%% Created by Maple 9.5, Linux
%% Source Worksheet: /var/www/html/mikesullivan/Courses/305/LectureNotes/HeatDisk3.mw
%% Generated: Fri Apr 21 11:57:19 CDT 2017
\documentclass{article}
\usepackage{maplestd2e}
\def\emptyline{\vspace{12pt}}
\DefineParaStyle{Maple Heading 4}
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\DefineParaStyle{Maple Text Output}
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\DefineParaStyle{Maple Heading 3}
\DefineParaStyle{Maple Heading 1}
\DefineParaStyle{Maple Title}
\DefineParaStyle{Maple Normal}
\DefineCharStyle{Maple 2D Input}
\DefineCharStyle{Maple Maple Input}
\DefineCharStyle{Maple 2D Output}
\DefineCharStyle{Maple 2D Math}
\DefineCharStyle{Maple Hyperlink}
\begin{document}
\pagestyle{empty}
\begin{center}\begin{Maple Normal}{\normalsize{}}{\LARGE{\underline{\textbf{Heat Equation Example}}}}\end{Maple Normal}\end{center}

\begin{Maple Normal}{\normalsize{
Consider unit disk, temp help to zero along circumference. Heat costant = 1. Initial temp dist is f(r,theta) = r*(1-r)*exp(r*sin(theta)).

I had to define my own version of polar coordinates where z is the dependent variable.}}\end{Maple Normal}

\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{addcoords(zcylindrical,[z,r,theta],[r*cos(theta),r*sin(theta),z],}\QTR{Maple Maple Input}{
[[1],[Pi],[0],[0..2,0..2*Pi,-1..1],[-4..4,-4..4,-4..4]]);}\end{Maple Normal}}{}
\end{mapleinput}
\begin{Maple Normal}{\normalsize{We define the init temp func and plot it.}}\end{Maple Normal}

\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{f := (r,theta) -\TEXTsymbol{>} r*(1-r)*exp(r*sin(theta));}\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\QTR{Maple 2D Output}{\mapleinline{inert}{2d}{f := proc (r, theta) options operator, arrow; r*(1-r)*exp(r*sin(theta)) end proc}{%
\[
f\, := \,( {r,\theta} )\mapsto r \left( 1-r \right) {e^{r\sin \left( \theta \right) }}
\]
}
}

\end{maplelatex}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{plot3d(f(r,theta),r=0..1,theta=0..2*Pi,coords=zcylindrical,color=white,view=-0.001..0.5); }\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{center}
\mapleplot{HeatDisk3plot3d1.eps}


\end{center}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{g := (n,m) -\TEXTsymbol{>} evalf(BesselJZeros(n,m));   # zeros of the Bessel functions}\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\QTR{Maple 2D Output}{\mapleinline{inert}{2d}{g := proc (n, m) options operator, arrow; evalf(BesselJZeros(n, m)) end proc}{%
\[
g\, := \,( {n,m} )\mapsto {\it BesselJZeros} \left( n,m \right) 
\]
}
}

\end{maplelatex}
\begin{Maple Normal}{\normalsize{Defining the Fourier-Bessel coeffs.}}\end{Maple Normal}

\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{a := (n,m) -\TEXTsymbol{>} evalf((1/Pi)*int(r*BesselJ(n,g(n,m)*r)*int( f(r,theta)*cos(n*theta),theta=0..2*Pi), r=0..1)/}\QTR{Maple Maple Input}{
               int(r*BesselJ(n,g(n,m)*r)\symbol{94}2,r=0..1));}\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\QTR{Maple 2D Output}{\mapleinline{inert}{2d}{a := proc (n, m) options operator, arrow; evalf(int(r*BesselJ(n, g(n, m)*r)*int(f(r, theta)*cos(n*theta), theta = 0 .. 2*Pi), r = 0 .. 1)/(Pi*int(r*BesselJ(n, g(n, m)*r)^2, r = 0 .. 1))) end proc}{%
$a\, := \,( {n,m} )\mapsto  0.1795871221\,{\frac {\int _{ 0.0}^{ 1.0}\!r{\it BesselJ} \left( n,{\it BesselJZeros} \left( n,m \right) r \right) \int _{ 0.0}^{ 6.283185308}\!r \left(  1.0- 1.0\,r \right) \\
\mbox{}{e^{r\sin \left( \theta \right) }}\cos \left( n\theta \right) {d\theta}{dr}\Gamma  \left(  2.0+n \right) \\
\mbox{}\Gamma  \left( n+ 1.0 \right) }{{ 2.0}^{- 2.0- 2.0\,n} \left( {\it BesselJZeros} \left( n,m \right)  \right) ^{n- 1.0}{\it StruveH} \left( n, 2.0\,{\it BesselJZeros} \left( n,m \right)  \right) \\
\mbox{}\Gamma  \left( n+ 1.500000000 \right) }}$}
}

