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<Text-field style="Heading 3" italic="false" layout="Heading 3"><Font italic="false">Example 5.3</Font></Text-field>
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<Text-field style="Text" layout="Normal">Consider the electrochemical discharge of a planar electrode.<Font superscript="true">[5]</Font> </Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
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<Text-field style="Text" layout="Normal" alignment="centred"></Text-field>
<Text-field style="Text" layout="Normal" alignment="centred">                                                                                         u(x,0) = 1                                                               (5.1.20)            </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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">LUkmbWZyYWNHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2KC1JJW1yb3dHRiQ2JS1JI21vR0YkNi1RKyZQYXJ0aWFsRDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRi1JI21pR0YkNiVRInVGJy8lJ2l0YWxpY0dGN0YyRjItRiw2JUYuLUZKNiVRInhGJ0ZNRjJGMi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGWS8lKWJldmVsbGVkR0Y3</Equation>(0,t) = 0 and <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation><Font encoding="UTF-8">(1,t) = -\316\264</Font></Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">where u is the dimensionless concentration and \357\201\244 is the dimensionless applied current density at the surface.  The electrochemical performance of the electrode depends on the concentration (u) at the surface.  When this problem is cast into finite differences we arrive at the matrix differential equation (equation (5.1.16) with a constant b vector.  Hence, we can use equation (5.1.18) to arrive at the semianalytical solution.  However, because of flux boundary conditions at both the ends, the A matrix becomes singular.  Hence, A cannot be inverted and equation (5.1.18) cannot be used.  It should be noted that if A is singular that does not mean that equation (5.1.16) does not have a solution.  One can prove that the solution (equation (5.1.18)) is independent of A-1 by using the series expansion for the exponential matrix[7]  [8]</Font></Text-field>
<Text-field style="Text" layout="Normal">	 	                                                                 <Image height="44" width="263" zoomable="false">[encoded data, 6712 characters]<Drawing-Root>LSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIiISEiIi0lKUJPVU5EU19ZR0YnLSUtQk9VTkRTX1dJRFRIRzYjJCIlSUVGKi0lLkJPVU5EU19IRUlHSFRHNiMkIiRTJUYqLSUpQ0hJTERSRU5HNiI=</Drawing-Root></Image>                              (5.1.21)</Text-field>
<Text-field style="Text" layout="Normal">Now the exponential matrix in the second term is represented as an infinite series:</Text-field>
<Text-field style="Text" layout="Normal">	 	                                    <Image height="51" width="480" zoomable="false">[encoded data, 10272 characters]<Drawing-Root>LSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIlU0ghIiItJSlCT1VORFNfWUc2IyQiIiFGKi0lLUJPVU5EU19XSURUSEc2IyQiJStbRiotJS5CT1VORFNfSEVJR0hURzYjJCIkNSZGKi0lKUNISUxEUkVORzYi</Drawing-Root></Image>     ( 5.1.22)</Text-field>
<Text-field style="Text" layout="Normal">where I is the identity matrix of order N x N.  Equation (5.1.22) can be further simplified by factoring out A-1b from the second and third terms.</Text-field>
