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<Text-field style="Heading 3" italic="false" layout="Heading 3"><Font italic="false">Example 5.2 </Font></Text-field>
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<Input>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">Consider the heat conduction/mass transfer problem in a cylinder.<Font superscript="true">[6] [9] [10] </Font> The governing equation in dimensionless form is</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal" alignment="centred"><Equation executable="false" style="Text" input-equation="" display="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">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</Equation></Text-field>
<Text-field style="Text" layout="Normal" alignment="centred"></Text-field>
<Text-field style="Text" layout="Normal" alignment="centred">                                                                                         u(x,0) = 0                                                               (5.1.19)            </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>(0,t) = 0 and u(1,t) = 1</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">The nonhomogeneous boundary condition at x = 1 contributes to the forcing function vector <Font bold="true">b</Font>.  However, in this case the <Font bold="true">b</Font> vector is a constant, and hence, equation (5.1.18) can be used.  (Note that equation (5.1.17) is valid even when the <Font bold="true">b</Font> vector is a constant.)  When the governing equation is applied at x = 0 in cylindrical or spherical coordinate, we have singularity at x = 0.<Font superscript="true">[11] [12] </Font> This singularity is avoided in our semianalytical technique as we use the boundary condition at x = 0 (symmetry boundary condition) to eliminate the dependent variable.  Equation (5.1.19) is solved in Maple below using the procedure described above.</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">restart;</Text-field>
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<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">with(linalg):with(plots):</Text-field>
</Input>
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<Group labelreference="L4" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">ge:=diff(u(x,t),t)=diff(u(x,t),x$2)+1/x*diff(u(x,t),x);</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjZ2VHNiIvLUklZGlmZkclKnByb3RlY3RlZEc2JC1JInVHRiQ2JEkieEdGJEkidEdGJEYuLCYtRic2JEYqLUkiJEdGKDYkRi0iIiMiIiIqJkYtISIiLUYnNiRGKkYtRjZGNg==">Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ1RzYiNiRJInhHRilJInRHRilGLCwmLUYkNiQtRiQ2JEYnRitGKyIiIiomRishIiJGMEYyRjI=</Equation></Text-field>
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<Group labelreference="L5" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc1:=diff(u(x,t),x);</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMxRzYiLUklZGlmZkclKnByb3RlY3RlZEc2JC1JInVHRiQ2JEkieEdGJEkidEdGJEYs">LUklZGlmZkclKnByb3RlY3RlZEc2JC1JInVHNiI2JEkieEdGKEkidEdGKEYq</Equation></Text-field>
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<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc2:=u(x,t)-1;</Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkYmMyRzYiLCYtSSJ1R0YkNiRJInhHRiRJInRHRiQiIiJGKyEiIg==">LCYtSSJ1RzYiNiRJInhHRiVJInRHRiUiIiIhIiJGKQ==</Equation></Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">IC:=u(x,0)=0;</Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjSUNHNiIvLUkidUdGJDYkSSJ4R0YkIiIhRio=">Ly1JInVHNiI2JEkieEdGJSIiIUYo</Equation></Text-field>
