{VERSION 5 0 "IBM INTEL NT" "5.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 
2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 
{CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 
0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 
1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }
{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 
2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 1 
{CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 
0 1 0 1 0 2 2 0 1 }}
{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 69 "We  calculate the biurcation diagram of the equilibr
ium solutions of " }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "diff(u(x,t),t) =
 diff(u(x,t),`$`(x,2))+lambda*f(u(x,t));" "6#/-%%diffG6$-%\"uG6$%\"xG%
\"tGF+,&-%%diffG6$-%\"uG6$%\"xG%\"tG-%\"$G6$%\"xG\"\"#\"\"\"*&%'lambda
GF:-%\"fG6#-F(6$F*F+F:F:" }}{PARA 0 "" 0 "" {TEXT -1 95 "with Dirichle
t boundary problem u(0,t)=u(1,t)=0. The equilibrium solutions satisfy \+
the equation" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "diff(u(x),`$`(x,2))+l
ambda*f(u) = 0;" "6#/,&-%%diffG6$-%\"uG6#%\"xG-%\"$G6$F+\"\"#\"\"\"*&%
'lambdaGF0-%\"fG6#F)F0F0\"\"!" }{TEXT -1 14 ", u(0)=u(1)=0." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "f(x) is
 the nonlinearity inn the equation, and F(x) is its antiderivative" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "b:=1; f:=x*(b-x); F1:=int(f,x); F:=
unapply(F1,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Solve the zeros
 of f(x) and F(x) to find the domain of the the time-mapping," }
{MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "If x belongs to the d
omain, then f(x)>0 and F(x)>0" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "fs
olve(f,x);fsolve(F1,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "The do
main of the time-mapping is (Initial, End)" }{MPLTEXT 1 0 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Initial:=0; End:=1; " }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 29 "The number of plotting points" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 15 "numplots:=100; " }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 33 "Define the array of data points (" }{XPPEDIT 18 0 "lambda
;" "6#%'lambdaG" }{TEXT -1 3 ",u)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
51 "lambda:=array(1..numplots); u:=array(1..numplots); " }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 26 "The bifurcation point is (" }{XPPEDIT 18 
0 "lambda" "6#%'lambdaG" }{TEXT -1 5 ",u)=(" }{XPPEDIT 18 0 "Pi^2;" "6
#*$)%#PiG\"\"#\"\"\"" }{TEXT -1 9 "/f'(0),0)" }{MPLTEXT 1 0 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "g1:=diff(f,x); g:=unapply(g1,x); " 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Initilize the sequence with the
 bifurcation point" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 38 "lambda[1]:=evalf(Pi^2/g(0)); u[1]:=0; " }{TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT -1 45 "Calculate the time-
mapping at plotting points" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "for n
 from 2 to numplots-1 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "   s:=I
nitial+(End-Initial)*(n-1)/numplots: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 45 "   evalf(Int(1/(sqrt(F(s)-F(x))), x=0..s));  " }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 32 "   lambda[n]:=2*(%^2); u[n]:=s; " }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 7 "end do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 71 "plot([seq([lambda[n],u[n]],n=1..numplots-1)],0..100,tickmarks=[1
0,11]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "for n from 1 to \+
numplots-1 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "  print(lambda[n],
evalf(u[n]));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end do:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "10 0 1" 45 }{VIEWOPTS 
1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }