%% Created by Maple 9.5, Linux %% Source Worksheet: /var/www/html/mikesullivan/Courses/305/LectureNotes/HeatDisk2.mw %% Generated: Sun Apr 23 16:53:36 CDT 2017 \documentclass{article} \usepackage{maplestd2e} \def\emptyline{\vspace{12pt}} \DefineParaStyle{Maple Heading 4} \DefineParaStyle{Maple Heading 2} \DefineParaStyle{Maple Text Output} \DefineParaStyle{Maple Bullet Item} \DefineParaStyle{Maple Warning} \DefineParaStyle{Maple Error} \DefineParaStyle{Maple Dash Item} \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{\textbf{Heat Equation Example: f(r,theta) = r(1-r\symbol{94}2)sin(3theta)\symbol{94}2.}}}\end{Maple Normal}\end{center} \begin{Maple Normal}{\normalsize{Switching over to cylindrical coords, but had to make my own with z=f(r,theta).}}\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{Setting up my own color scheme b/c maple will change it to whatever it wants. }}\end{Maple Normal} \begin{mapleinput} \mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{B:= z -\TEXTsymbol{>} piecewise(z<0.6, 1.0, -z/0.4 + 2.5 );}\end{Maple Normal}}{} \end{mapleinput} \mapleresult \begin{maplelatex} \QTR{Maple 2D Output}{\mapleinline{inert}{2d}{B := proc (z) options operator, arrow; piecewise(z < .6, 1.0, -z/.4+2.5) end proc}{% \[ B\, := \,z\mapsto \cases{ 1.0&$z< 0.6$\cr - 2.500000000\,z+ 2.5&otherwise\cr} \] } } \end{maplelatex} \begin{mapleinput} \mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{R:= z -\TEXTsymbol{>} piecewise(z<0.4, z/0.4, 1.0);}\end{Maple Normal}}{} \end{mapleinput} \mapleresult \begin{maplelatex} \QTR{Maple 2D Output}{\mapleinline{inert}{2d}{R := proc (z) options operator, arrow; piecewise(z < .4, z/.4, 1.0) end proc}{% \[ R\, := \,z\mapsto \cases{ 2.500000000\,z&$z< 0.4$\cr 1.0&otherwise\cr} \] } } \end{maplelatex} \begin{mapleinput} \mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{plot([R(z),B(z)],z=0..1,color=[red,blue]);}\end{Maple Normal}}{} \end{mapleinput} \mapleresult \begin{center} \mapleplot{HeatDisk2plot1.eps} \end{center} \begin{Maple Normal}{\normalsize{Plotting z=f(r,theta), polar coords, my colors. }}\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\symbol{94}2)*sin(3*theta)\symbol{94}2;}\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^2)*sin(3*theta)^2 end proc}{% \[ f\, := \,( {r,\theta} )\mapsto r \left( 1-{r}^{2} \right) \left( \sin \left( 3\,\theta \right) \right) ^{2} \] } } \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,view=0..0.5,coords=zcylindrical,numpoints=10000,}\QTR{Maple Maple Input}{ color=[R(f(r,theta)/0.4) ,0, B(f(r,theta)/0.4)]);}\end{Maple Normal}}{} \end{mapleinput} \mapleresult \begin{center} \mapleplot{HeatDisk2plot3d2.eps} \end{center} \begin{Maple Normal}{\normalsize{Working out the Fourier-Bessel series coeffs. }}\end{Maple Normal} \begin{mapleinput} \mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{fr := (r) -\TEXTsymbol{>} r*(1-r\symbol{94}2);}\end{Maple Normal}}{} \end{mapleinput} \mapleresult \begin{maplelatex} \QTR{Maple 2D Output}{\mapleinline{inert}{2d}{fr := proc (r) options operator, arrow; r*(1-r^2) end