\end{maplelatex}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{b := (n,m) -\TEXTsymbol{>} evalf((1/Pi)*int(int( f(r,theta)*sin(n*theta)*r*BesselJ(n,g(n,m)*r),theta=0..2*Pi),r=0..1)/}\QTR{Maple Maple Input}{
               int(r*BesselJ(n,g(n,m)*r)\symbol{94}2,r=0..1)) ;}\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\QTR{Maple 2D Output}{\mapleinline{inert}{2d}{b := proc (n, m) options operator, arrow; evalf(int(int(f(r, theta)*sin(n*theta)*r*BesselJ(n, g(n, m)*r), theta = 0 .. 2*Pi), r = 0 .. 1)/(Pi*int(r*BesselJ(n, g(n, m)*r)^2, r = 0 .. 1))) end proc}{%
$b\, := \,( {n,m} )\mapsto  0.1795871221\,{\frac {\int _{ 0.0}^{ 1.0}\!\int _{ 0.0}^{ 6.283185308}\!{r}^{2} \left(  1.0- 1.0\,r \right) \\
\mbox{}{e^{r\sin \left( \theta \right) }}\sin \left( n\theta \right) {\it BesselJ} \left( n,{\it BesselJZeros} \left( n,m \right) r \right) \\
\mbox{}{d\theta}\,{dr}\\
\mbox{}\Gamma  \left(  2.0+n \right) \Gamma  \left( n+ 1.0 \right) }{{ 2.0}^{- 2.0- 2.0\,n}\\
\mbox{} \left( {\it BesselJZeros} \left( n,m \right)  \right) ^{- 1.0+n}{\it StruveH} \left( n, 2.0\,{\it BesselJZeros} \left( n,m \right)  \right) \\
\mbox{}\Gamma  \left( n+ 1.500000000 \right) }}$}
}

\end{maplelatex}

\mapleresult
\begin{maplettyout}
\QTR{Maple 2D Output}{}

\end{maplettyout}
\begin{Maple Normal}{\normalsize{Store coeffs into matricies so they do not have to be recomputed each time I run a plot or reload this worksheet.}}\end{Maple Normal}

\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{#A:=Matrix([[a(0,1),a(0,2),a(0,3),a(0,4),a(0,5),a(0,6)],}\QTR{Maple Maple Input}{
           [0,0,0,0,0,0],}\QTR{Maple Maple Input}{
           [a(2,1),a(2,2),a(2,3),a(2,4),a(2,5),a(2,6)],}\QTR{Maple Maple Input}{
           [0,0,0,0,0,0],}\QTR{Maple Maple Input}{
           [a(4,1),a(4,2),a(4,3),a(4,4),a(4,5),a(4,6)]]);}\end{Maple Normal}}{}
\end{mapleinput}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{#B:=Matrix([[0,0,0,0,0,0],}\QTR{Maple Maple Input}{
           [b(1,1),b(1,2),b(1,3),b(1,4),b(1,5),b(1,6)],}\QTR{Maple Maple Input}{
           [0,0,0,0,0,0],}\QTR{Maple Maple Input}{
           [b(3,1),b(3,2),b(3,3),b(3,4),b(3,5),b(3,6)],}\QTR{Maple Maple Input}{
           [0,0,0,0,0,0]]);}\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\QTR{Maple 2D Output}{\mapleinline{inert}{2d}{A := Matrix([[.6954751486, -.4951037620, 0.2522196891e-1, -0.9475954779e-1, 0.3038642443e-3, -0.3669013092e-1], [0, 0, 0, 0, 0, 0], [-0.5315429379e-1, 0.2517204028e-1, -0.9849377859e-2, 0.6053404618e-2, -0.3486761734e-2, 0.2469523976e-2], [0, 0, 0, 0, 0, 0], [0.7751847099e-3, -0.3706824374e-3, 0.1898884847e-3, -0.1167463015e-3, 0.7661691311e-4, -0.5400306453e-4]])}{%
\[
A\, := \, \left[ \begin {array}{cccccc}  0.6954751486&- 0.4951037620& 0.02522196891&- 0.09475954779& 0.0003038642443&- 0.03669013092\\\noalign{\medskip}0&0&0&0&0&0\\\noalign{\medskip}- 0.05315429379& 0.02517204028&- 0.009849377859& 0.006053404618&- 0.003486761734& 0.002469523976\\\noalign{\medskip}0&0&0&0&0&0\\\noalign{\medskip} 0.0007751847099&- 0.0003706824374& 0.0001898884847&- 0.0001167463015& 0.00007661691311&- 0.00005400306453\end {array} \right] 
\]
}
}

