{VERSION 5 0 "IBM INTEL LINUX" "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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning " 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {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 "restart;" }}{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/,$*&F,F0F.!\"#F/7$,$F,F0F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Fixed points of reaction eqn" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "_EnvAllSolutions := true:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "fpts := solve(\{evalm(R)[1]=0,evalm(R)[2]=0\},\{u,v\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%fptsG<$/%\"vG*&\"\"\"F)*$)%\" bG\"\"#F)!\"\"/%\"uG*&F)F)F,F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Evaluate Jacobian at fixed points" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 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!\"\"*$)%#nbGF)F*F*F*F**&F0F*%\"dGF*F* F*F(F*F*F-F**(F-F*F0F*F3F*F.F/F**&)F1\"\"%F*F3F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "In order to have unstabilities we need: Re(lambda )>0 or h <0" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "nbTOn := nb = n*Pi/L :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "hb:= 4*a0*a2;\nh:= simplify(su bs(nbTOn,hb));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#hbG,*%\"bG\"\"%** F'\"\"\"F&F))%#nbG\"\"#F)%\"dGF)!\"\"*&F'F)F*F)F)*(F'F))F+F'F)F-F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG,$*&,**&%\"bG\"\"\")%\"LG\"\"% F*!\"\"*,F)F*)%\"nG\"\"#F*)%#PiGF2F*%\"dGF*)F,F2F*F**(F0F*F3F*F6F*F.*( )F1F-F*)F4F-F*F5F*F.F*F,!\"%F;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Find minimum of h(nb):" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "solve(diff(hb,nb)=0,nb);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"!,$*(%\"dG!\"\"\"\"##\"\"\"F(*&F&F*,&*&%\"bGF*F&F*F *F*F'F*F)F),$F%#F'F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "Take the \+ 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 1 0 18 "(solve(hmin=0,d));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&nbTOn G/%#nbG*(%\"nG\"\"\"%#PiGF)%\"LG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%nminG,$*,%\"dG!\"\"\"\"##\"\"\"F)*&F'F+,&*&%\"bGF+F'F+F+F+F(F+F* %\"LGF+%#PiGF(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%hminG,$*&,(*&% \"bG\"\"\"%\"dGF*!\"'*&)F)\"\"#F*)F+F/F*F*F*F*F*F+!\"\"F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*&,&!\"$\"\"\"*&\"\"#F'-%%sqrtG6#F)F'F'F'%\"b G!\"\"F.*&,&\"\"$F'*&F)F'F*F'F'F'F-F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 150 "Plot the regions of unstability: \nBLUE: hmin>0 => (Re( lambda<0)) => STABLE for ALL modes\nRED: hmin<0 => (Re(lambda>0)) \+ => UNSTABLE for SOME modes" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with( plots):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "contourplot(hmin,b=0..5, d=0.001..5,grid=[20,50],filled=true,contours=[0], coloring=[red,blue]) ;" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has b een redefined\n" }}{PARA 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=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$\"+CrUGe!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%&nmin0G$\"+Ee#Q+\"!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#d1G$\" +#ee=r&!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#d0G$\"+CrUGe!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#d2G$\"+mc*\\%f!\"*" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6%7Z7$$\"\"!F)$\" \"%F)7$$\"3I019zkFjI!#=$\"3KdF4OMt#*R!#<7$$\"3:<0RkphGdF/$\"3l%4h'omhu RF27$$\"38OLJ*4rgs)F/$\"3,*=?eLF 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