%% Created by Maple 9.5, Linux
%% Source Worksheet: /var/www/html/mikesullivan/Courses/305/LectureNotes/HeatDIsk4.mw
%% Generated: Sun Apr 23 18:18:55 CDT 2017
\documentclass{article}
\usepackage{maplestd2e}
\def\emptyline{\vspace{12pt}}
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\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{\textbf{Heat Equation on a Disk Example with Discontinuous init condition}}}\end{Maple Normal}\end{center}

\begin{Maple Normal}{\normalsize{}}\end{Maple Normal}
\begin{Maple Normal}{\normalsize{Problem: On unit disk init temp is 1 inside circle of radius 1/2, center (0,1/2), that is r<sin(theta). Temp held t one along circumference, r=1. }}\end{Maple Normal}

\begin{Maple Normal}{\normalsize{}}\end{Maple Normal}
\begin{Maple Normal}{\normalsize{}}\end{Maple Normal}
\begin{Maple Normal}{\normalsize{Set up cylindrical coords with dependent variable z. }}\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{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{}\end{Maple Normal}}{}
\end{mapleinput}
\begin{Maple Normal}{\normalsize{Define f(r,theta) the graph it. I st itas a procedure and this changes the format of the plotting and integration commands.}}\end{Maple Normal}

\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{f := proc(r, theta) }\QTR{Maple Maple Input}{
     if ((r)<= (sin(theta))) then 1.0 else 0.0 fi;end;}\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\QTR{Maple 2D Output}{\mapleinline{inert}{2d}{f := proc (r, theta) if r <= sin(theta) then 1.0 else 0. end if end proc}{%
\[
f\, := \,\textbf{proc} (rtheta) \\
\mapleIndent{1} \textbf{if} \,r&le\mathit{sin} (theta) \, \textbf{then} \\
\mapleIndent{2} 1.0\\
\mapleIndent{1} \textbf{else}\\
\mapleIndent{2} 0.\\
\mapleIndent{1} \textbf{end\ if}; \\
\textbf{end\ proc};
\]
}
}

\end{maplelatex}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{}\end{Maple Normal}}{}
\end{mapleinput}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{plot3d( f,0..1,0..2*Pi,coords=zcylindrical,numpoints=50000); #The default coloring has no physical significance.  }\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{center}
\mapleplot{HeatDIsk4plot3d1.eps}


\end{center}
\begin{Maple Normal}{\normalsize{I did not use my f procedure to do the integrals. I just integrate 1 over r=0..sin(theta) for 0<= theta <= pi, since f is zero elsewhere. Here is a quick example where I find teh volume under f(r,theta). }}\end{Maple Normal}

\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{int(int(r,r=0..sin(theta)),theta=0..Pi);   # I will do the integrals like this. }\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\QTR{Maple 2D Output}{\mapleinline{inert}{2d}{1/4*Pi}{%
\[
1/4\,\pi 
\]
}
}

\end{maplelatex}
\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{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{a := (n,m) -\TEXTsymbol{>} evalf((1/Pi)*evalf(Int(evalf(Int( 1*r*BesselJ(n,g(n,m)*r)*cos(n*theta),r=0..sin(theta))), theta=0..Pi))/}\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(evalf(Int(evalf(Int(r*BesselJ(n, g(n, m)*r)*cos(n*theta), r = 0 .. sin(theta))), theta = 0 .. Pi))/(Pi*int(r*BesselJ(n, g(n, m)*r)^2, r = 0 .. 1))) end proc}{%
$a\, := \,( {n,m} )\mapsto  0.1795871221\,{\frac {\int _{ 0.0}^{ 3.141592654}\!\int _{ 0.0}^{\sin \left( \theta \right) }\!r{\it BesselJ} \left( n,{\it BesselJZeros} \left( n,m \right) r \right) \cos \left( n\theta \right) {dr}\,{d\theta}\\
\mbox{}\Gamma  \left(  2.0+n \right) \\
\mbox{}\Gamma  \left(  1.0+n \right) }{{ 2.0}^{- 2.0- 2.0\,n} \left( {\it BesselJZeros} \left( n,m \right)  \right) ^{- 1.0+n}\\
\mbox{}{\it StruveH} \left( n, 2.0\,{\it BesselJZeros} \left( n,m \right)  \right) \Gamma  \left( n+ 1.500000000 \right) \\
\mbox{}}}$}
}