<Text-field style="Text" layout="Normal">	 	                                     <Image height="60" width="492" zoomable="false">[encoded data, 10040 characters]<Drawing-Root>LSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIiISEiIi0lKUJPVU5EU19ZR0YnLSUtQk9VTkRTX1dJRFRIRzYjJCIlP1xGKi0lLkJPVU5EU19IRUlHSFRHNiMkIiQrJ0YqLSUpQ0hJTERSRU5HNiI=</Drawing-Root></Image>(5.1.23)</Text-field>
<Text-field style="Text" layout="Normal">The identity matrices inside the parenthesis get cancelled.  Next, A-1 outside the parenthesis is taken inside the parenthesis to obtain:</Text-field>
<Text-field style="Text" layout="Normal">	                                                      <Image height="60" width="397" zoomable="false">[encoded data, 8628 characters]<Drawing-Root>LSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIiISEiIi0lKUJPVU5EU19ZR0YnLSUtQk9VTkRTX1dJRFRIRzYjJCIlcVJGKi0lLkJPVU5EU19IRUlHSFRHNiMkIiQrJ0YqLSUpQ0hJTERSRU5HNiI=</Drawing-Root></Image>                         (5.1.24)</Text-field>
<Text-field style="Text" layout="Normal">We observe that equation (5.1.24) is independent of A-1 and, hence, we can obtain a solution for equation (5.1.16) even when A is singular.  The infinite series in equation (5.1.23) is difficult to calculate.  Alternatively, we use equation (5.1.17), which is valid even when A is singular to obtain the semianalytical solutions.  Equation (5.1.19) is solved below in Maple using the procedure described above with equation (5.1.17)a for the nonhomogeneous part.</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">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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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">ge:=diff(u(x,t),t)=diff(u(x,t),x$2);</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc1:=diff(u(x,t),x);</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">bc2:=diff(u(x,t),x)+delta;</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">#Digits:=50;</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">IC:=u(x,0)=1;</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">N:=4;</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 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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<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>
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<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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<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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<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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<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>
</Input>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq[0]:=bc1; </Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq[N+1]:=bc2; </Text-field>
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<Group labelreference="L19" 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>
</Input>
</Group>
<Group labelreference="L20" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">u[0](t):=(solve(eq[0],u[0](t)));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi0mSSJ1RzYiNiMiIiE2I0kidEdGJiwmKiYjIiIiIiIkRi4tJkYlNiMiIiNGKUYuISIiKiYjIiIlRi9GLi0mRiU2I0YuRilGLkYu">LCYqJiMiIiIiIiRGJS0mSSJ1RzYiNiMiIiM2I0kidEdGKkYlISIiKiYjIiIlRiZGJS0mRik2I0YlRi1GJUYl</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L21" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">u[N+1](t):=solve(eq[N+1],u[N+1](t));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi0mSSJ1RzYiNiMiIiY2I0kidEdGJiwoKiYjIiIiIiIkRi4tJkYlNiNGL0YpRi4hIiIqJiMiIiVGL0YuLSZGJTYjRjZGKUYuRi4qKCMiIiNGL0YuSSZkZWx0YUdGJkYuSSJoR0YmRi5GMw==">LCgqJiMiIiIiIiRGJS0mSSJ1RzYiNiNGJjYjSSJ0R0YqRiUhIiIqJiMiIiVGJkYlLSZGKTYjRjFGLEYlRiUqKCMiIiNGJkYlSSZkZWx0YUdGKkYlSSJoR0YqRiVGLg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L22" 