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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 1 "\n" }{TEXT 256 16 "SOLUTI
ON TO HW#5" }{TEXT -1 80 "\n\nCode to compute the unstability window f
or a system of\nReaction-Diffusion PDEs" }{MPLTEXT 1 0 0 "" }}{PARA 0
"> " 0 "" {MPLTEXT 1 0 8 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0
13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protect
ed names norm and trace have been redefined and unprotected\n" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Define RHS of
the reaction" }
{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "R := vector( [u
^2/v-b*u, u^2-v] );\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG-%'vect
orG6#7$,&*&%\"uG\"\"#%\"vG!\"\"\"\"\"*&%\"bGF/F+F/F.,&*$)F+F,F/F/F-F.
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Jacobian" }{MPLTEXT 1 0 0 "" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "J:=jacobian(R, [u,v]);" }}{PARA
11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%'matrixG6#7$7$,&*&%\"uG\"\"\"%\"vG!
\"\"\"\"#%\"bGF/,$*&F,F0F.!\"#F/7$,$F,F0F/" }}}{EXCHG {PARA 0 "" 0 ""
{TEXT -1 28 "Fixed points of reaction eqn" }}{PARA 0 "> " 0 ""
{MPLTEXT 1 0 25 "_EnvAllSolutions := true:" }}{PARA 0 "> " 0 ""
{MPLTEXT 1 0 51 "fpts := solve(\{evalm(R)[1]=0,evalm(R)[2]=0\},\{u,v\}
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%fptsG "
0 "" {MPLTEXT 1 0 36 "J0 := simplify(subs(fpts,evalm(J)));" }}{PARA
11 "" 1 "" {XPPMATH 20 "6#>%#J0G-%'matrixG6#7$7$%\"bG,$*$)F*\"\"#\"\"
\"!\"\"7$,$*&F/F/F*F0F.F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "Det
erminant for solutions of the foram sum a_n exp(lamba t) cos(n pi x/L)
to exist:\nREMEMBER THAT:
nbar=nb=n*pi/L" }}{PARA 0 "> " 0 ""
{MPLTEXT 1 0 117 "Lambda:=Matrix([[lambda,0],[0,lambda]]):\nD0:=Matrix
([[1,0],[0,d]]):\nM:= Lambda-evalm(J0) + (nb)^2*D0;\ndetM := det(M):"
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG,(-%'matrixG6#7$7$%\"bG,$*$)F+
\"\"#\"\"\"!\"\"7$,$*&F0F0F+F1F/F1F1*&)%#nbGF/F0-%'RTABLEG6$\"*)GF\\8-
%'MATRIXG6#7$7$F0\"\"!7$FA%\"dGF0F0-F96$\"*)oL\\8-F=6#7$7$%'lambdaGFA7
$FAFKF0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "DetM = 0 implies solut
ions of the form sum a_n exp(lamba t) cos(n pi x/L) to exists" }
{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "Pl := collect(
detM,lambda);\na2 := coeff(Pl,lambda,2):\na1 := coeff(Pl,lambda,1):\na
0 := coeff(Pl,lambda,0):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#PlG,.*$
)%'lambdaG\"\"#\"\"\"F**&,*%\"bG!\"\"*$)%#nbGF)F*F*F*F**&F0F*%\"dGF*F*
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"" 0 "" {TEXT -1 60 "In order to have unstabilities we need: Re(lambda
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:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "hb:= 4*a0*a2;\nh:= simplify(su
bs(nbTOn,hb));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#hbG,*%\"bG\"\"%**
F'\"\"\"F&F))%#nbG\"\"#F)%\"dGF)!\"\"*&F'F)F*F)F)*(F'F))F+F'F)F-F)F)"
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F*!\"\"*,F)F*)%\"nG\"\"#F*)%#PiGF2F*%\"dGF*)F,F2F*F**(F0F*F3F*F6F*F.*(
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{MPLTEXT 1 0 24 "solve(diff(hb,nb)=0,nb);" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6%\"\"!,$*(%\"dG!\"\"\"\"##\"\"\"F(*&F&F*,&*&%\"bGF*F&F*F
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nontrivial minimum, compute the minimum of h and find hmin=0:" }
{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "nbTOn := nb = n
*Pi/L;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "nbmin := 1/2/d*2^(1/2)*(d
*(b*d-1))^(1/2):\nnmin := nbmin*L/Pi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1
0 33 "hmin := simplify(subs(n=nmin,h));" }}{PARA 0 "> " 0 "" {MPLTEXT
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\\`v;$\"\"\"!\"$Ff]_l" 1 2 0 1 10 0 2 9 1 4 2 1..000000
45. 0 "Curve 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 62 "For example: from the plot it is clear that b=d=4
hmin " 0 "" {MPLTEXT 1 0 21 "subs(
\{d=4,b=4\},hmin);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!$h\"\"\"%" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Define h for a particular choice o
f d,b and L" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "
choice0 := d=d,b=1,L=49;\n#choice0 := d=d,b=1,L=46.345;\n#choice0 := d
=d,b=.1,L=93;\nh0 := subs(\{choice0\},h);\nh0n := subs(choice0,(subs(n
bTOn,h)));\nh0min:=subs(choice0,hmin);" }}{PARA 11 "" 1 "" {XPPMATH
20 "6#>%(choice0G6%/%\"dGF'/%\"bG\"\"\"/%\"LG\"#\\" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#h0G,*\"\"%\"\"\"*&#F&\"%,CF'*()%\"nG\"\"#F')%#PiGF
.F'%\"dGF'F'!\"\"*(#F&F*F'F,F'F/F'F'**#F&\"(,[w&F')F-F&F')F0F&F'F1F'F'
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$h0nG,*\"\"%\"\"\"*&#F&\"%,CF'*(
)%\"nG\"\"#F')%#PiGF.F'%\"dGF'F'!\"\"*(#F&F*F'F,F'F/F'F'**#F&\"(,[w&F'
)F-F&F')F0F&F'F1F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&h0minG,$*&,
(%\"dG!\"'*$)F(\"\"#\"\"\"F-F-F-F-F(!\"\"F." }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 11 "Plot h vs n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "dc :=
solve(h0min=0,d);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "dc0 := evalf(
dc[2]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "nmin0 := subs(choice0,ev
alf(subs(d=dc0,nmin)));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "d1:=dc0
-dc0/50;p1:=subs(\{d=d1\},h0n):\nd0:=dc0;
p2:=subs(\{d=d0\},h0n)
:\nd2:=dc0+dc0/50;p3:=subs(\{d=d2\},h0n):" }}{PARA 0 "> " 0 ""
{MPLTEXT 1 0 58 "interv:= n=0..nmin0*1.4:\n#interv:= n=nmin0*0.9..nmin
0*1.1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "pp1:=plot(p1,interv,colo
r=[green],thickness=3):\npp2:=plot(p2,interv,color=[red],thickness=3):
\npp3:=plot(p3,interv,color=[blue],thickness=3):\nplots[display]([pp1,
pp2,pp3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dcG6$,&\"\"$\"\"\"*&
\"\"#F(-%%sqrtG6#F*F(!\"\",&F'F(*&F*F(F+F(F(" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#>%$dc0G$\"+CrUGe!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
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{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#d2G$\"+mc*\\%f!\"*" }}{PARA 13 ""
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