\end{maplelatex}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{}\end{Maple Normal}}{}
\end{mapleinput}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{b := (n,m) -\TEXTsymbol{>} evalf((1/Pi)*evalf(Int(evalf(Int( 1*r*BesselJ(n,g(n,m)*r)*sin(n*theta),r=0..sin(theta))), theta=0..Pi))/}\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(evalf(Int(evalf(Int(r*BesselJ(n, g(n, m)*r)*sin(n*theta), r = 0 .. sin(theta))), theta = 0 .. Pi))/(Pi*int(r*BesselJ(n, g(n, m)*r)^2, r = 0 .. 1))) end proc}{%
$b\, := \,( {n,m} )\mapsto  0.1795871221\,{\frac {\int _{ 0.0}^{ 3.141592654}\!\int _{ 0.0}^{\sin \left( \theta \right) }\!r{\it BesselJ} \left( n,{\it BesselJZeros} \left( n,m \right) r \right) \sin \left( n\theta \right) {dr}\,{d\theta}\\
\mbox{}\Gamma  \left(  2.0+n \right) \Gamma  \left(  1.0+n \right) }{{ 2.0}^{- 2.0- 2.0\,n}\\
\mbox{} \left( {\it BesselJZeros} \left( n,m \right)  \right) ^{n- 1.0}{\it StruveH} \left( n, 2.0\,{\it BesselJZeros} \left( n,m \right)  \right) \\
\mbox{}\Gamma  \left( n+ 1.500000000 \right) }}$}
}

\end{maplelatex}
\begin{Maple Normal}{\normalsize{Next we define U(r,theta,0). }}\end{Maple Normal}