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<Group labelreference="L8" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">N:=10;</Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkiTkc2IiIjNQ==">IiM1</Equation></Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">L:=1;</Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkiTEc2IiIiIg==">IiIi</Equation></Text-field>
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<Group labelreference="L10" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">dydxf:=1/2*(-u[2](t)-3*u[0](t)+4*u[1](t))/h:</Text-field>
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<Group labelreference="L11" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">dydxb:=1/2*(u[N-1](t)+3*u[N+1](t)-4*u[N](t))/h:</Text-field>
</Input>
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<Group labelreference="L12" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">dydx:=1/2/h*(u[m+1](t)-u[m-1](t)):</Text-field>
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<Group labelreference="L13" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">d2ydx2:=1/h^2*(u[m-1](t)-2*u[m](t)+u[m+1](t)):</Text-field>
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<Group labelreference="L14" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc1:=subs(diff(u(x,t),x)=dydxf,u(x,t)=u[0](t),x=0,bc1):</Text-field>
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<Group labelreference="L15" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc2:=subs(diff(u(x,t),x)=dydxb,u(x,t)=u[N+1](t),x=1,bc2):</Text-field>
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<Group labelreference="L16" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq[0]:=bc1; </Text-field>
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<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq[N+1]:=bc2; </Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI2VxRzYiNiMiIzYsJi0mSSJ1R0YlRiY2I0kidEdGJSIiIkYuISIi">LCYtJkkidUc2IjYjIiM2NiNJInRHRiYiIiIhIiJGKw==</Equation></Text-field>
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<Group labelreference="L18" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">for i from 1 to N do eq[i]:=diff(u[i](t),t)= subs(diff(u(x,t),x$2) = subs(m=i,d2ydx2),diff(u(x,t),x) = subs(m=i,dydx),u(x,t)=u[i](t),x=i*h,rhs(ge));od:</Text-field>
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<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">u[0](t):=(solve(eq[0],u[0](t)));</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi0mSSJ1RzYiNiMiIiE2I0kidEdGJiwmKiYjIiIiIiIkRi4tJkYlNiMiIiNGKUYuISIiKiYjIiIlRi9GLi0mRiU2I0YuRilGLkYu">LCYtJkkidUc2IjYjIiIjNiNJInRHRiYjISIiIiIkLSZGJTYjIiIiRikjIiIlRi0=</Equation></Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">u[N+1](t):=solve(eq[N+1],u[N+1](t));</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi0mSSJ1RzYiNiMiIzY2I0kidEdGJiIiIg==">IiIi</Equation></Text-field>
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<Group labelreference="L21" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">for i from 1 to N do eq[i]:=eval(eq[i]);od;</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI2VxRzYiNiMiIiIvLUklZGlmZkclKnByb3RlY3RlZEc2JC0mSSJ1R0YlRiY2I0kidEdGJUYxLCYqJilJImhHRiUiIiMhIiIsJiomI0Y2IiIkRictJkYvNiNGNkYwRidGJyomRjpGJ0YtRidGN0YnRicqKCNGJ0Y2RidGNEY3LCYqJiMiIiVGO0YnRjxGJ0YnKiZGREYnRi1GJ0Y3RidGJw==">Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtJkkidUc2IjYjIiIiNiNJInRHRipGLiwmKiZJImhHRiohIiMsJi0mRik2IyIiI0YtI0Y3IiIkRicjRjJGOUYsRiwqJkYxRjIsJkY0IyIiJUY5RicjISIlRjlGLCNGLEY3</Equation></Text-field>
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<Output>
