{VERSION 2 3 "DEC ALPHA UNIX" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 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 }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 }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 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "As an example of Euler's method for syst ems of first-order equations,\nconsider the following nonlinear predat or-prey model.\n" }{XPPEDIT 18 0 "dF/dt = (-0.03 + 0.001*R)*F" "/*&%#d FG\"\"\"%#dtG!\"\"*&,&\$\"\"\$!\"#F'*&\$\"\"\"!\"\$F%%\"RGF%F%F%%\"FGF%" } {TEXT -1 14 " " }{XPPEDIT 18 0 "F(0) = 20" "/-%\"FG6#\"\" !\"#?" }{MPLTEXT 1 0 1 "\n" }{XPPEDIT 18 0 "dR/dt = (0.05 - 0.001*F)*R " "/*&%#dRG\"\"\"%#dtG!\"\"*&,&\$\"\"&!\"#F%*&\$\"\"\"!\"\$F%%\"FGF%F'F%% \"RGF%" }{TEXT -1 90 " R(0) = 50\n\nFirst, define the \+ functions that give the value of the slope:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f := (F,R,t) -> (-0.03 + 0.001*R)*F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6%%\"FG%\"RG%\"tG6\"6\$%)operatorG%&ar rowGF**&,&\$!\"\$!\"#\"\"\"9%\$F3F1F39\$F3F*F*" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 36 "g := (F,R,t) -> (0.05 - 0.001*F)*R;\n" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"gG:6%%\"FG%\"RG%\"tG6\"6\$%)operatorG%&arrowG F**&,&\$\"\"&!\"#\"\"\"9\$\$!\"\"!\"\$F39%F3F*F*" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 78 "Then, define the step size and the initial values for e ach variable and for t." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 7 "h := 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG\" \"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "t[0] := 0; F[0] := 20; R[0] := 50;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"tG6#\"\"!F' " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"FG6#\"\"!\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"RG6#\"\"!\"#]" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 136 "Then for each variable, the new value is ``old value + s tep size times slope.''\nSo after one step, the values are calculated \+ as follows:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "t[1] := t[0] + h:\nF[1] := F[0] + h*f(F[0],R[0],t[0]):\nR[1] \+ := R[0] + h*g(F[0],R[0],t[0]):\n[F[1],R[1],t[1]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\$\"&+/#!\"\$\$\"&+:&F&\"\"\"" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 77 "Usually you will want to perform a large number of iter ations, and to do that" }}{PARA 0 "" 0 "" {TEXT -1 205 "it is easiest \+ to write a for-loop to do all the calculations at once.\n\nThe for loo p in this example creates three sequences of points: one in the t-F p lane,\none in the t-R plane and one in the F-R plane. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Start by defining the the step size and n umber of steps:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "N := 200; h := 1 ;" }}{PARA 0 "" 0 "" {TEXT -1 50 " Next define the i nitial values:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "t[0] := 0; F[0] := 20; R[0] := 50;" }}{PARA 0 "" 0 "" {TEXT -1 78 " Then define the first point in each of the three s equences." }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eul er_ptsF := [t[0],F[0]];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "euler_pt sR := [t[0],R[0]];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "euler_ptsFR : = [F[0],R[0]];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for n from 0 to N -1 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " t[n+1] := t[n] + h;" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " F[n+1] := F[n] + h*f(F[n],R[n], t[n]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " R[n+1] := R[n] + h*g(F [n],R[n],t[n]);" }}{PARA 0 "" 0 "" {TEXT -1 60 " ne w value = old value + stepsize * slope" }{MPLTEXT 1 0 0 "" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 33 " new_ptF := [ t[n+1], F[n+1] ];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " new_ptR := [ t[n+1], R[n+1] ];" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " new_ptFR := [ F[n+1], R[n+1] ]; " }}{PARA 0 "" 0 "" {TEXT -1 78 " define the newest points in the three sequences of points." }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " euler_ptsF := euler_ptsF, new_ptF;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " euler_ptsR := euler_ptsR, new_pt R;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " euler_ptsFR := euler_ptsFR , new_ptFR;" }}{PARA 0 "" 0 "" {TEXT -1 90 " modify each of the three sequences by adding the new point at the end." } {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"\$ +#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG\"\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>&%\"tG6#\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> &%\"FG6#\"\"!\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"RG6#\"\"!\"# ]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+euler_ptsFG7\$\"\"!\"#?" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%+euler_ptsRG7\$\"\"!\"#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,euler_ptsFRG7\$\"#?\"#]" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 246 "The colon at the end of the for loop suppresses a ll output from calculations\nwithin the for-loop. We can see the list of points by executing, for example,\n``euler_ptsF;'' to see the list of points in the t-F plane. A graph is probably\nuseful. 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