proc}{% \[ {\it fr}\, := \,r\mapsto r \left( 1-{r}^{2} \right) \] } } \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));}\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}{b0 := m -\TEXTsymbol{>} (int(r*fr(r)*BesselJ(0,g(0,m)*r) ,r=0..1))/(int(r*BesselJ(0,g(0,m)*r)\symbol{94}2 ,r=0..1));}\end{Maple Normal}}{} \end{mapleinput} \mapleresult \begin{maplelatex} \QTR{Maple 2D Output}{\mapleinline{inert}{2d}{b0 := proc (m) options operator, arrow; int(r*fr(r)*BesselJ(0, g(0, m)*r), r = 0 .. 1)/int(r*BesselJ(0, g(0, m)*r)^2, r = 0 .. 1) end proc}{% ${\it b0}\, := \,m\mapsto 2.0\, \left( 4.0\,{\frac { 0.2500000000\, \left( {\it BesselJZeros} \left( 0.0,m \right) \right) ^{2}\\ \mbox{}{\it BesselJ} \left( 1.0,{\it BesselJZeros} \left( 0.0,m \right) \right) + 0.3926990817\,{\it BesselJZeros} \left( 0.0,m \right) \left( {\it BesselJ} \left( 0.0,{\it BesselJZeros} \left( 0.0,m \right) \right) {\it StruveH} \left( 1.0,{\it BesselJZeros} \left( 0.0,m \right) \right) \\ \mbox{}- 1.0\,{\it BesselJ} \left( 1.0,{\it BesselJZeros} \left( 0.0,m \right) \right) \\ \mbox{}{\it StruveH} \left( 0.0,{\it BesselJZeros} \left( 0.0,m \right) \right) \right) \\ \mbox{}}{ \left( {\it BesselJZeros} \left( 0.0\\ \mbox{},m \right) \right) ^{3}}}- 16.0\,{\frac { 0.1875000000\, \left( {\it BesselJZeros} \left( 0.0,m \right) \right) ^{3}\\ \mbox{}{\it BesselJ} \left( 0.0,{\it BesselJZeros} \left( 0.0,m \right) \right) + 0.01250000000\, \left( {\it BesselJZeros} \left( 0.0,m \right) \right) ^{2}\\ \mbox{} \left( - 45.0+ 5.0\, \left( {\it BesselJZeros} \left( 0.0,m \right) \right) ^{2} \right) {\it BesselJ} \left( 1.0,{\it BesselJZeros} \left( 0.0,m \right) \right) \\ \mbox{}- 0.8835729338\,{\it BesselJZeros} \left( 0.0,m \right) \left( {\it BesselJ} \left( 0.0,{\it BesselJZeros} \left( 0.0,m \right) \right) {\it StruveH} \left( 1.0,{\it BesselJZeros} \left( 0.0,m \right) \right) \\ \mbox{}- 1.0\,{\it BesselJ} \left( 1.0,{\it BesselJZeros} \left( 0.0,m \right) \right) \\ \mbox{}{\it StruveH} \left( 0.0,{\it BesselJZeros} \left( 0.0,m \right) \right) \right) \\ \mbox{}}{ \left( {\it BesselJZeros} \left( 0.0,m \right) \right) ^{5}}} \right) \left( {\it BesselJ} \left( 1.0\\ \mbox{},{\it BesselJZeros} \left( 0.0,m \right) \right) \right) ^{-2}$} } \end{maplelatex} \begin{mapleinput} \mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{b6 := m -\TEXTsymbol{>} (int(r*fr(r)*BesselJ(6,g(6,m)*r) ,r=0..1))/(int(r*BesselJ(6,g(6,m)*r)\symbol{94}2 ,r=0..1));}\end{Maple Normal}}{} \end{mapleinput} \mapleresult \begin{maplelatex} \QTR{Maple 2D Output}{\mapleinline{inert}{2d}{b6 := proc (m) options operator, arrow; int(r*fr(r)*BesselJ(6, g(6, m)*r), r = 0 .. 1)/int(r*BesselJ(6, g(6, m)*r)^2, r = 0 .. 