\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(0,m)),m=1..M)/2 + add(add(BesselJ(n,g(n,m)*r)*(a(n,m)*cos(n*theta) + b(n,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(0, m), m = 1 .. M)+add(add(BesselJ(n, g(n, m)*r)*(a(n, m)*cos(n*theta)+b(n, m)*sin(n*theta)), m = 1 .. M), n = 1 .. N) end proc}{%
${\it U0}\, := \,( {r,\theta} )\mapsto  0.5156824565\,{\it BesselJ} \left( 0, 2.404825558\\
\mbox{}\,r \right) - 0.1113929242\,{\it BesselJ} \left( 0, 5.520078110\\
\mbox{}\,r \right) + 0.1008134545\,{\it BesselJ} \left( 0, 8.653727913\\
\mbox{}\,r \right) - 0.05608369495\,{\it BesselJ} \left( 0, 11.79153444\\
\mbox{}\,r \right) +{\it BesselJ} \left( 1, 3.831705970\,r \right)  \left( -{ 4.894645637\times 10^{-13}}\\
\mbox{}\,\cos \left( \theta \right) + 1.085142765\,\sin \left( \theta \right)  \right) \\
\mbox{}+{\it BesselJ} \left( 1, 7.015586670\,r \right)  \left( { 2.910783727\times 10^{-13}}\\
\mbox{}\,\cos \left( \theta \right) + 0.05692516541\,\sin \left( \theta \right)  \right) +{\it BesselJ} \left( 1, 10.17346814\,r \right)  \left( -{ 1.183349384\times 10^{-13}}\\
\mbox{}\,\cos \left( \theta \right) + 0.3560094996\,\sin \left( \theta \right)  \right) \\
\mbox{}+{\it BesselJ} \left( 1, 13.32369194\,r \right)  \left( { 7.468139225\times 10^{-14}}\\
\mbox{}\,\cos \left( \theta \right) + 0.03585756165\,\sin \left( \theta \right)  \right) +{\it BesselJ} \left( 2, 5.135622302\,r \right)  \left( - 0.7366894221\,\cos \left( 2\,\theta \right) -{ 9.743289496\times 10^{-13}}\\
\mbox{}\,\sin \left( 2\,\theta \right)  \right) \\
\mbox{}+{\it BesselJ} \left( 2, 8.417244140\,r \right)  \left(  0.1407327261\,\cos \left( 2\,\theta \right) +{ 1.710065840\times 10^{-13}}\\
\mbox{}\,\sin \left( 2\,\theta \right)  \right) +{\it BesselJ} \left( 2, 11.61984117\,r \right)  \left( - 0.1985662729\,\cos \left( 2\,\theta \right) -{ 1.882969489\times 10^{-13}}\\
\mbox{}\,\sin \left( 2\,\theta \right)  \right) \\
\mbox{}+{\it BesselJ} \left( 2, 14.79595178\,r \right)  \left(  0.08618480866\,\cos \left( 2\,\theta \right) +{ 4.580286703\times 10^{-14}}\\
\mbox{}\,\sin \left( 2\,\theta \right)  \right) +{\it BesselJ} \left( 3, 6.380161896\,r \right)  \left( { 9.111113021\times 10^{-13}}\\
\mbox{}\,\cos \left( 3\,\theta \right) - 0.3192128606\,\sin \left( 3\,\theta \right)  \right) \\
\mbox{}+{\it BesselJ} \left( 3, 9.761023130\,r \right)  \left( -{ 1.027147557\times 10^{-13}}\\
\mbox{}\,\cos \left( 3\,\theta \right) + 0.3943303382\,\sin \left( 3\,\theta \right)  \right) +{\it BesselJ} \left( 3, 13.01520072\,r \right)  \left( { 2.558707092\times 10^{-13}}\\
\mbox{}\,\cos \left( 3\,\theta \right) - 0.01837299547\,\sin \left( 3\,\theta \right)  \right) \\
\mbox{}+{\it BesselJ} \left( 3, 16.22346616\,r \right)  \left( -{ 7.209292945\times 10^{-14}}\\
\mbox{}\,\cos \left( 3\,\theta \right) + 0.2306629256\,\sin \left( 3\,\theta \right)  \right) +{\it BesselJ} \left( 4, 7.588342435\,r \right)  \left(  0.01388718603\,\cos \left( 4\,\theta \right) +{ 1.238785795\times 10^{-12}}\\
\mbox{}\,\sin \left( 4\,\theta \right)  \right) \\
\mbox{}+{\it BesselJ} \left( 4, 11.06470949\,r \right)  \left( - 0.4514662294\,\cos \left( 4\,\theta \right) +{ 3.080365453\times 10^{-15}}\\
\mbox{}\,\sin \left( 4\,\theta \right)  \right) +{\it BesselJ} \left( 4, 14.37253667\,r \right)  \left(  0.01790188414\,\cos \left( 4\,\theta \right) +{ 3.601078903\times 10^{-13}}\\
\mbox{}\,\sin \left( 4\,\theta \right)  \right) \\
\mbox{}+{\it BesselJ} \left( 4, 17.61596605\,r \right)  \left( - 0.2118676576\,\cos \left( 4\,\theta \right) -{ 6.045489514\times 10^{-14}}\\
\mbox{}\,\sin \left( 4\,\theta \right)  \right) $}
}

\end{maplelatex}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{N:=4: M:=4:}\end{Maple Normal}}{}
\end{mapleinput}
\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,color=white); }\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{center}
\mapleplot{HeatDIsk4plot3d2.eps}


\end{center}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{N:=5: M:=5:}\end{Maple Normal}}{}
\end{mapleinput}
\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,color=white); }\QTR{Maple Maple Input}{
}\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{center}
\mapleplot{HeatDIsk4plot3d3.eps}


\end{center}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{N:=6: M:=6:}\QTR{Maple Maple Input}{
plot3d(U0(r,theta),r=0..1,theta=0..2*Pi,coords=zcylindrical,color=white); }\QTR{Maple Maple Input}{
}\QTR{Maple Maple Input}{
 }\QTR{Maple Maple Input}{
}\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{center}
\mapleplot{HeatDIsk4plot3d4.eps}


\end{center}

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

\end{maplettyout}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{N:=8: M:=6:}\QTR{Maple Maple Input}{
plot3d(U0(r,theta),r=0..1,theta=0..2*Pi,coords=zcylindrical,color=white); }\QTR{Maple Maple Input}{
}\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{center}
\mapleplot{HeatDIsk4plot3d5.eps}