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<Output>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI2VxRzYiNiMiIiUvLUklZGlmZkclKnByb3RlY3RlZEc2JC0mSSJ1R0YlRiY2I0kidEdGJUYxLCYqJilJImhHRiUiIiMhIiIsKC0mRi82IyIiJEYwIiIiKiZGNkY9Ri1GPUY3LSZGLzYjIiImRjBGPUY9Rj0qKCNGPSIiKUY9RjRGNywmRj9GPUY5RjdGPUY9">Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtJkkidUc2IjYjIiIlNiNJInRHRipGLiwmKiZJImhHRiohIiMsKC0mRik2IyIiJEYtIiIiRidGMi0mRik2IyIiJkYtRjhGOEY4KiZGMUYyLCZGOUY4RjQhIiJGOCNGOCIiKQ==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI2VxRzYiNiMiIiYvLUklZGlmZkclKnByb3RlY3RlZEc2JC0mSSJ1R0YlRiY2I0kidEdGJUYxLCYqJilJImhHRiUiIiMhIiIsKC0mRi82IyIiJUYwIiIiKiZGNkY9Ri1GPUY3LSZGLzYjIiInRjBGPUY9Rj0qKCNGPSIjNUY9RjRGNywmRj9GPUY5RjdGPUY9">Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtJkkidUc2IjYjIiImNiNJInRHRipGLiwmKiZJImhHRiohIiMsKC0mRik2IyIiJUYtIiIiRidGMi0mRik2IyIiJ0YtRjhGOEY4KiZGMUYyLCZGOUY4RjQhIiJGOCNGOCIjNQ==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI2VxRzYiNiMiIicvLUklZGlmZkclKnByb3RlY3RlZEc2JC0mSSJ1R0YlRiY2I0kidEdGJUYxLCYqJilJImhHRiUiIiMhIiIsKC0mRi82IyIiJkYwIiIiKiZGNkY9Ri1GPUY3LSZGLzYjIiIoRjBGPUY9Rj0qKCNGPSIjN0Y9RjRGNywmRj9GPUY5RjdGPUY9">Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtJkkidUc2IjYjIiInNiNJInRHRipGLiwmKiZJImhHRiohIiMsKC0mRik2IyIiJkYtIiIiRidGMi0mRik2IyIiKEYtRjhGOEY4KiZGMUYyLCZGOUY4RjQhIiJGOCNGOCIjNw==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI2VxRzYiNiMiIigvLUklZGlmZkclKnByb3RlY3RlZEc2JC0mSSJ1R0YlRiY2I0kidEdGJUYxLCYqJilJImhHRiUiIiMhIiIsKC0mRi82IyIiJ0YwIiIiKiZGNkY9Ri1GPUY3LSZGLzYjIiIpRjBGPUY9Rj0qKCNGPSIjOUY9RjRGNywmRj9GPUY5RjdGPUY9">Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtJkkidUc2IjYjIiIoNiNJInRHRipGLiwmKiZJImhHRiohIiMsKC0mRik2IyIiJ0YtIiIiRidGMi0mRik2IyIiKUYtRjhGOEY4KiZGMUYyLCZGOUY4RjQhIiJGOCNGOCIjOQ==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI2VxRzYiNiMiIikvLUklZGlmZkclKnByb3RlY3RlZEc2JC0mSSJ1R0YlRiY2I0kidEdGJUYxLCYqJilJImhHRiUiIiMhIiIsKC0mRi82IyIiKEYwIiIiKiZGNkY9Ri1GPUY3LSZGLzYjIiIqRjBGPUY9Rj0qKCNGPSIjO0Y9RjRGNywmRj9GPUY5RjdGPUY9">Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtJkkidUc2IjYjIiIpNiNJInRHRipGLiwmKiZJImhHRiohIiMsKC0mRik2IyIiKEYtIiIiRidGMi0mRik2IyIiKkYtRjhGOEY4KiZGMUYyLCZGOUY4RjQhIiJGOCNGOCIjOw==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI2VxRzYiNiMiIiovLUklZGlmZkclKnByb3RlY3RlZEc2JC0mSSJ1R0YlRiY2I0kidEdGJUYxLCYqJilJImhHRiUiIiMhIiIsKC0mRi82IyIiKUYwIiIiKiZGNkY9Ri1GPUY3LSZGLzYjIiM1RjBGPUY9Rj0qKCNGPSIjPUY9RjRGNywmRj9GPUY5RjdGPUY9">Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtJkkidUc2IjYjIiIqNiNJInRHRipGLiwmKiZJImhHRiohIiMsKC0mRik2IyIiKUYtIiIiRidGMi0mRik2IyIjNUYtRjhGOEY4KiZGMUYyLCZGOUY4RjQhIiJGOCNGOCIjPQ==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI2VxRzYiNiMiIzUvLUklZGlmZkclKnByb3RlY3RlZEc2JC0mSSJ1R0YlRiY2I0kidEdGJUYxLCYqJilJImhHRiUiIiMhIiIsKC0mRi82IyIiKkYwIiIiKiZGNkY9Ri1GPUY3Rj1GPUY9Rj0qKCNGPSIjP0Y9RjRGNywmRj1GPUY5RjdGPUY9">Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtJkkidUc2IjYjIiM1NiNJInRHRipGLiwmKiZJImhHRiohIiMsKC0mRik2IyIiKkYtIiIiRidGMkY4RjhGOEY4KiZGMUYyLCZGOEY4RjQhIiJGOCNGOCIjPw==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L22" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eqs:=[seq(rhs(eq[j]),j=1..N)]:</Text-field>
</Input>
</Group>
<Group labelreference="L23" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Y:=[seq(u[i](t),i=1..N)];</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkiWUc2IjcsLSZJInVHRiQ2IyIiIjYjSSJ0R0YkLSZGKDYjIiIjRistJkYoNiMiIiRGKy0mRig2IyIiJUYrLSZGKDYjIiImRistJkYoNiMiIidGKy0mRig2IyIiKEYrLSZGKDYjIiIpRistJkYoNiMiIipGKy0mRig2IyIjNUYr">NywtJkkidUc2IjYjIiIiNiNJInRHRiYtJkYlNiMiIiNGKS0mRiU2IyIiJEYpLSZGJTYjIiIlRiktJkYlNiMiIiZGKS0mRiU2IyIiJ0YpLSZGJTYjIiIoRiktJkYlNiMiIilGKS0mRiU2IyIiKkYpLSZGJTYjIiM1Rik=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L24" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">A:=genmatrix(eqs,Y,'b1'):</Text-field>