1) end proc}{% ${\it b6}\, := \,m\mapsto 3.544907701\, \left( 4.0\, \left( 0.0006172839506\,{\frac { \left( 6480.0\, \left( {\it BesselJZeros} \left( 6.0,m \right) \right) ^{2}+ 259200.0\\ \mbox{} \right) \\ \mbox{}{\it BesselJ} \left( 0.0,{\it BesselJZeros} \left( 6.0,m \right) \right) }{{\it BesselJZeros} \left( 6.0,m \right) }}\\ \mbox{}+ 0.001234567901\,{\frac { \left( - 202.5000000\, \left( {\it BesselJZeros} \left( 6.0,m \right) \right) ^{4}- 259200.0\\ \mbox{}+ 25920.0\, \left( {\it BesselJZeros} \left( 6.0,m \right) \right) ^{2} \right) {\it BesselJ} \left( 1.0,{\it BesselJZeros} \left( 6.0,m \right) \right) \\ \mbox{}}{ \left( {\it BesselJZeros} \left( 6.0,m \right) \right) ^{2}}}- 13.74446786\,{\it BesselJZeros} \left( 6.0,m \right) \left( {\it BesselJ} \left( 0.0,{\it BesselJZeros} \left( 6.0,m \right) \right) {\it StruveH} \left( 1.0,{\it BesselJZeros} \left( 6.0,m \right) \right) \\ \mbox{}- 1.0\,{\it BesselJ} \left( 1.0,{\it BesselJZeros} \left( 6.0,m \right) \right) \\ \mbox{}{\it StruveH} \left( 0.0,{\it BesselJZeros} \left( 6.0,m \right) \right) \right) \\ \mbox{} \right) \left( {\it BesselJZeros} \left( 6.0,m \right) \right) ^{-3}- 16.0\,{\frac { 0.0001262626263\,{\it BesselJZeros} \left( 6.0,m \right) \\ \mbox{} \left( - 10395.0\, \left( {\it BesselJZeros} \left( 6.0,m \right) \right) ^{2}+ 950400.0\\ \mbox{} \right) {\it BesselJ} \left( 0.0,{\it BesselJZeros} \left( 6.0,m \right) \right) + 0.0002525252525\, \left( 51232.50000\, \left( {\it BesselJZeros} \left( 6.0,m \right) \right) ^{2}- 950400.0\\ \mbox{}- 247.5000000\, \left( {\it BesselJZeros} \left( 6.0,m \right) \right) ^{4} \right) {\it BesselJ} \left( 1.0,{\it BesselJZeros} \left( 6.0,m \right) \right) \\ \mbox{}- 92.77515805\,{\it BesselJZeros} \left( 6.0,m \right) \left( {\it BesselJ} \left( 0.0,{\it BesselJZeros} \left( 6.0,m \right) \right) {\it StruveH} \left( 1.0,{\it BesselJZeros} \left( 6.0,m \right) \right) \\ \mbox{}- 1.0\,{\it BesselJ} \left( 1.0,{\it BesselJZeros} \left( 6.0,m \right) \right) \\ \mbox{}{\it StruveH} \left( 0.0,{\it BesselJZeros} \left( 6.0,m \right) \right) \right) \\ \mbox{}}{ \left( {\it BesselJZeros} \left( 6.0,m \right) \right) ^{5}}}\\ \mbox{} \right) \left( {\it BesselJZeros} \left( 6.0,m \right) \right) ^{2} \left( 0.0000000001192482338\,{\frac { \left( 14863564800.0\, \left( {\it BesselJZeros} \left( 6.0,m \right) \right) ^{8}- 45660871070000.0\, \left( {\it BesselJZeros} \left( 6.0,m \right) \right) ^{4}\\ \mbox{}+ 1369826132000000.0\, \left( {\it BesselJZeros} \left( 6.0\\ \mbox{},m \right) \right) ^{2}- 10958609060000000.0 \right) \\ \mbox{} \left( {\it BesselJ} \left( 0.0,{\it BesselJZeros} \left( 6.0,m \right) \right) \right) ^{2}}{ \left( {\it BesselJZeros} \left( 6.0,m \right) \right) ^{6}}}\\ \mbox{}+ 0.0000000001192482338\,{\frac { \left( - 10958609060000000.0\, \left( {\it BesselJZeros} \left( 6.0\\ \mbox{},m \right) \right) ^{2}+ 639252194900000.0\, \left( {\it BesselJZeros} \left( 6.0,m \right) \\ \mbox{} \right) ^{4}- 8561413325000.0\, \left( {\it BesselJZeros} \left( 6.0,m \right) \right) ^{6}\\ \mbox{}+ 43834436220000000.0 \right) \\ \mbox{}{\it BesselJ} \left( 0.0,{\it BesselJZeros} \left( 6.0,m \right) \right) {\it BesselJ} \left( 1.0,{\it BesselJZeros} \left( 6.0,m \right) \right) \\ \mbox{}}{ \left( {\it BesselJZeros} \left( 6.0,m \right) \right) ^{7}}}+ 0.0000000001192482338\,{\frac { \left( 16437913580000000.0\, \left( {\it