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>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI2VxRzYiNiMiIiIvLUklZGlmZkclKnByb3RlY3RlZEc2JC0mSSJ1R0YlRiY2I0kidEdGJUYxKiYpSSJoR0YlIiIjISIiLCYqJiNGNSIiJEYnLSZGLzYjRjVGMEYnRicqJkY5RidGLUYnRjZGJw==">Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtJkkidUc2IjYjIiIiNiNJInRHRipGLiomKUkiaEdGKiIiIyEiIiwmKiYjRjIiIiRGLC0mRik2I0YyRi1GLEYsKiZGNkYsRidGLEYzRiw=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI2VxRzYiNiMiIiMvLUklZGlmZkclKnByb3RlY3RlZEc2JC0mSSJ1R0YlRiY2I0kidEdGJUYxKiYpSSJoR0YlRichIiIsKC0mRi82IyIiIkYwRjoqJkYnRjpGLUY6RjUtJkYvNiMiIiRGMEY6Rjo=">Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtJkkidUc2IjYjIiIjNiNJInRHRipGLiomKUkiaEdGKkYsISIiLCgtJkYpNiMiIiJGLUY3KiZGLEY3RidGN0YyLSZGKTYjIiIkRi1GN0Y3</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI2VxRzYiNiMiIiQvLUklZGlmZkclKnByb3RlY3RlZEc2JC0mSSJ1R0YlRiY2I0kidEdGJUYxKiYpSSJoR0YlIiIjISIiLCgtJkYvNiNGNUYwIiIiKiZGNUY7Ri1GO0Y2LSZGLzYjIiIlRjBGO0Y7">Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtJkkidUc2IjYjIiIkNiNJInRHRipGLiomKUkiaEdGKiIiIyEiIiwoLSZGKTYjRjJGLSIiIiomRjJGOEYnRjhGMy0mRik2IyIiJUYtRjhGOA==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI2VxRzYiNiMiIiUvLUklZGlmZkclKnByb3RlY3RlZEc2JC0mSSJ1R0YlRiY2I0kidEdGJUYxKiYpSSJoR0YlIiIjISIiLCgqJiNGNSIiJCIiIi0mRi82I0Y6RjBGO0Y7KiZGOUY7Ri1GO0Y2KihGOUY7SSZkZWx0YUdGJUY7RjRGO0Y2Rjs=">Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtJkkidUc2IjYjIiIlNiNJInRHRipGLiomKUkiaEdGKiIiIyEiIiwoKiYjRjIiIiQiIiItJkYpNiNGN0YtRjhGOComRjZGOEYnRjhGMyooRjZGOEkmZGVsdGFHRipGOEYxRjhGM0Y4</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L23" 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="L24" 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="PkkiWUc2IjcmLSZJInVHRiQ2IyIiIjYjSSJ0R0YkLSZGKDYjIiIjRistJkYoNiMiIiRGKy0mRig2IyIiJUYr">NyYtJkkidUc2IjYjIiIiNiNJInRHRiYtJkYlNiMiIiNGKS0mRiU2IyIiJEYpLSZGJTYjIiIlRik=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L25" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">A:=genmatrix(eqs,Y,'b1'):</Text-field>
</Input>
</Group>
<Group labelreference="L26" 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="LUknbWF0cml4RzYiNiM3JjcjIiIhRidGJzcjLCQqKCMiIiMiIiQiIiJJImhHRiQhIiJJJmRlbHRhR0YkRi9GMQ==">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</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L27" 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="PkkiaEc2IiMiIiIiIiY=">IyIiIiIiJg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L28" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">A:=map(eval,A):</Text-field>
</Input>
</Group>
<Group labelreference="L29" 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="L30" 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="LUknbWF0cml4RzYiNiM3JjcmIyEjXSIiJCMiI11GKiIiIUYtNyYiI0RGKUYvRi03JkYtRi9GKUYvNyZGLUYtRitGKA==">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</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L31" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">det(A);</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="L62" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">We observe that the <Font bold="true">A</Font> matrix is singular..</Text-field>
</Input>
</Group>