</Input>
</Group>
<Group labelreference="L25" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">b:=matrix(N,1):for i to N do b[i,1]:=-eval(b1[i]);od:evalm(b);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="LUknbWF0cml4RzYiNiM3LDcjIiIhRidGJ0YnRidGJ0YnRidGJzcjLCQqJiMiI0AiIz8iIiIpSSJoR0YkIiIjISIiRi8=">PTYiNiQ7IiIiIiM1O0YmRiZFXFtsKzYkRidGJiwkKiRJImhHRiMhIiMjIiNAIiM/NiQiIiVGJiIiITYkIiInRiZGNDYkIiIkRiZGNDYkIiIjRiZGNDYkIiIoRiZGNDYkIiIqRiZGNDYkIiImRiZGNDYkIiIpRiZGNDYkRiZGJkY0</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L26" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">h:=eval(L/(N+1));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkiaEc2IiMiIiIiIzY=">IyIiIiIjNg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L27" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">A:=map(eval,A):</Text-field>
</Input>
</Group>
<Group labelreference="L28" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">if N &gt; 4 then A:=map(evalf,A);end: </Text-field>
</Input>
</Group>
<Group labelreference="L29" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">evalm(A);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="LUknbWF0cml4RzYiNiM3LDcsJCErTExMODshIigkIitMTEw4O0YqJCIiIUYuRi1GLUYtRi1GLUYtRi03LCQiKysrK3YhKiEiKSQhJFUjRi4kIisrK103OkYqRi1GLUYtRi1GLUYtRi03LEYtJCIrTExMMzVGKkYzJCIrbm1tNjlGKkYtRi1GLUYtRi1GLTcsRi1GLSQiKysrdmU1RipGMyQiKysrRGg4RipGLUYtRi1GLUYtNyxGLUYtRi0kIisrKysqMyJGKkYzJCIrKysrSjhGKkYtRi1GLUYtNyxGLUYtRi1GLSQiK25tOzQ2RipGMyQiK0xMJDNKIkYqRi1GLUYtNyxGLUYtRi1GLUYtJCIrSDlkQjZGKkYzJCIrciZHa0giRipGLUYtNyxGLUYtRi1GLUYtRi0kIisrXVBNNkYqRjMkIisrXWkmRyJGKkYtNyxGLUYtRi1GLUYtRi1GLSQiK3l4eFU2RipGMyQiK0FBQXg3Rio3LEYtRi1GLUYtRi1GLUYtRi0kIisrK11cNkYqRjM=">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</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L30" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">mat:=exponential(A,t):</Text-field>
</Input>
</Group>
<Group labelreference="L31" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">mat:=map(evalf,mat):</Text-field>
</Input>
</Group>
<Group labelreference="L32" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Y0:=matrix(N,1):for i from 1 to N do Y0[i,1]:=evalf(subs(x=i*h,rhs(IC)));od:evalm(Y0);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="LUknbWF0cml4RzYiNiM3LDcjJCIiIUYpRidGJ0YnRidGJ0YnRidGJ0Yn">PTYiNiQ7IiIiIiM1O0YmRiZFXFtsKzYkRidGJiQiIiFGLDYkIiIlRiZGKzYkIiInRiZGKzYkIiIkRiZGKzYkIiIjRiZGKzYkIiIoRiZGKzYkIiIqRiZGKzYkIiImRiZGKzYkIiIpRiZGKzYkRiZGJkYr</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L58" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The nonhomogeneous solution (equation (5.17b)) is found and added to the homogeneous solution.</Text-field>
</Input>
</Group>
<Group labelreference="L33" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">s1:=evalm(Y0+inverse(A)&amp;*b):</Text-field>
</Input>
</Group>
<Group labelreference="L34" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Y:=evalm(mat&amp;*s1-inverse(A)&amp;*b):</Text-field>
</Input>
</Group>
<Group labelreference="L35" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Y:=map(simplify,Y):</Text-field>
</Input>
</Group>
<Group labelreference="L59" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The number of digits is decreased to five for brevity.  However, for accuracy, one has to use a minimum of 10 digits, which is the default in Maple.</Text-field>