BesselJZeros} \left( 6.0\\ \mbox{},m \right) \right) ^{2}+ 14863564800.0\, \left( {\it BesselJZeros} \left( 6.0,m \right) \right) ^{10}- 1780773972000000.0\, \left( {\it BesselJZeros} \left( 6.0\\ \mbox{},m \right) \right) ^{4}\\ \mbox{}+ 45660871070000.0\, \left( {\it BesselJZeros} \left( 6.0,m \right) \right) ^{6}\\ \mbox{}- 535088332800.0\, \left( {\it BesselJZeros} \left( 6.0,m \right) \right) ^{8}- 43834436220000000.0\\ \mbox{} \right) \left( {\it BesselJ} \left( 1.0,{\it BesselJZeros} \left( 6.0,m \right) \right) \right) ^{2}\\ \mbox{}}{ \left( {\it BesselJZeros} \left( 6.0,m \right) \right) ^{8}}} \right) ^{-1}$} } \end{maplelatex} \begin{Maple Normal}{\normalsize{Setting up the Fourier-Bessel series for t=0. }}\end{Maple Normal} \begin{mapleinput} \mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{M:=10: # Experiment with different values for M. }\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{>} 1/2*sum( b0(m)*BesselJ(0,g(0,m)*r) ,m=1..M) - 1/2*cos(6*theta)*sum( b6(m)*BesselJ(6,g(6,m)*r), m=1..M);}\end{Maple Normal}}{} \end{mapleinput} \mapleresult \begin{maplelatex} \QTR{Maple 2D Output}{\mapleinline{inert}{2d}{U0 := proc (r, theta) options operator, arrow; 1/2*(sum(b0(m)*BesselJ(0, g(0, m)*r), m = 1 .. M))-1/2*cos(6*theta)*(sum(b6(m)*BesselJ(6, g(6, m)*r), m = 1 .. M)) end proc}{% ${\it U0}\, := \,( {r,\theta} )\mapsto 0.2438167972\,{\it BesselJ} \left( 0.0\\ \mbox{}, 2.404825558\,r \right) - 0.2017381014\,{\it BesselJ} \left( 0.0, 5.520078110\\ \mbox{}\,r \right) \\ \mbox{}+ 0.02141230345\,{\it BesselJ} \left( 0.0, 8.653727913\\ \mbox{}\,r \right) - 0.03285803925\,{\it BesselJ} \left( 0.0, 11.79153444\\ \mbox{}\,r \right) + 0.004257616168\,{\it BesselJ} \left( 0.0, 14.93091771\\ \mbox{}\,r \right) \\ \mbox{}- 0.01213982986\,{\it BesselJ} \left( 0.0, 18.07106397\\ \mbox{}\,r \right) + 0.001267098004\,{\it BesselJ} \left( 0.0, 21.21163663\\ \mbox{}\,r \right) - 0.006108240625\,{\it BesselJ} \left( 0.0, 24.35247153\\ \mbox{}\,r \right) \\ \mbox{}+ 0.0004237165844\,{\it BesselJ} \left( 0.0, 27.49347913\\ \mbox{}\,r \right) - 0.003617684813\,{\it BesselJ} \left( 0.0, 30.63460647\\ \mbox{}\,r \right) -1/2\,\cos \left( 6\,\theta \right) \left( 1.096101644\,{\it BesselJ} \left( 6.0, 9.936109524\\ \mbox{}\,r \right) + 0.5287527864\,{\it BesselJ} \left( 6.0, 13.58929017\\ \mbox{}\,r \right) + 0.3875710380\,{\it BesselJ} \left( 6.0, 17.00381967\\ \mbox{}\,r \right) + 0.2484018464\,{\it BesselJ} \left( 6.0, 20.32078921\\ \mbox{}\,r \right) \\ \mbox{}+ 0.2024095590\,{\it BesselJ} \left( 6.0, 23.58608444\\ \mbox{}\,r \right) + 0.1451958925\,{\it BesselJ} \left( 6.0, 26.82015198\\ \mbox{}\,r \right) + 0.1249173155\,{\it BesselJ} \left( 6.0, 30.03372239\\ \mbox{}\,r \right) \\ \mbox{}+ 0.09545720492\,{\it BesselJ} \left( 6.0, 33.23304176\\ \mbox{}\,r \right) + 0.08490443426\,{\it BesselJ} \left( 6.0, 36.42201967\\ \mbox{}\,r \right) + 0.06760848949\,{\it BesselJ} \left( 6.0, 39.60323942\\ \mbox{}\,r \right) \\ \mbox{} \right) $} } \end{maplelatex} \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,view=0..0.5,}\QTR{Maple