\end{center}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{
N:=9: M:=9:}\QTR{Maple Maple Input}{
plot3d(U0(r,theta),r=0..1,theta=0..2*Pi,coords=zcylindrical,color=white); }\QTR{Maple Maple Input}{
}\QTR{Maple Maple Input}{
 }\QTR{Maple Maple Input}{
}\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{center}
\mapleplot{HeatDIsk4plot3d6.eps}


\end{center}
\begin{Maple Normal}{\normalsize{Next we define the solution u(r,theta,t) and plot it for several values of t. }}\end{Maple Normal}

\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{N:=4: M:=4:}\end{Maple Normal}}{}
\end{mapleinput}
\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}{Ut := (r,theta,t) -\TEXTsymbol{>} add(BesselJ(0,g(0,m)*r)*exp(-g(0,m)\symbol{94}2*t)*(a(0,m)),m=1..M)/2 + add(add(BesselJ(n,g(n,m)*r)*exp(-g(n,m)\symbol{94}2*t)*(a(n,m)*cos(n*theta) + b(n,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}{Ut := proc (r, theta, t) options operator, arrow; 1/2*add(BesselJ(0, g(0, m)*r)*exp(-g(0, m)^2*t)*a(0, m), m = 1 .. M)+add(add(BesselJ(n, g(n, m)*r)*exp(-g(n, m)^2*t)*(a(n, m)*cos(n*theta)+b(n, m)*sin(n*theta)), m = 1 .. M), n = 1 .. N) end proc}{%
${\it Ut}\, := \,( {r,\theta,t} )\mapsto  0.5156824565\,{\it BesselJ} \left( 0, 2.404825558\,r \right) {e^{- 5.783185964\,t}}\\
\mbox{}- 0.1113929242\,{\it BesselJ} \left( 0, 5.520078110\,r \right) {e^{- 30.47126234\,t}}\\
\mbox{}+ 0.1008134545\,{\it BesselJ} \left( 0, 8.653727913\,r \right) {e^{- 74.88700679\,t}}\\
\mbox{}- 0.05608369495\,{\it BesselJ} \left( 0, 11.79153444\,r \right) {e^{- 139.0402844\,t}}\\
\mbox{}+{\it BesselJ} \left( 1, 3.831705970\,r \right) {e^{- 14.68197064\,t}} \left( -{ 4.894645637\times 10^{-13}}\\
\mbox{}\,\cos \left( \theta \right) + 1.085142765\,\sin \left( \theta \right)  \right) \\
\mbox{}+{\it BesselJ} \left( 1, 7.015586670\,r \right) {e^{- 49.21845632\,t}} \left( { 2.910783727\times 10^{-13}}\\
\mbox{}\,\cos \left( \theta \right) + 0.05692516541\,\sin \left( \theta \right)  \right) +{\it BesselJ} \left( 1, 10.17346814\,r \right) {e^{- 103.4994540\,t}} \left( -{ 1.183349384\times 10^{-13}}\\
\mbox{}\,\cos \left( \theta \right) + 0.3560094996\,\sin \left( \theta \right)  \right) \\
\mbox{}+{\it BesselJ} \left( 1, 13.32369194\,r \right) {e^{- 177.5207669\,t}} \left( { 7.468139225\times 10^{-14}}\\
\mbox{}\,\cos \left( \theta \right) + 0.03585756165\,\sin \left( \theta \right)  \right) +{\it BesselJ} \left( 2, 5.135622302\,r \right) {e^{- 26.37461643\,t}} \left( - 0.7366894221\,\cos \left( 2\,\theta \right) -{ 9.743289496\times 10^{-13}}\\
\mbox{}\,\sin \left( 2\,\theta \right)  \right) \\
\mbox{}+{\it BesselJ} \left( 2, 8.417244140\,r \right) {e^{- 70.84999891\,t}} \left(  0.1407327261\,\cos \left( 2\,\theta \right) +{ 1.710065840\times 10^{-13}}\\
\mbox{}\,\sin \left( 2\,\theta \right)  \right) +{\it BesselJ} \left( 2, 11.61984117\,r \right) {e^{- 135.0207088\,t}} \left( - 0.1985662729\,\cos \left( 2\,\theta \right) -{ 1.882969489\times 10^{-13}}\\