</Input>
</Group>
<Group labelreference="L36" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Digits:=5;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkknRGlnaXRzRzYiIiIm">IiIm</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L37" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">for i from 1 to N do u[i](t):=evalf((Y[i,1]));od:</Text-field>
</Input>
</Group>
<Group labelreference="L38" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">for i from 0 to N+1 do u[i](t):=eval(u[i](t));od;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LDgtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJ0R0YoJCEmdHUlISIjJCImaWIiISInLUYkNiMsJEYrJCEmPFclRi4kISZYKWZGMS1GJDYjLCRGKyQhJl8mUkYuJCImMEgiISImLUYkNiMsJEYrJCEmYUwkRi4kISZHPyNGQC1GJDYjLCRGKyQhJi1rI0YuJCImZkkkRkAtRiQ2IywkRiskISZAJD5GLiQhJip6WEZALUYkNiMsJEYrJCEmP0YiRi4kIiYpUWdGQC1GJDYjLCRGKyQhJlA4KCEiJCQhJiwheUZALUYkNiMsJEYrJCEmXilIRlxvJCImSC4iISIlLUYkNiMsJEYrJCEmb3YmRmZvJCEmUWYiRmZvJCImKCoqKipGQCIiIg==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LDgtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJ0R0YoJCEmdHUlISIjJCImbCZ5ISIoLUYkNiMsJEYrJCEmPFclRi4kISYyNyQhIictRiQ2IywkRiskISZfJlJGLiQiJj01KEY5LUYkNiMsJEYrJCEmYUwkRi4kISZUSSIhIiYtRiQ2IywkRiskISYtayNGLiQiJiJSQEZILUYkNiMsJEYrJCEmQCQ+Ri4kISZLRiRGSC1GJDYjLCRGKyQhJj9GIkYuJCImP3klRkgtRiQ2IywkRiskISZQOCghIiQkISYjKXonRkgtRiQ2IywkRiskISZeKUhGXW8kIiYoRygqRkgtRiQ2IywkRiskISZvdiYhIiUkISZeZCJGXHAkIiYrKyJGXHAiIiI=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LDgtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJ0R0YoJCEmdHUlISIjJCEmaV8iISInLUYkNiMsJEYrJCEmPFclRi4kIiY0WiZGMS1GJDYjLCRGKyQhJl8mUkYuJCEmNC4iISImLUYkNiMsJEYrJCEmYUwkRi4kIiY/UiJGQC1GJDYjLCRGKyQhJi1rI0YuJCEmOU8iRkAtRiQ2IywkRiskISZAJD5GLiQiJnBZJ0YxLUYkNiMsJEYrJCEmP0YiRi4kIiY8LCJGQC1GJDYjLCRGKyQhJlA4KCEiJCQhJkJ6JEZALUYkNiMsJEYrJCEmXilIRlxvJCImJ0d6RkAtRiQ2IywkRiskISZvdiYhIiUkISYqPTpGW3AkIiYrKyJGW3AiIiI=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LDgtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJ0R0YoJCEmdHUlISIjJCImcSg9ISInLUYkNiMsJEYrJCEmPFclRi4kISYtVyZGMS1GJDYjLCRGKyQhJl8mUkYuJCImRD8nRjEtRiQ2IywkRiskISZhTCRGLiQhJk4rJyEiKC1GJDYjLCRGKyQhJi1rI0YuJCEmYDMiISImLUYkNiMsJEYrJCEmQCQ+Ri4kIiZFPCNGTy1GJDYjLCRGKyQhJj9GIkYuJCEmODUjRk8tRiQ2IywkRiskISZQOCghIiQkISYzKyNGMS1GJDYjLCRGKyQhJl4pSEZdbyQiJlBHJkZPLUYkNiMsJEYrJCEmb3YmISIlJCEmdVUiRlxwJCImKysiRlxwIiIi</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LDgtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJ0R0YoJCEmdHUlISIjJCEmVysjISInLUYkNiMsJEYrJCEmPFclRi4kIiZKKVFGMS1GJDYjLCRGKyQhJl8mUkYuJCImLj0nISIoLUYkNiMsJEYrJCEmYUwkRi4kISZOYipGMS1GJDYjLCRGKyQhJi1rI0YuJCImPDkiISImLUYkNiMsJEYrJCEmQCQ+Ri4kIiYsKkdGMS1GJDYjLCRGKyQhJj9GIkYuJCEmOlYjRk8tRiQ2IywkRiskISZQOCghIiQkIiZwWSNGTy1GJDYjLCRGKyQhJl4pSEZdbyQiJnNGI0ZPLUYkNiMsJEYrJCEmb3YmISIlJCEmUUkiRlxwJCImKysiRlxwIiIi</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LDgtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJ0R0YoJCEmdHUlISIjJCImcic+ISInLUYkNiMsJEYrJCEmPFclRi4kISZkYCJGMS1GJDYjLCRGKyQhJl8mUkYuJCEmOF8mRjEtRiQ2IywkRiskISZhTCRGLiQiJjsqb0YxLUYkNiMsJEYrJCEmLWsjRi4kIiZVZidGMS1GJDYjLCRGKyQhJkAkPkYuJCEmaWUiISImLUYkNiMsJEYrJCEmP0YiRi4kISZAOyVGMS1GJDYjLCRGKyQhJlA4KCEiJCQiJiVbS0ZVLUYkNiMsJEYrJCEmXilIRlxvJCEmYmcmRjEtRiQ2IywkRiskISZvdiYhIiUkISZEOiJGW3AkIiYrKyJGW3AiIiI=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LDgtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJ0R0YoJCEmdHUlISIjJCEmJyp6IiEiJy1GJDYjLCRGKyQhJjxXJUYuJCEmVVcpISIoLUYkNiMsJEYrJCEmXyZSRi4kIiZGJ2VGMS1GJDYjLCRGKyQhJmFMJEYuJCImbjIkRjEtRiQ2IywkRiskISYtayNGLiQhJksvIiEiJi1GJDYjLCRGKyQhJkAkPkYuJCEmJHoiKUYxLUYkNiMsJEYrJCEmP0YiRi4kIiYvaiJGTy1GJDYjLCRGKyQhJlA4KCEiJCQiJm85I0ZPLUYkNiMsJEYrJCEmXilIRl1vJCEmbXYjRk8tRiQ2IywkRiskISZvdiYhIiUkISYhKnkqRk8kIiYrKyJGXHAiIiI=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LDgtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJ0R0YoJCEmdHUlISIjJCImMWAiISInLUYkNiMsJEYrJCEmPFclRi4kIiY+ZyNGMS1GJDYjLCRGKyQhJl8mUkYuJCEmVz4jRjEtRiQ2IywkRiskISZhTCRGLiQhJip6ekYxLUYkNiMsJEYrJCEmLWsjRi4kISZ2I1FGMS1GJDYjLCRGKyQhJkAkPkYuJCImeC4iISImLUYkNiMsJEYrJCEmP0YiRi4kIiYseSJGVS1GJDYjLCRGKyQhJlA4KCEiJCQiJidRWSEiKC1GJDYjLCRGKyQhJl4pSEZcbyQhJnIpUkZVLUYkNiMsJEYrJCEmb3YmISIlJCEmLCp5RlUkIiYrKyJGXHAiIiI=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LDgtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJ0R0YoJCEmdHUlISIjJCEmIyk9IiEiJy1GJDYjLCRGKyQhJjxXJUYuJCEmYUskRjEtRiQ2IywkRiskISZfJlJGLiQhJkNbI0YxLUYkNiMsJEYrJCEmYUwkRi4kIiYkb0hGMS1GJDYjLCRGKyQhJi1rI0YuJCImOHAqRjEtRiQ2IywkRiskISZAJD5GLiQiJiUqNCIhIiYtRiQ2IywkRiskISY/RiJGLiQiJkFqIkYxLUYkNiMsJEYrJCEmUDgoISIkJCEmJip6IkZVLUYkNiMsJEYrJCEmXilIRlxvJCEmYjglRlUtRiQ2IywkRiskISZvdiYhIiUkISZTKmVGVSQiJisrIkZbcCIiIg==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LDgtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJ0R0YoJCEmdHUlISIjJCImTSspISIoLUYkNiMsJEYrJCEmPFclRi4kIiZPJEghIictRiQ2IywkRiskISZfJlJGLiQiJjIhXEY5LUYkNiMsJEYrJCEmYUwkRi4kIiZ3I1xGOS1GJDYjLCRGKyQhJi1rI0YuJCImd3IiRjktRiQ2IywkRiskISZAJD5GLiQhJk8pXEY5LUYkNiMsJEYrJCEmP0YiRi4kISZcVSIhIiYtRiQ2IywkRiskISZQOCghIiQkISYoSENGZ24tRiQ2IywkRiskISZeKUhGXW8kISZoSSRGZ24tRiQ2IywkRiskISZvdiYhIiUkISYqb1FGZ24kIiYrKyJGXHAiIiI=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi0mSSJ1RzYiNiMiIzU2I0kidEdGJiw4KiYkIiZKJlIhIigiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2IywkKiYkIiZ0dSUhIiNGMEYqRjAhIiJGMEY8KiYkIiYhbzshIidGMC1GMjYjLCQqJiQiJjxXJUY7RjBGKkYwRjxGMEY8KiYkIiYkcE9GQEYwLUYyNiMsJComJCImXyZSRjtGMEYqRjBGPEYwRjwqJiQiJnc9J0ZARjAtRjI2IywkKiYkIiZhTCRGO0YwRipGMEY8RjBGPComJCImcycqKUZARjAtRjI2IywkKiYkIiYtayNGO0YwRipGMEY8RjBGPComJCImVDwiISImRjAtRjI2IywkKiYkIiZAJD5GO0YwRipGMEY8RjBGPComJCImb1UiRl9vRjAtRjI2IywkKiYkIiY/RiJGO0YwRipGMEY8RjBGPComJCImbGoiRl9vRjAtRjI2IywkKiYkIiZQOCghIiRGMEYqRjBGPEYwRjwqJiQiJjl6IkZfb0YwLUYyNiMsJComJCImXilIRmhwRjBGKkYwRjxGMEY8KiYkIiZEKT1GX29GMC1GMjYjLCQqJiQiJm92JiEiJUYwRipGMEY8RjBGPCQiJisrIkZbckYw">LDgtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJ0R0YoJCEmdHUlISIjJCEmSiZSISIoLUYkNiMsJEYrJCEmPFclRi4kISYhbzshIictRiQ2IywkRiskISZfJlJGLiQhJiRwT0Y5LUYkNiMsJEYrJCEmYUwkRi4kISZ3PSdGOS1GJDYjLCRGKyQhJi1rI0YuJCEmcycqKUY5LUYkNiMsJEYrJCEmQCQ+Ri4kISZUPCIhIiYtRiQ2IywkRiskISY/RiJGLiQhJm9VIkZWLUYkNiMsJEYrJCEmUDgoISIkJCEmbGoiRlYtRiQ2IywkRiskISZeKUhGXW8kISY5eiJGVi1GJDYjLCRGKyQhJm92JiEiJSQhJkQpPUZWJCImKysiRlxwIiIi</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi0mSSJ1RzYiNiMiIzY2I0kidEdGJiIiIg==">IiIi</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L60" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Hence, we have obtained a semianalytical solution (analytical in time and numerical in x).  Next, plots are made:</Text-field>
</Input>
</Group>
<Group labelreference="L65" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">for i from 0 to N+1 do
  p[i]:=plot(u[i](t),t=0..0.4,color=COLOR(HUE,i/(N+2)),thickness=3):
end do:</Text-field>
</Input>
</Group>
<Group labelreference="L66" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">arw:=arrow(&lt;0.26,0.6&gt;,&lt;-0.2,0.3&gt;,width=[1/300,relative],head_width=[5,relative],head_length=[1/20,relative]):