Maple Input}{ color=[R(U0(r,theta)/0.4), 0, B (U0(r,theta)/0.4)],numpoints=10000);}\end{Maple Normal}}{} \end{mapleinput} \mapleresult \begin{center} \mapleplot{HeatDisk2plot3d3.eps} \end{center} \begin{Maple Normal}{\normalsize{Setting up Fourier-Bessel series for any time t. }}\end{Maple Normal} \begin{mapleinput} \mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{M:=10:}\end{Maple Normal}}{} \end{mapleinput} \begin{mapleinput} \mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{U := (r,theta,t) -\TEXTsymbol{>} 1/2*sum( b0(m)*BesselJ(0,g(0,m)*r)*exp(-g(0,m)\symbol{94}2*t) ,m=1..M) - 1/2*cos(6*theta)*sum( b6(m)*BesselJ(6,g(6,m)*r)*exp(-g(6,m)\symbol{94}2*t), m=1..M);}\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*(sum(b0(m)*BesselJ(0, g(0, m)*r)*exp(-g(0, m)^2*t), m = 1 .. M))-1/2*cos(6*theta)*(sum(b6(m)*BesselJ(6, g(6, m)*r)*exp(-g(6, m)^2*t), m = 1 .. M)) end proc}{% $U\, := \,( {r,\theta,t} )\mapsto 0.2438167972\,{\it BesselJ} \left( 0.0, 2.404825558\,r \right) \\ \mbox{}{e^{- 5.783185964\,t}}- 0.2017381014\,{\it BesselJ} \left( 0.0, 5.520078110\,r \right) {e^{- 30.47126234\,t}}\\ \mbox{}+ 0.02141230345\,{\it BesselJ} \left( 0.0, 8.653727913\,r \right) \\ \mbox{}{e^{- 74.88700679\,t}}\\ \mbox{}- 0.03285803925\,{\it BesselJ} \left( 0.0, 11.79153444\,r \right) \\ \mbox{}{e^{- 139.0402844\,t}}+ 0.004257616168\,{\it BesselJ} \left( 0.0, 14.93091771\,r \right) \\ \mbox{}{e^{- 222.9323037\,t}}\\ \mbox{}- 0.01213982986\,{\it BesselJ} \left( 0.0, 18.07106397\,r \right) \\ \mbox{}{e^{- 326.5633530\,t}}+ 0.001267098004\,{\it BesselJ} \left( 0.0, 21.21163663\,r \right) \\ \mbox{}{e^{- 449.9335285\,t}}\\ \mbox{}- 0.006108240625\,{\it BesselJ} \left( 0.0, 24.35247153\,r \right) \\ \mbox{}{e^{- 593.0428696\,t}}+ 0.0004237165844\,{\it BesselJ} \left( 0.0, 27.49347913\,r \right) \\ \mbox{}{e^{- 755.8913947\,t}}\\ \mbox{}- 0.003617684813\,{\it BesselJ} \left( 0.0, 30.63460647\,r \right) \\ \mbox{}{e^{- 938.4791136\,t}}-1/2\,\cos \left( 6\,\theta \right) \left( 1.096101644\,{\it BesselJ} \left( 6.0, 9.936109524\,r \right) {e^{- 98.72627247\,t}}\\ \mbox{}+ 0.5287527864\,{\it BesselJ} \left( 6.0, 13.58929017\,r \right) {e^{- 184.6688073\,t}}\\ \mbox{}+ 0.3875710380\,{\it BesselJ} \left( 6.0, 17.00381967\,r \right) {e^{- 289.1298834\,t}}\\ \mbox{}+ 0.2484018464\,{\it BesselJ} \left( 6.0, 20.32078921\,r \right) {e^{- 412.9344741\,t}}\\ \mbox{}+ 0.2024095590\,{\it BesselJ} \left( 6.0, 23.58608444\,r \right) {e^{- 556.3033792\,t}}\\ \mbox{}+ 0.1451958925\,{\it BesselJ} \left( 6.0, 26.82015198\,r \right) {e^{- 719.3205522\,t}}\\ \mbox{}+ 0.1249173155\,{\it BesselJ} \left( 6.0, 30.03372239\,r \right) {e^{- 902.0244806\,t}}\\ \mbox{}+ 0.09545720492\,{\it BesselJ} \left( 6.0, 33.23304176\,r \right) \\ \mbox{}{e^{- 1104.435065\,t}}+ 0.08490443426\,{\it BesselJ} \left( 6.0, 36.42201967\,r \right) \\ \mbox{}{e^{- 1326.563517\,t}}\\ \mbox{}+ 0.06760848949\,{\it BesselJ} \left( 6.0, 39.60323942\,r \right) \\ \mbox{}{e^{- 