{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 Output" 2 20 "" 0 1 0 0 255 1 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 "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times " 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 } {PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 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 "Title" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 24 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "restart;\nwith(plots ):\nwith(inttrans):\nalias(Y(s)=laplace(y(t),t,s)):" }}{PARA 7 "" 1 " " {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }} }{EXCHG {PARA 256 "" 0 "" {TEXT -1 52 "Cold pills: Periodic impulses a nd Laplace transforms" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 20 "About t his worksheet" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 15 "Author and Date " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Matthias Kawski\nArizona State University\nhttp://math.asu.edu/~kawski" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "O riginal version MAPLE release 8. \nJune 2007. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 24 "Content, Pu rpose and Use" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 176 "This worksheet i llustrates a naive applications of Laplace transforms to solve\nlinear differential equations driven by a periodic pulses -- here interprete d as\ntaking medicine." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "A good part of the worksheet exper iments with MAPLE's capabilities of " }}{PARA 0 "" 0 "" {TEXT -1 155 " working with infinite series, or simply long series --- but in the end we take\na pragmatic approach, and type in the expected solution by h and when MAPLE's" }}{PARA 0 "" 0 "" {TEXT -1 179 "invlaplace balks.\nS uitable for demos, inviting simple modifications ... and some question s such as\nhow will the final level change when one doubles the daily \+ dose, or takes twice " }}{PARA 0 "" 0 "" {TEXT -1 26 "the dose every o ther day. " }{MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 33 "Updates and log of modifications." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "none yet.." }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 39 "Starting \+ from the differential equation" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 " A finite number of days in the treatment" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "de0a:=diff(y(t),t)=sum(Dirac(t-'k'),'k'=1..7):de0a; \nAE0a:=subs(y(0)=0,laplace(de0a,t,s)):\nAE0a;\nY(s)=solve(AE0a,Y(s)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"yG6#%\"tGF*,0-%&Dir acG6#,&F*\"\"\"F0!\"\"F0-F-6#,&F*F0\"\"#F1F0-F-6#,&F*F0\"\"$F1F0-F-6#, &F*F0\"\"%F1F0-F-6#,&F*F0\"\"&F1F0-F-6#,&F*F0\"\"'F1F0-F-6#,&F*F0\"\"( F1F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"sG\"\"\"-%\"YG6#F%F&,0-% $expG6#,$F%!\"\"F&-F,6#,$*&\"\"#F&F%F&F/F&-F,6#,$*&\"\"$F&F%F&F/F&-F,6 #,$*&\"\"%F&F%F&F/F&-F,6#,$*&\"\"&F&F%F&F/F&-F,6#,$*&\"\"'F&F%F&F/F&-F ,6#,$*&\"\"(F&F%F&F/F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"YG6#%\" sG*&,0-%$expG6#,$F'!\"\"\"\"\"-F+6#,$*&\"\"#F/F'F/F.F/-F+6#,$*&\"\"$F/ F'F/F.F/-F+6#,$*&\"\"%F/F'F/F.F/-F+6#,$*&\"\"&F/F'F/F.F/-F+6#,$*&\"\"' F/F'F/F.F/-F+6#,$*&\"\"(F/F'F/F.F/F/F'F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "The infinite treatment" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "de0:=diff(y(t),t)=sum(Dirac(t-'k'),'k'=1..infinity):de0;\nAE0:=su bs(y(0)=0,laplace(de0,t,s)):\nAE0;\nY(s)=solve(AE0,Y(s));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"yG6#%\"tGF*-%$sumG6$-%&DiracG6#, &F*\"\"\"%\"kG!