pt:=textplot([[0.3,0.55,&quot;Follow the arrow:&quot;],seq([0.26,0.5-i*0.05,typeset(u[i],&quot;(t)&quot;)],i=0..2),[0.26,0.35,&quot;...&quot;],[0.265,0.3,typeset(u[11],&quot;(t)&quot;)]]):</Text-field>
</Input>
</Group>
<Group labelreference="L40" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">display([seq(p[i],i=0..11),arw,pt],axes=boxed,labels=[t,&quot;u&quot;],title=&quot;Figure Exp. 5.5.&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, 40628 characters]</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L41" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">tf:=0.2;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjdGZHNiIkIiIjISIi">JCIiIyEiIg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L42" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">M:=30;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkiTUc2IiIjSQ==">IiNJ</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L43" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">T1:=[seq(tf*i/M,i=0..M)];</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjVDFHNiI3QSQiIiFGJyQiJm5tJyEiKCQiJkxMIiEiJyQiJisrI0YtJCImbm0jRi0kIiZMTCRGLSQiJisrJUYtJCImbm0lRi0kIiZMTCZGLSQiJisrJ0YtJEYpRi0kIiZMTChGLSQiJisrKUYtJCImbm0pRi0kIiZMTCpGLSQiJisrIiEiJiQiJm4xIkZHJCImTDgiRkckIiYrPyJGRyQiJm5FIkZHJEYsRkckIiYrUyJGRyQiJm5ZIkZHJCImTGAiRkckIiYrZyJGRyQiJm5tIkZHJCImTHQiRkckIiYrIT1GRyQiJm4nPUZHJCImTCQ+RkckRi9GRw==">N0EkIiIhRiQkIiZubSchIigkIiZMTCIhIickIiYrKyNGKiQiJm5tI0YqJCImTEwkRiokIiYrKyVGKiQiJm5tJUYqJCImTEwmRiokIiYrKydGKiRGJkYqJCImTEwoRiokIiYrKylGKiQiJm5tKUYqJCImTEwqRiokIiYrKyIhIiYkIiZuMSJGRCQiJkw4IkZEJCImKz8iRkQkIiZuRSJGRCRGKUZEJCImK1MiRkQkIiZuWSJGRCQiJkxgIkZEJCImK2ciRkQkIiZubSJGRCQiJkx0IkZEJCImKyE9RkQkIiZuJz1GRCQiJkwkPkZEJEYsRkQ=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L44" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">PP:=matrix(N+2,M+1);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjUFBHNiItSSZhcnJheUclKnByb3RlY3RlZEc2JTsiIiIiIzc7RioiI0o3Ig==">PTYiNiQ7IiIiIiM3O0YmIiNKRVxbbCE=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L45" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">for i from 1 to N+2 do PP[i,1]:=evalf(subs(x=(i-1)*h,rhs(IC)));od:</Text-field>
</Input>
</Group>
<Group labelreference="L46" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">for i from 1 to N+2 do for j from 2 to M+1 do PP[i,j]:=evalf(subs(t=T1[j],u[i-1](t)));od;od:</Text-field>
</Input>
</Group>
<Group labelreference="L47" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">plotdata := [seq([ seq([(i-1)*h,T1[j],PP[i,j]], i=1..N+2)], j=1..M+1)]:</Text-field>
</Input>
</Group>
<Group labelreference="L48" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">surfdata( plotdata, axes=boxed, title=&quot;Figure Exp. 5.6.&quot;,labels=[x,t,u],orientation=[-145,45]);</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, 14380 characters]</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L62" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">It is some times important to know how the temperature or concentration at the center of the cylinder changes with time.  Analytical solutions for cylinders involve Bessel functions and an infinite series.  With our semianalytical method, we can find how the temperature varies analytically with time.  The time dependent variable at the center of the cylinder varies with time as:</Text-field>
</Input>
</Group>