<Group labelreference="L32" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">mat:=exponential(A,t):</Text-field>
</Input>
</Group>
<Group labelreference="L33" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">mat:=map(evalf,mat):</Text-field>
</Input>
</Group>
<Group labelreference="L34" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">mat:=map(simplify,mat):</Text-field>
</Input>
</Group>
<Group labelreference="L35" 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="LUknbWF0cml4RzYiNiM3JjcjJCIiIiIiIUYnRidGJw==">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</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L36" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">b2:=subs(t=tau,evalm(b));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjYjJHNiItSSdtYXRyaXhHRiQ2IzcmNyMiIiFGKUYpNyMsJCooIyIiIyIiJCIiIkkiaEdGJCEiIkkmZGVsdGFHRiRGMUYz">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</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L37" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">mat2:=subs(t=tau,evalm(mat)):</Text-field>
</Input>
</Group>
<Group labelreference="L38" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">mat3:=evalm(mat2&amp;*b2):</Text-field>
</Input>
</Group>
<Group labelreference="L39" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">mat4:=map(int,mat3,tau=0..t):</Text-field>
</Input>
</Group>
<Group labelreference="L40" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Y:=evalm(mat&amp;*Y0+mat4):</Text-field>
</Input>
</Group>
<Group labelreference="L41" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">#Y:=map(simplify,Y):</Text-field>
</Input>
</Group>
<Group labelreference="L42" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">#Digits:=20;</Text-field>
</Input>
</Group>
<Group labelreference="L43" 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="L44" 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="Pi0mSSJ1RzYiNiMiIiE2I0kidEdGJiwuJCIrLCsrKzUhIioiIiIqJiQiKysrK1M6ISM1Ri9JJmRlbHRhR0YmRi9GLyooJCIrT0xMTEghIzZGL0Y0Ri8tSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2IywkKiYkIitubW1tVCEiKUYvRipGLyEiIkYvRi8qKCQiK2wiXDYnWyEjN0YvRjRGLy1GOjYjLCQqJiQiKyMqW0xWIilGQ0YvRipGL0ZERi9GRCooJCIrVD1zJXkiRjNGL0Y0Ri8tRjo2IywkKiYkIit0PExCNUZDRi9GKkYvRkRGL0ZEKihGLEYvRjRGL0YqRi9GRA==">LC4kIissKysrNSEiKiIiIiomJCIrKysrUzohIzVGJkkmZGVsdGFHNiJGJkYmKigkIitPTExMSCEjNkYmRitGJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGLDYjLCQqJiQiK25tbW1UISIpRiZJInRHRixGJiEiIkYmRiYqKCQiK2wiXDYnWyEjN0YmRitGJi1GMjYjLCQqJiQiKyMqW0xWIilGO0YmRjxGJkY9RiZGPSooJCIrVD1zJXkiRipGJkYrRiYtRjI2IywkKiYkIit0PExCNUY7RiZGPEYmRj1GJkY9KihGI0YmRitGJkY8RiZGPQ==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi0mSSJ1RzYiNiMiIiI2I0kidEdGJiwuJCIrLCsrKzUhIipGKComJCIrKysrUzghIzVGKEkmZGVsdGFHRiZGKEYoKigkIisuSUdcPSEjN0YoRjNGKC1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjLCQqJiQiKyMqW0xWIikhIilGKEYqRighIiJGKEZDKigkIitxcl0iWyJGMkYoRjNGKC1GOTYjLCQqJiQiK3Q8TEI1RkJGKEYqRihGQ0YoRkMqKCQiKywrKys7ISM2RihGM0YoLUY5NiMsJComJCIrbm1tbVRGQkYoRipGKEZDRihGKCooRixGKEYzRihGKkYoRkM=">LC4kIissKysrNSEiKiIiIiomJCIrKysrUzghIzVGJkkmZGVsdGFHNiJGJkYmKigkIisuSUdcPSEjN0YmRitGJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGLDYjLCQqJiQiKyMqW0xWIikhIilGJkkidEdGLEYmISIiRiZGPSooJCIrcXJdIlsiRipGJkYrRiYtRjI2IywkKiYkIit0PExCNUY7RiZGPEYmRj1GJkY9KigkIissKysrOyEjNkYmRitGJi1GMjYjLCQqJiQiK25tbW1URjtGJkY8RiZGPUYmRiYqKEYjRiZGK0YmRjxGJkY9</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi0mSSJ1RzYiNiMiIiM2I0kidEdGJiwuJCIrLCsrKzUhIioiIiIqJiQiKywrKyt1ISM2Ri9JJmRlbHRhR0YmRi9GLyooJCIrLSsrK0NGM0YvRjRGLy1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjLCQqJiQiK25tbW1UISIpRi9GKkYvISIiRi9GQyooJCIrIls6aj0oISM3Ri9GNEYvLUY5NiMsJComJCIrIypbTFYiKUZCRi9GKkYvRkNGL0YvKigkIitaOmo9ZEYzRi9GNEYvLUY5NiMsJComJCIrdDxMQjVGQkYvRipGL0ZDRi9GQyooRixGL0Y0Ri9GKkYvRkM=">LC4kIissKysrNSEiKiIiIiomJCIrLCsrK3UhIzZGJkkmZGVsdGFHNiJGJkYmKigkIistKysrQ0YqRiZGK0YmLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YsNiMsJComJCIrbm1tbVQhIilGJkkidEdGLEYmISIiRiZGPCooJCIrIls6aj0oISM3RiZGK0YmLUYxNiMsJComJCIrIypbTFYiKUY6RiZGO0YmRjxGJkYmKigkIitaOmo9ZEYqRiZGK0YmLUYxNiMsJComJCIrdDxMQjVGOkYmRjtGJkY8RiZGPCooRiNGJkYrRiZGO0YmRjw=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi0mSSJ1RzYiNiMiIiQ2I0kidEdGJiwuJCIrLCsrKzUhIioiIiIqJiQiKygqKioqKipmIyEjNkYvSSZkZWx0YUdGJkYvISIiKigkIistKysrQ0YzRi9GNEYvLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiMsJComJCIrbm1tbVQhIilGL0YqRi9GNUYvRjUqKCQiKyJbOmo9KCEjN0YvRjRGLy1GOjYjLCQqJiQiKyMqW0xWIilGQ0YvRipGL0Y1Ri9GNSooJCIrWjpqPWRGM0YvRjRGLy1GOjYjLCQqJiQiK3Q8TEI1RkNGL0YqRi9GNUYvRi8qKEYsRi9GNEYvRipGL0Y1">LC4kIissKysrNSEiKiIiIiomJCIrKCoqKioqKmYjISM2RiZJJmRlbHRhRzYiRiYhIiIqKCQiKy0rKytDRipGJkYrRiYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiw2IywkKiYkIitubW1tVCEiKUYmSSJ0R0YsRiZGLUYmRi0qKCQiKyJbOmo9KCEjN0YmRitGJi1GMjYjLCQqJiQiKyMqW0xWIilGO0YmRjxGJkYtRiZGLSooJCIrWjpqPWRGKkYmRitGJi1GMjYjLCQqJiQiK3Q8TEI1RjtGJkY8RiZGLUYmRiYqKEYjRiZGK0YmRjxGJkYt</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi0mSSJ1RzYiNiMiIiU2I0kidEdGJiwuJCIrLCsrKzUhIioiIiIqJiQiKysrK2c7ISM1Ri9JJmRlbHRhR0YmRi8hIiIqKEYsRi9GNEYvRipGL0Y1KigkIissKysrOyEjNkYvRjRGLy1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjLCQqJiQiK25tbW1UISIpRi9GKkYvRjVGL0YvKigkIisuSUdcPSEjN0YvRjRGLy1GPDYjLCQqJiQiKyMqW0xWIilGRUYvRipGL0Y1Ri9GLyooJCIrcXJdIlsiRjNGL0Y0Ri8tRjw2IywkKiYkIit0PExCNUZFRi9GKkYvRjVGL0Yv">LC4kIissKysrNSEiKiIiIiomJCIrKysrZzshIzVGJkkmZGVsdGFHNiJGJiEiIiooRiNGJkYrRiZJInRHRixGJkYtKigkIissKysrOyEjNkYmRitGJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGLDYjLCQqJiQiK25tbW1UISIpRiZGL0YmRi1GJkYmKigkIisuSUdcPSEjN0YmRitGJi1GNTYjLCQqJiQiKyMqW0xWIilGPkYmRi9GJkYtRiZGJiooJCIrcXJdIlsiRipGJkYrRiYtRjU2IywkKiYkIit0PExCNUY+RiZGL0YmRi1GJkYm</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi0mSSJ1RzYiNiMiIiY2I0kidEdGJiwuJCIrLCsrKzUhIioiIiIqJiQiKyoqKioqKmZNISM1Ri9JJmRlbHRhR0YmRi8hIiIqKCQiK09MTExIISM2Ri9GNEYvLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiMsJComJCIrbm1tbVQhIilGL0YqRi9GNUYvRi8qKCQiK2wiXDYnWyEjN0YvRjRGLy1GOzYjLCQqJiQiKyMqW0xWIilGREYvRipGL0Y1Ri9GLyooJCIrVD1zJXkiRjNGL0Y0Ri8tRjs2IywkKiYkIit0PExCNUZERi9GKkYvRjVGL0YvKihGLEYvRjRGL0YqRi9GNQ==">LC4kIissKysrNSEiKiIiIiomJCIrKioqKioqZk0hIzVGJkkmZGVsdGFHNiJGJiEiIiooJCIrT0xMTEghIzZGJkYrRiYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiw2IywkKiYkIitubW1tVCEiKUYmSSJ0R0YsRiZGLUYmRiYqKCQiK2wiXDYnWyEjN0YmRitGJi1GMzYjLCQqJiQiKyMqW0xWIilGPEYmRj1GJkYtRiZGJiooJCIrVD1zJXkiRipGJkYrRiYtRjM2IywkKiYkIit0PExCNUY8RiZGPUYmRi1GJkYmKihGI0YmRitGJkY9RiZGLQ==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L63" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">We obtain the semianalytical solution for the concentration profile as a function of dimensionless current density, \316\264.  Concentration profiles can be plotted specifying values for \316\264.</Font></Text-field>
</Input>
</Group>