\end{maplelatex}

\mapleresult
\begin{maplelatex}
\QTR{Maple 2D Output}{\mapleinline{inert}{2d}{B := Matrix([[0, 0, 0, 0, 0, 0], [.2669643913, -.1367586324, 0.3651543590e-1, -0.2740526925e-1, 0.1180889830e-1, -0.1026528673e-1], [0, 0, 0, 0, 0, 0], [-0.7312801665e-2, 0.3439768703e-2, -0.1598315629e-2, 0.9649278828e-3, -0.6067110055e-3, 0.4226439297e-3], [0, 0, 0, 0, 0, 0]])}{%
\[
B\, := \, \left[ \begin {array}{cccccc} 0&0&0&0&0&0\\\noalign{\medskip} 0.2669643913&- 0.1367586324& 0.03651543590&- 0.02740526925& 0.01180889830&- 0.01026528673\\\noalign{\medskip}0&0&0&0&0&0\\\noalign{\medskip}- 0.007312801665& 0.003439768703&- 0.001598315629& 0.0009649278828&- 0.0006067110055& 0.0004226439297\\\noalign{\medskip}0&0&0&0&0&0\end {array} \right] 
\]
}
}

\end{maplelatex}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{A := Matrix([[.6954751486, -.4951037620, 0.2522196891e-1, -0.9475954779e-1, 0.3038642443e-3, -0.3669013092e-1], [0, 0, 0, 0, 0, 0], [-0.5315429379e-1, 0.2517204028e-1, -0.9849377859e-2, 0.6053404618e-2, -0.3486761734e-2, 0.2469523976e-2], [0, 0, 0, 0, 0, 0], [0.7751847099e-3, -0.3706824374e-3, 0.1898884847e-3, -0.1167463015e-3, 0.7661691311e-4, -0.5400306453e-4]]);}\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\QTR{Maple 2D Output}{\mapleinline{inert}{2d}{A := Matrix([[.6954751486, -.4951037620, 0.2522196891e-1, -0.9475954779e-1, 0.3038642443e-3, -0.3669013092e-1], [0, 0, 0, 0, 0, 0], [-0.5315429379e-1, 0.2517204028e-1, -0.9849377859e-2, 0.6053404618e-2, -0.3486761734e-2, 0.2469523976e-2], [0, 0, 0, 0, 0, 0], [0.7751847099e-3, -0.3706824374e-3, 0.1898884847e-3, -0.1167463015e-3, 0.7661691311e-4, -0.5400306453e-4]])}{%
\[
A\, := \, \left[ \begin {array}{cccccc}  0.6954751486&- 0.4951037620& 0.02522196891&- 0.09475954779& 0.0003038642443&- 0.03669013092\\\noalign{\medskip}0&0&0&0&0&0\\\noalign{\medskip}- 0.05315429379& 0.02517204028&- 0.009849377859& 0.006053404618&- 0.003486761734& 0.002469523976\\\noalign{\medskip}0&0&0&0&0&0\\\noalign{\medskip} 0.0007751847099&- 0.0003706824374& 0.0001898884847&- 0.0001167463015& 0.00007661691311&- 0.00005400306453\end {array} \right] 
\]
}
}