<Group labelreference="L49" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">u[0](t);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY5LUYjNictSSNtbkdGJDYkUSkwLjAxNTU2MkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJ0YxLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHRjovJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkktSSVtc3VwR0YkNiUtRjU2LVEvJkV4cG9uZW50aWFsRTtGJ0YxRjhGO0Y9Rj9GQUZDRkVGRy9GS1EsMC4xMTExMTExZW1GJy1GIzYmLUY1Ni1RKiZ1bWludXMwO0YnRjFGOEY7Rj1GP0ZBRkNGRS9GSFEsMC4yMjIyMjIyZW1GJy9GS0ZaLUYjNictRi42JFEnNDc0LjczRidGMUY0LUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnL0YyUSdpdGFsaWNGJy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRidGMUZkb0YxLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Zkb0YxLUY1Ni1RKCZtaW51cztGJ0YxRjhGO0Y9Rj9GQUZDRkVGWUZlbi1GIzYnLUYuNiRRKTAuMDU5ODQ1RidGMUY0LUZNNiVGTy1GIzYmRlYtRiM2Jy1GLjYkUSc0NDQuMTdGJ0YxRjRGW29GZG9GMUZkb0YxRmdvRmRvRjEtRjU2LVEiK0YnRjFGOEY7Rj1GP0ZBRkNGRUZZRmVuLUYjNictRi42JFEoMC4xMjkwNUYnRjFGNC1GTTYlRk8tRiM2JkZWLUYjNictRi42JFEnMzk1LjUyRidGMUY0RltvRmRvRjFGZG9GMUZnb0Zkb0YxRmpvLUYjNictRi42JFEoMC4yMjAyOEYnRjFGNC1GTTYlRk8tRiM2JkZWLUYjNictRi42JFEnMzMzLjU0RidGMUY0RltvRmRvRjFGZG9GMUZnb0Zkb0YxRltxLUYjNictRi42JFEoMC4zMzA1OUYnRjFGNC1GTTYlRk8tRiM2JkZWLUYjNictRi42JFEnMjY0LjAyRidGMUY0RltvRmRvRjFGZG9GMUZnb0Zkb0YxRmpvLUYjNictRi42JFEoMC40NTc5OUYnRjFGNC1GTTYlRk8tRiM2JkZWLUYjNictRi42JFEnMTkzLjIxRidGMUY0RltvRmRvRjFGZG9GMUZnb0Zkb0YxRltxLUYjNictRi42JFEoMC42MDM4OEYnRjFGNC1GTTYlRk8tRiM2JkZWLUYjNictRi42JFEnMTI3LjIwRidGMUY0RltvRmRvRjFGZG9GMUZnb0Zkb0YxRmpvLUYjNictRi42JFEoMC43ODAwMUYnRjFGNC1GTTYlRk8tRiM2JkZWLUYjNictRi42JFEnNzEuMzM3RidGMUY0RltvRmRvRjFGZG9GMUZnb0Zkb0YxRltxLUYjNictRi42JFEnMS4wMzI5RidGMUY0LUZNNiVGTy1GIzYmRlYtRiM2Jy1GLjYkUScyOS44NTFGJ0YxRjRGW29GZG9GMUZkb0YxRmdvRmRvRjFGam8tRiM2Jy1GLjYkUScxLjU5MzhGJ0YxRjQtRk02JUZPLUYjNiZGVi1GIzYnLUYuNiRRJzUuNzU2OEYnRjFGNEZbb0Zkb0YxRmRvRjFGZ29GZG9GMUZbcS1GLjYkUSgwLjk5OTk3RidGMUZkb0Yx">LDgtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJ0R0YoJCEmdHUlISIjJCImaWIiISInLUYkNiMsJEYrJCEmPFclRi4kISZYKWZGMS1GJDYjLCRGKyQhJl8mUkYuJCImMEgiISImLUYkNiMsJEYrJCEmYUwkRi4kISZHPyNGQC1GJDYjLCRGKyQhJi1rI0YuJCImZkkkRkAtRiQ2IywkRiskISZAJD5GLiQhJip6WEZALUYkNiMsJEYrJCEmP0YiRi4kIiYpUWdGQC1GJDYjLCRGKyQhJlA4KCEiJCQhJiwheUZALUYkNiMsJEYrJCEmXilIRlxvJCImSC4iISIlLUYkNiMsJEYrJCEmb3YmRmZvJCEmUWYiRmZvJCImKCoqKipGQCIiIg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L63" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">We can find the time taken for the center of the cylinder to reach 0.5 (i.e., 50% of the steady state):</Text-field>
</Input>
</Group>
<Group labelreference="L50" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">fsolve(u[0](t)-0.5,t=0..2);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">JCImWysjISIm</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L64" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">In this example temperature (or concentration) at the surface is fixed (u(1,t) = 1).  With our semianalytical solution we can find how the flux at the surface varies as a function of time.  This flux is given bya three point backward difference at x = 1.</Text-field>
</Input>
</Group>
<Group labelreference="L51" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">flux:=dydxb;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LDYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJ0R0YoJCEmdHUlISIjJCImKTQ4ISImLUYkNiMsJEYrJCEmPFclRi4kIiZJRyZGMS1GJDYjLCRGKyQhJl8mUkYuJCImbzIiISIlLUYkNiMsJEYrJCEmYUwkRi4kIiZCaiJGQC1GJDYjLCRGKyQhJi1rI0YuJCImdDEjRkAtRiQ2IywkRiskISZAJD5GLiQiJiozQkZALUYkNiMsJEYrJCEmP0YiRi4kIiZfTiNGQC1GJDYjLCRGKyQhJlA4KCEiJCQiJlNFI0ZALUYkNiMsJEYrJCEmXilIRlxvJCImRjcjRkAtRiQ2IywkRiskISZvdiZGQCQiJk8sI0ZA</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L52" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"></Text-field>
</Input>
</Group>
</Worksheet>