<Group labelreference="L69" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">pp:=plot([seq(subs(delta=1,u[i](t)),i=0..N+1)],t=0..0.4);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="[encoded data, 9272 characters]">[encoded data, 9240 characters]</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L72" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">pt:=textplot([[0.25,evalf(subs(t=0.25,delta=1,u[0](t))),typeset(u[0],&quot;(t)&quot;),align=above],seq([0.25,evalf(subs(t=0.25,delta=1,u[i](t))),typeset(u[i],&quot;(t)&quot;),align=below],i=1..N+1)]);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjcHRHNiItSS9JTlRFUkZBQ0VfUExPVEdGJDYoLUklVEVYVEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYlNyQkIiNEISIjJCIrXzEiPSEqKSEjNS1JKV9UWVBFU0VUR0YkNiQmSSJ1R0YkNiMiIiFRJCh0KUYkSStBTElHTkFCT1ZFR0YqLUYpNiU3JEYvJCIrJXkmR0QoKUY0LUY2NiQmRjk2IyIiIkY8SStBTElHTkJFTE9XR0YqLUYpNiU3JEYvJCIrejZyJj4pRjQtRjY2JCZGOTYjIiIjRjxGSC1GKTYlNyRGLyQiK3JXRiVHKEY0LUY2NiQmRjk2IyIiJEY8RkgtRik2JTckRi8kIis1UXNhZkY0LUY2NiQmRjk2IyIiJUY8RkgtRik2JTckRi8kIitCcD95VEY0LUY2NiQmRjk2IyIiJkY8Rkg=">LSUlUExPVEc2KC1JJVRFWFRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JTckJCIjRCEiIyQiK18xIj0hKikhIzUtSSlfVFlQRVNFVEdGKjYkJkkidUdGKjYjIiIhUSQodClGKkkrQUxJR05BQk9WRUdGJy1GJjYlNyRGLSQiKyV5JkdEKClGMi1GNDYkJkY3NiMiIiJGOkkrQUxJR05CRUxPV0dGJy1GJjYlNyRGLSQiK3o2ciY+KUYyLUY0NiQmRjc2IyIiI0Y6RkYtRiY2JTckRi0kIityV0YlRyhGMi1GNDYkJkY3NiMiIiRGOkZGLUYmNiU3JEYtJCIrNVFzYWZGMi1GNDYkJkY3NiMiIiVGOkZGLUYmNiU3JEYtJCIrQnA/eVRGMi1GNDYkJkY3NiMiIiZGOkZG</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L46" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">display([pp,pt],thickness=4,title=&quot;Figure Exp. 5.7&quot;,axes=boxed,labels=[t,&quot;u&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, 31116 characters]</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L70" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">pp:=plot([seq(subs(delta=0.1,u[i](t)),i=0..N+1)],t=0..0.4);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="[encoded data, 9472 characters]">[encoded data, 9440 characters]</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L73" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">pt:=textplot([[0.25,evalf(subs(t=0.25,delta=0.1,u[0](t))),typeset(u[0],&quot;(t)&quot;),align=above],seq([0.25,evalf(subs(t=0.25,delta=0.1,u[i](t))),typeset(u[i],&quot;(t)&quot;),align=below],i=1..N+1)]);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjcHRHNiItSS9JTlRFUkZBQ0VfUExPVEdGJDYoLUklVEVYVEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYlNyQkIiNEISIjJCIrdTU9ISopKiEjNS1JKV9UWVBFU0VUR0YkNiQmSSJ1R0YkNiMiIiFRJCh0KUYkSStBTElHTkFCT1ZFR0YqLUYpNiU3JEYvJCIrKWVHRCgpKkY0LUY2NiQmRjk2IyIiIkY8SStBTElHTkJFTE9XR0YqLUYpNiU3JEYvJCIrRjZkPikqRjQtRjY2JCZGOTYjIiIjRjxGSC1GKTYlNyRGLyQiK2R1VUcoKkY0LUY2NiQmRjk2IyIiJEY8RkgtRik2JTckRi8kIishUnNhZipGNC1GNjYkJkY5NiMiIiVGPEZILUYpNiU3JEYvJCIrLTIjeVQqRjQtRjY2JCZGOTYjIiImRjxGSA==">LSUlUExPVEc2KC1JJVRFWFRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JTckJCIjRCEiIyQiK3U1PSEqKSohIzUtSSlfVFlQRVNFVEdGKjYkJkkidUdGKjYjIiIhUSQodClGKkkrQUxJR05BQk9WRUdGJy1GJjYlNyRGLSQiKyllR0QoKSpGMi1GNDYkJkY3NiMiIiJGOkkrQUxJR05CRUxPV0dGJy1GJjYlNyRGLSQiK0Y2ZD4pKkYyLUY0NiQmRjc2IyIiI0Y6RkYtRiY2JTckRi0kIitkdVVHKCpGMi1GNDYkJkY3NiMiIiRGOkZGLUYmNiU3JEYtJCIrIVJzYWYqRjItRjQ2JCZGNzYjIiIlRjpGRi1GJjYlNyRGLSQiKy0yI3lUKkYyLUY0NiQmRjc2IyIiJkY6RkY=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L64" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">display([pp,pt],thickness=4,title=&quot;Figure Exp. 5.8&quot;,axes=boxed,labels=[t,&quot;u&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, 31264 characters]</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L48" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">tf:=0.4;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjdGZHNiIkIiIlISIi">JCIiJSEiIg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L65" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">We observe that as \316\264 decreases, the time taken for discharge (the concentration to decrease from 1 to 0) increases.  A three dimensional plot for the concentration profile can be made as follows:</Font></Text-field>
