group>
<Group labelreference="L31" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">r(x):=x^2;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi1JInJHNiI2I0kieEdGJSokKUYnIiIjIiIi">KiRJInhHNiIiIiM=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L55" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Next, the integrals are calculated to obtain A<Font subscript="true">n</Font>:</Text-field>
</Input>
</Group>
<Group labelreference="L32" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">I1:=int(phi[n]^2*r(x),x=0..1):</Text-field>
</Input>
</Group>
<Group labelreference="L33" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">IC;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">Ly1JInVHNiI2JEkieEdGJSIiISIiIg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L34" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">I2:=int(rhs(IC)*phi[n]*r(x),x=0..1):</Text-field>
</Input>
</Group>
<Group labelreference="L35" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">vars:={sin(n*Pi)=0}:</Text-field>
</Input>
</Group>
<Group labelreference="L36" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">I1:=subs(vars,I1):</Text-field>
</Input>
</Group>
<Group labelreference="L37" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">I2:=subs(vars,I2):</Text-field>
</Input>
</Group>
<Group labelreference="L38" 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="PiZJIkFHNiI2I0kibkdGJSomLUkqUElFQ0VXSVNFR0YlNiQ3JCIiIS9GJ0YtNyQsJCooRichIiJJI1BpRyUqcHJvdGVjdGVkR0YyLUkkY29zRzYkRjRJKF9zeXNsaWJHRiU2IyomRiciIiJGM0Y7RjtGMkkqb3RoZXJ3aXNlR0YlRjstRio2JEYsNyQjRjsiIiNGPEYy">KiYtSSpwaWVjZXdpc2VHJSpwcm90ZWN0ZWRHNiUvSSJuRzYiIiIhRiosJCooRighIiJJI1BpR0YlRi0tSSRjb3NHNiRGJUkoX3N5c2xpYkdGKTYjKiZGKCIiIkYuRjVGNUYtRjUtRiQ2JUYnRiojRjUiIiNGLQ==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L56" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The dimensionless concentration profile is given by:</Text-field>
</Input>
</Group>
<Group labelreference="L39" 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="L40" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">u(x,t):=subs(infinity=N,u(x,t)):</Text-field>
</Input>
</Group>
<Group labelreference="L41" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">ua:=subs(N=20,u(x,t)):</Text-field>
</Input>
</Group>
<Group labelreference="L42" 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="PkkjdXVHNiItSSpQSUVDRVdJU0VHRiQ2JDckIiIiL0kidEdGJCIiITckLUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiQqLC1GJjYkNyRGLC9JIm5HRiRGLDckLCQqKEY5ISIiSSNQaUdGMUY9LUkkY29zR0YwNiMqJkY5RilGPkYpRilGPUkqb3RoZXJ3aXNlR0YkRiktRiY2JEY3NyQjRikiIiNGQ0Y9LUkkc2luR0YwNiMqKEY5RilGPkYpSSJ4R0YkRilGKUZNRj0tSSRleHBHRjA2IywkKigpRjlGSEYpKUY+RkhGKUYrRilGPUYpL0Y5O0YpIiM/MkYsRis=">LUkqcGllY2V3aXNlRyUqcHJvdGVjdGVkRzYmL0kidEc2IiIiISIiIjJGKUYnLUkkU3VtRzYkRiRJKF9zeXNsaWJHRig2JCosLUYjNiUvSSJuR0YoRilGKSwkKihGNSEiIkkjUGlHRiRGOC1JJGNvc0dGLjYjKiZGNUYqRjlGKkYqRjhGKi1GIzYlRjRGKSNGKiIiI0Y4LUkkc2luR0YuNiMqKEY1RipGOUYqSSJ4R0YoRipGKkZGRjgtSSRleHBHRi42IywkKihGNUZBRjlGQUYnRipGOEYqL0Y1O0YqIiM/</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L43" 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. 7.7.&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="L44" 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. 7.8.&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, 16208 characters]</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L49" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"><Equation executable="true" style="2D Input" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRicvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation></Text-field>
</Input>
</Group>
</Worksheet>