\end{maplelatex}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{B := Matrix([[0, 0, 0, 0, 0, 0], [.2669643913, -.1367586324, 0.3651543590e-1, -0.2740526925e-1, 0.1180889830e-1, -0.1026528673e-1], [0, 0, 0, 0, 0, 0], [-0.7312801665e-2, 0.3439768703e-2, -0.1598315629e-2, 0.9649278828e-3, -0.6067110055e-3, 0.4226439297e-3], [0, 0, 0, 0, 0, 0]]);}\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\QTR{Maple 2D Output}{\mapleinline{inert}{2d}{B := Matrix([[0, 0, 0, 0, 0, 0], [.2669643913, -.1367586324, 0.3651543590e-1, -0.2740526925e-1, 0.1180889830e-1, -0.1026528673e-1], [0, 0, 0, 0, 0, 0], [-0.7312801665e-2, 0.3439768703e-2, -0.1598315629e-2, 0.9649278828e-3, -0.6067110055e-3, 0.4226439297e-3], [0, 0, 0, 0, 0, 0]])}{%
\[
B\, := \, \left[ \begin {array}{cccccc} 0&0&0&0&0&0\\\noalign{\medskip} 0.2669643913&- 0.1367586324& 0.03651543590&- 0.02740526925& 0.01180889830&- 0.01026528673\\\noalign{\medskip}0&0&0&0&0&0\\\noalign{\medskip}- 0.007312801665& 0.003439768703&- 0.001598315629& 0.0009649278828&- 0.0006067110055& 0.0004226439297\\\noalign{\medskip}0&0&0&0&0&0\end {array} \right] 
\]
}
}

\end{maplelatex}
\begin{Maple Normal}{\normalsize{Now we plot the Fourier-Bessel approximation to the init temp dist, that is we plot u(r,theta,0) and compare to the given f(r,theta).}}\end{Maple Normal}

\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{N:=2: M:=2:}\end{Maple Normal}}{}
\end{mapleinput}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{U0:= (r,theta) -\TEXTsymbol{>} add(BesselJ(0,g(0,m)*r)*A[1,m] ,m=1..M)/2 + add(add(BesselJ(n,g(n,m)*r)*(A[n+1,m]*cos(n*theta) + B[n+1,m]*sin(n*theta)),m=1..M ), n=1..N );}\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\QTR{Maple 2D Output}{\mapleinline{inert}{2d}{U0 := proc (r, theta) options operator, arrow; 1/2*add(BesselJ(0, g(0, m)*r)*A[1, m], m = 1 .. M)+add(add(BesselJ(n, g(n, m)*r)*(A[n+1, m]*cos(n*theta)+B[n+1, m]*sin(n*theta)), m = 1 .. M), n = 1 .. N) end proc}{%
${\it U0}\, := \,( {r,\theta} )\mapsto  0.3477375743\,{\it BesselJ} \left( 0, 2.404825558\\
\mbox{}\,r \right) - 0.2475518810\,{\it BesselJ} \left( 0, 5.520078110\\
\mbox{}\,r \right) + 0.2669643913\,{\it BesselJ} \left( 1, 3.831705970\,r \right) \sin \left( \theta \right) \\
\mbox{}- 0.1367586324\,{\it BesselJ} \left( 1, 7.015586670\,r \right) \sin \left( \theta \right) - 0.05315429379\,{\it BesselJ} \left( 2, 5.135622302\,r \right) \cos \left( 2\,\theta \right) \\
\mbox{}+ 0.02517204028\,{\it BesselJ} \left( 2, 8.417244140\,r \right) \cos \left( 2\,\theta \right) $}
}

\end{maplelatex}
\begin{Maple Normal}{\normalsize{As you can see below, it looks pretty good with just a few terms.}}\end{Maple Normal}

\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{plot3d(U0(r,theta),r=0..1,theta=0..2*Pi,coords=zcylindrical,numpoints=1000,color=white,view=-0.001..0.5); }\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{center}
\mapleplot{HeatDisk3plot3d2.eps}


\end{center}
\begin{Maple Normal}{\normalsize{NOw we plot u(r,theta,t) for several values of t.}}\end{Maple Normal}

