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{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 57 "b:=0.1; f:=x*(x-1)*(b-x); F1:=int(f
,x); F:=unapply(F1,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Solve t
he 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 th
e domain, then f(x)>0 and F(x)>0" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 
"fsolve(f,x);fsolve(F1,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "The
 domain of the time-mapping is (Initial, End)" }{MPLTEXT 1 0 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Initial:=0.16; End:=0.97; " }}}
{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 0 "" }{TEXT -1 45 "Calculate the ti
me-mapping at plotting points" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "fo
r n from 1 to numplots-1 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "   s
:=Initial+(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..200
,tickmarks=[10,11]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "for
 n from 1 to numplots-1 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "  pri
nt(lambda[n],evalf(u[n]));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end do
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "12" 0 }
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