1568.416573\,t}} \right) $} } \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(U(r,theta,0.001),r=0..1,theta=0..2*Pi,coords=zcylindrical,view=0..0.5,}\QTR{Maple Maple Input}{ color=[R(U(r,theta,0.001)/0.4) , 0,B( U(r,theta,0.001)/0.4) ],numpoints=10000,title="t=0.001");}\end{Maple Normal}}{} \end{mapleinput} \mapleresult \begin{center} \mapleplot{HeatDisk2plot3d4.eps} \end{center} \begin{mapleinput} \mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{plot3d(U(r,theta,0.003),r=0..1,theta=0..2*Pi,coords=zcylindrical,view=0..0.5,}\QTR{Maple Maple Input}{ color=[R(U(r,theta,0.003)/0.4) , 0,B( U(r,theta,0.003)/0.4) ],numpoints=10000,title="t=0.003");}\end{Maple Normal}}{} \end{mapleinput} \mapleresult \begin{center} \mapleplot{HeatDisk2plot3d5.eps} \end{center} \begin{mapleinput} \mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{plot3d(U(r,theta,0.005),r=0..1,theta=0..2*Pi,coords=zcylindrical,view=0..0.5,}\QTR{Maple Maple Input}{ color=[R(U(r,theta,0.005)/0.4) , 0,B( U(r,theta,0.005)/0.4) ],numpoints=10000,title="t=0.005");}\end{Maple Normal}}{} \end{mapleinput} \mapleresult \begin{center} \mapleplot{HeatDisk2plot3d6.eps} \end{center} \begin{mapleinput} \mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{plot3d(U(r,theta,0.01),r=0..1,theta=0..2*Pi,coords=zcylindrical,view=0..0.5,}\QTR{Maple Maple Input}{ color=[R(U(r,theta,0.01)/0.4) , 0,B( U(r,theta,0.01)/0.4) ],numpoints=10000,title="t=0.01");}\end{Maple Normal}}{} \end{mapleinput} \mapleresult \begin{center} \mapleplot{HeatDisk2plot3d7.eps} \end{center} \begin{mapleinput} \mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{plot3d(U(r,theta,0.02),r=0..1,theta=0..2*Pi,coords=zcylindrical,view=0..0.5,}\QTR{Maple Maple Input}{ color=[R(U(r,theta,0.02)/0.4) , 0,B( U(r,theta,0.02)/0.4) ],numpoints=10000,title="t=0.02");}\end{Maple Normal}}{} \end{mapleinput} \mapleresult \begin{center} \mapleplot{HeatDisk2plot3d8.eps} \end{center} \begin{mapleinput} \mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{plot3d(U(r,theta,0.05),r=0..1,theta=0..2*Pi,coords=zcylindrical,view=0..0.5,color=[R(U(r,theta,0.05)/0.4) , 0,B( U(r,theta,0.05)/0.4) ],numpoints=10000,title="t=0.05");}\end{Maple Normal}}{} \end{mapleinput} \mapleresult \begin{center} \mapleplot{HeatDisk2plot3d9.eps} \end{center} \begin{mapleinput} \mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{plot3d(U(r,theta,0.1),r=0..1,theta=0..2*Pi,coords=zcylindrical,view=0..0.5,color=[R(U(r,theta,0.1)/0.4) , 0,B( U(r,theta,0.1)/0.4) ],numpoints=10000,title="t=0.1");}\end{Maple Normal}}{} \end{mapleinput} \mapleresult \begin{center} \mapleplot{HeatDisk2plot3d10.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} \mapleresult \begin{maplettyout} \begin{Maple Warning}{\normalsize{Warning, the name changecoords has been redefined}}\end{Maple Warning} \end{maplettyout} \begin{mapleinput} \mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{animate(plot3d,[U(r,theta,t),r=0..1,theta=0..2*Pi,coords=zcylindrical,view=0..0.5,color=[R(U(r,theta,t)/0.4) , 0,B( U(r,theta,t)/0.4) ],numpoints=1000 ],t=0..0.1); }\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