\mbox{}\,\sin \left( 2\,\theta \right)  \right) \\
\mbox{}+{\it BesselJ} \left( 2, 14.79595178\,r \right) {e^{- 218.9201891\,t}} \left(  0.08618480866\,\cos \left( 2\,\theta \right) +{ 4.580286703\times 10^{-14}}\\
\mbox{}\,\sin \left( 2\,\theta \right)  \right) +{\it BesselJ} \left( 3, 6.380161896\,r \right) {e^{- 40.70646582\,t}} \left( { 9.111113021\times 10^{-13}}\\
\mbox{}\,\cos \left( 3\,\theta \right) - 0.3192128606\,\sin \left( 3\,\theta \right)  \right) \\
\mbox{}+{\it BesselJ} \left( 3, 9.761023130\,r \right) {e^{- 95.27757254\,t}} \left( -{ 1.027147557\times 10^{-13}}\\
\mbox{}\,\cos \left( 3\,\theta \right) + 0.3943303382\,\sin \left( 3\,\theta \right)  \right) +{\it BesselJ} \left( 3, 13.01520072\,r \right) {e^{- 169.3954498\,t}} \left( { 2.558707092\times 10^{-13}}\\
\mbox{}\,\cos \left( 3\,\theta \right) - 0.01837299547\,\sin \left( 3\,\theta \right)  \right) \\
\mbox{}+{\it BesselJ} \left( 3, 16.22346616\,r \right) {e^{- 263.2008542\,t}} \left( -{ 7.209292945\times 10^{-14}}\\
\mbox{}\,\cos \left( 3\,\theta \right) + 0.2306629256\,\sin \left( 3\,\theta \right)  \right) +{\it BesselJ} \left( 4, 7.588342435\,r \right) {e^{- 57.58294091\,t}} \left(  0.01388718603\,\cos \left( 4\,\theta \right) +{ 1.238785795\times 10^{-12}}\\
\mbox{}\,\sin \left( 4\,\theta \right)  \right) \\
\mbox{}+{\it BesselJ} \left( 4, 11.06470949\,r \right) {e^{- 122.4277961\,t}} \left( - 0.4514662294\,\cos \left( 4\,\theta \right) +{ 3.080365453\times 10^{-15}}\\
\mbox{}\,\sin \left( 4\,\theta \right)  \right) +{\it BesselJ} \left( 4, 14.37253667\,r \right) {e^{- 206.5698103\,t}} \left(  0.01790188414\,\cos \left( 4\,\theta \right) +{ 3.601078903\times 10^{-13}}\\
\mbox{}\,\sin \left( 4\,\theta \right)  \right) \\
\mbox{}+{\it BesselJ} \left( 4, 17.61596605\,r \right) {e^{- 310.3222599\,t}} \left( - 0.2118676576\,\cos \left( 4\,\theta \right) -{ 6.045489514\times 10^{-14}}\\
\mbox{}\,\sin \left( 4\,\theta \right)  \right) $}
}

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

\mapleresult
\begin{center}
\mapleplot{HeatDIsk4plot3d7.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(Ut(r,theta,t),r=0..1,theta=0..2*Pi,coords=zcylindrical,color=white);}\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{center}
\mapleplot{HeatDIsk4plot3d8.eps}


\end{center}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{t:=0.03;}\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\QTR{Maple 2D Output}{\mapleinline{inert}{2d}{t := 0.3e-1}{%
\[
t\, := \, 0.03
\]
}
}

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

\mapleresult
\begin{center}
\mapleplot{HeatDIsk4plot3d9.eps}


\end{center}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{t:=0.06;}\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\QTR{Maple 2D Output}{\mapleinline{inert}{2d}{t := 0.6e-1}{%
\[
t\, := \, 0.06
\]
}
}

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

\mapleresult
\begin{center}
\mapleplot{HeatDIsk4plot3d10.eps}


\end{center}
\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