\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{U := (r,theta,t) -\TEXTsymbol{>} add(BesselJ(0,g(0,m)*r)*A[1,m]*exp(-g(0,m)\symbol{94}2*t) ,m=1..M)/2 + add(add(BesselJ(n,g(n,m)*r)*exp(-g(n,m)\symbol{94}2*t)*(A[n+1,m]*cos(n*theta) + B[n+1,m]*sin(n*theta)),m=1..M ), n=1..N );}\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\QTR{Maple 2D Output}{\mapleinline{inert}{2d}{U := proc (r, theta, t) options operator, arrow; 1/2*add(BesselJ(0, g(0, m)*r)*A[1, m]*exp(-g(0, m)^2*t), m = 1 .. M)+add(add(BesselJ(n, g(n, m)*r)*exp(-g(n, m)^2*t)*(A[n+1, m]*cos(n*theta)+B[n+1, m]*sin(n*theta)), m = 1 .. M), n = 1 .. N) end proc}{%
$U\, := \,( {r,\theta,t} )\mapsto  0.3477375743\,{\it BesselJ} \left( 0, 2.404825558\,r \right) {e^{- 5.783185964\,t}}\\
\mbox{}- 0.2475518810\,{\it BesselJ} \left( 0, 5.520078110\,r \right) {e^{- 30.47126234\,t}}\\
\mbox{}+ 0.2669643913\,{\it BesselJ} \left( 1, 3.831705970\,r \right) {e^{- 14.68197064\,t}}\sin \left( \theta \right) \\
\mbox{}- 0.1367586324\,{\it BesselJ} \left( 1, 7.015586670\,r \right) {e^{- 49.21845632\,t}}\sin \left( \theta \right) - 0.05315429379\,{\it BesselJ} \left( 2, 5.135622302\,r \right) {e^{- 26.37461643\,t}}\cos \left( 2\,\theta \right) \\
\mbox{}+ 0.02517204028\,{\it BesselJ} \left( 2, 8.417244140\,r \right) {e^{- 70.84999891\,t}}\cos \left( 2\,\theta \right) $}
}

\end{maplelatex}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{t:=0.01;}\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\QTR{Maple 2D Output}{\mapleinline{inert}{2d}{t := 0.1e-1}{%
\[
t\, := \, 0.01
\]
}
}

\end{maplelatex}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{plot3d(U(r,theta,t),r=0..1,theta=0..2*Pi,coords=zcylindrical,numpoints=1000,color=white,view=-0.001..0.5); }\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{center}
\mapleplot{HeatDisk3plot3d3.eps}


\end{center}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{t:=0.02;}\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\QTR{Maple 2D Output}{\mapleinline{inert}{2d}{t := 0.2e-1}{%
\[
t\, := \, 0.02
\]
}
}

\end{maplelatex}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{plot3d(U(r,theta,t),r=0..1,theta=0..2*Pi,coords=zcylindrical,numpoints=1000,color=white,view=-0.001..0.5); }\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{center}
\mapleplot{HeatDisk3plot3d4.eps}


\end{center}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{t:=0.05:}\end{Maple Normal}}{}
\end{mapleinput}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{plot3d(U(r,theta,t),r=0..1,theta=0..2*Pi,coords=zcylindrical,numpoints=1000,color=white,view=-0.001..0.5); }\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{center}
\mapleplot{HeatDisk3plot3d5.eps}


\end{center}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{t:=0.1:}\end{Maple Normal}}{}
\end{mapleinput}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{plot3d(U(r,theta,t),r=0..1,theta=0..2*Pi,coords=zcylindrical,numpoints=1000,color=white,view=-0.001..0.5); }\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{center}
\mapleplot{HeatDisk3plot3d6.eps}


\end{center}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{with(plots):}\end{Maple Normal}}{}
\end{mapleinput}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{animate(plot3d ,[ U(r,theta,a),r=0..1,theta=0..2*Pi,coords=zcylindrical,numpoints=1000,color=white,view=-0.001..0.5 ] ,a=0.0..0.2,frames=200);}\end{Maple Normal}}{}
\end{mapleinput}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{}\end{Maple Normal}}{}
\end{mapleinput}


\end{document}
%% End of Maple 9.5 Output