</Input>
</Group>
<Group labelreference="L49" 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="L50" 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="PkkjVDFHNiI3QSQiIiFGJyQiK0xMTEw4ISM2JCIrbm1tbUVGKiQiKysrKytTRiokIitMTExMYEYqJCIrbm1tbW1GKiQiKysrKyshKUYqJCIrTExMTCQqRiokIitubW1tNSEjNSQiKysrKys3RjkkRilGOSQiK25tbW05RjkkIisrKysrO0Y5JCIrTExMTDxGOSQiK25tbW09RjkkIisrKysrP0Y5JCIrTExMTEBGOSQiK25tbW1BRjkkIisrKysrQ0Y5JCIrTExMTERGOSRGLEY5JCIrKysrK0dGOSQiK0xMTExIRjkkIitubW1tSUY5JCIrKysrK0tGOSQiK0xMTExMRjkkIitubW1tTUY5JCIrKysrK09GOSQiK0xMTExQRjkkIitubW1tUUY5JEYuRjk=">N0EkIiIhRiQkIitMTExMOCEjNiQiK25tbW1FRickIisrKysrU0YnJCIrTExMTGBGJyQiK25tbW1tRickIisrKysrISlGJyQiK0xMTEwkKkYnJCIrbm1tbTUhIzUkIisrKysrN0Y2JEYmRjYkIitubW1tOUY2JCIrKysrKztGNiQiK0xMTEw8RjYkIitubW1tPUY2JCIrKysrKz9GNiQiK0xMTExARjYkIitubW1tQUY2JCIrKysrK0NGNiQiK0xMTExERjYkRilGNiQiKysrKytHRjYkIitMTExMSEY2JCIrbm1tbUlGNiQiKysrKytLRjYkIitMTExMTEY2JCIrbm1tbU1GNiQiKysrKytPRjYkIitMTExMUEY2JCIrbm1tbVFGNiRGK0Y2</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L51" 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="PkkjUFBHNiItSSZhcnJheUclKnByb3RlY3RlZEc2JTsiIiIiIic7RioiI0o3Ig==">PTYiNiQ7IiIiIiInO0YmIiNKRVxbbCE=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L52" 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="L53" 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],subs(delta=1,u[i-1](t))));od;od:</Text-field>
</Input>
</Group>
<Group labelreference="L54" 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="L55" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">surfdata( plotdata, axes=boxed, title=&quot;Figure Exp. 5.9&quot;,labels=[x,t,u],orientation=[45,45]);</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, 8896 characters]</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L66" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Electrochemical behavior of the electrode depends on the surface concentration given by:</Text-field>
</Input>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">u[N+1](t);</Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LC4kIissKysrNSEiKiIiIiomJCIrKioqKioqZk0hIzVGJkkmZGVsdGFHNiJGJiEiIiooJCIrT0xMTEghIzZGJkYrRiYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiw2IywkKiYkIitubW1tVCEiKUYmSSJ0R0YsRiZGLUYmRiYqKCQiK2wiXDYnWyEjN0YmRitGJi1GMzYjLCQqJiQiKyMqW0xWIilGPEYmRj1GJkYtRiZGJiooJCIrVD1zJXkiRipGJkYrRiYtRjM2IywkKiYkIit0PExCNUY8RiZGPUYmRi1GJkYmKihGI0YmRitGJkY9RiZGLQ==</Equation></Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">plot3d(u[N+1](t),delta=0..1,t=0..1,axes=boxed,title=&quot;Figure Exp. 5.10&quot;,view=[0..5,0..1,0..1],orientation=[0,90],labels=[delta,t,&quot;us&quot;]);</Text-field>
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<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, 13804 characters]</Plot></Text-field>
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<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">For low values of \316\264, the surface concentration remains close to 1, and as \316\264 decrease, the surface concentration depletes faster.  Accuracy can be increased by increasing the number of node points.</Font></Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal"></Text-field>
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