\"\"/F3;F2%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/*&%\"sG\"\"\"-%\"YG6#F%F&*&F&F&,&-%$expGF)F&F&!\"\"F." }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%\"YG6#%\"sG*&\"\"\"F)*&F'F),&-%$expGF&F)F)!\" \"F)F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Adding decay (excretion ) to the model" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "de1:=diff(y(t),t )-y(t)=sum(Dirac(t-'k'),'k'=1..7):\nde1;\nAE1:=subs(y(0)=0,laplace(de1 ,t,s)):\nAE1;\nY(s)=solve(AE1,Y(s));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/,&-%%diffG6$-%\"yG6#%\"tGF+\"\"\"F(!\"\",0-%&DiracG6#,&F+F,F,F-F,-F 06#,&F+F,\"\"#F-F,-F06#,&F+F,\"\"$F-F,-F06#,&F+F,\"\"%F-F,-F06#,&F+F, \"\"&F-F,-F06#,&F+F,\"\"'F-F,-F06#,&F+F,\"\"(F-F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&%\"sG\"\"\"-%\"YG6#F&F'F'F(!\"\",0-%$expG6#,$F&F+F '-F.6#,$*&\"\"#F'F&F'F+F'-F.6#,$*&\"\"$F'F&F'F+F'-F.6#,$*&\"\"%F'F&F'F +F'-F.6#,$*&\"\"&F'F&F'F+F'-F.6#,$*&\"\"'F'F&F'F+F'-F.6#,$*&\"\"(F'F&F 'F+F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"YG6#%\"sG*&,0-%$expG6#,$ F'!\"\"\"\"\"-F+6#,$*&\"\"#F/F'F/F.F/-F+6#,$*&\"\"$F/F'F/F.F/-F+6#,$*& \"\"%F/F'F/F.F/-F+6#,$*&\"\"&F/F'F/F.F/-F+6#,$*&\"\"'F/F'F/F.F/-F+6#,$ *&\"\"(F/F'F/F.F/F/,&F'F/F/F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 44 "Restart in the frequency domain (sign err or)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "This is what the geometric \+ series (almost) correctly sums to ... did not want to bother" }}{PARA 0 "" 0 "" {TEXT -1 68 "to try to get MAPLE do it to above formula (but made a sign mistake)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Y1:=1/s/(1 +exp(-s));\ny1:=invlaplace(Y1,s,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#Y1G*&\"\"\"F&*&%\"sGF&,&F&F&-%$expG6#,$F(!\"\"F&F&F." }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#y1G,\"\"\"\"\"#F'*F'F(F')!\"\",&F'F,-%&fl oorG6#%\"tGF,F'F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "Just for il lustration, we make MAPLE expand the periodic delays into a series\n-- the sign erro becomes obvious here." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "Y1=subs(exp(-s)=x,Y1);\nseries(subs(exp(-s)=x,Y1),x=0,12);\nY1= simplify(subs(x=exp(-1),series(subs(exp(-s)=x,Y1),x=0,7)));\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&\"\"\"F%*&%\"sGF%,&F%F%-%$expG6#,$F '!\"\"F%F%F-*&F%F%*&F'F%,&F%F%%\"xGF%F%F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+=%\"xG*&\"\"\"F&%\"sG!\"\"\"\"!,$F%F(F&F%\"\"#F*\"\"$F %\"\"%F*\"\"&F%\"\"'F*\"\"(F%\"\")F*\"\"*F%\"#5F*\"#6-%\"OG6#F&\"#7" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&\"\"\"F%*&%\"sGF%,&F%F%-%$expG6#,$ F'!\"\"F%F%F-,$*&,2F%F--F*6#F-F%-F*6#!\"#F--F*6#!\"$F%-F*6#!\"%F--F*6# !\"&F%-F*6#!\"'F-*&-%\"OG6#-F*6#!\"(F%F'F%F-F%F'F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "display([\nplot(y1,t=0..12,-3..3,thickne ss=3,discont=true),\nplot(y1,t=0..12,-3..3,thickness=1)\n],xtickmarks= 2,ytickmarks=3);" }}{PARA 13 "" 1 "" {GLPLOT2D 411 411 411 {PLOTDATA 2 "6(-%'CURVESG607S7$$\"37+++++++?!#E$\"\"\"\"\"!7$$\"3s/y%zg:(z@!#>F+ 7$$\"3_O#G&HVFwSF1F+7$$\"3Lb+$o'H:4iF1F+7$$\"33))>C3(phN)F1F+7$$\"3]Sh Q0#)H\\5!#=F+7$$\"3EMq#H13uC\"F>F+7$$\"3[U)RaMRDX\"F>F+7$$\"3+@zD6kok; F>F+7$$\"3a(QX(=J:w=F>F+7$$\"3yU'e*>En$4#F>F+7$$\"3+6c-:RE&G#F>F+7$$\" 3o)>'*\\L]4]#F>F+7$$\"355*HTQAvr#F>F+7$$\"3E\"3&H$oHi#HF>F+7$$\"3nUOX0 fv:JF>F+7$$\"3y\\)o\\\"47TLF>F+7$$\"3*[>0#RY.KNF>F+7$$\"3))[NB'o7Tv$F> F+7$$\"3Uv0`(Q*o]RF>F+7$$\"3_RXe%=lj;%F>F+7$$\"31UIEY&R
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