{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 "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 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 } } {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "# Set up the ODE and boundary cond itions:\node := eps*diff(u(x),x,x)+(x-1/2)*diff(u(x),x)-(x-1/2)*u(x)=0 ;\nbc := \{u(0)=-2, u(1)=exp(1.)\};\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/,(*&%$epsG\"\"\"-%%diffG6$-%\"uG6#%\"xG-%\"$G6$F0\"\"#F)F)* &,&F0F)#F)F4!\"\"F)-F+6$F-F0F)F)*&F6F)F-F)F8\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bcG<$/-%\"uG6#\"\"!!\"#/-F(6#\"\"\"$\"+G=G=F!\"*" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "# The uniform asymptotic e xpansion is given by:\nunif := exp(x)*(-1/2+3/2*sqrt(2/Pi)*\nint(exp(- s^2/2), s=0..(x-1/2)/eps^(1/2)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %%unifG*&-%$expG6#%\"xG\"\"\",&#F*\"\"#!\"\"*&#\"\"$F-F*-%$erfG6#,$** \"\"%F.F-#F*F-,&*&F-F*F)F*F*F*F.F*%$epsG#F.F-F*F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "eps := 0.01;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$epsG$\"\"\"!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "# Solve the problem numerically:\npp := dsolve(\{ode \} union bc, numeric);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ppGf*6#%& x_bvpG6(%$resG%%dataG%)solnprocG%&ndsolG%)outpointG%\"iG6#%inCopyright ~(c)~2000~by~Waterloo~Maple~Inc.~All~rights~reserved.G6\"C)>%8_EnvDSNu mericSaveDigitsG%'DigitsG>F5\"#9@%/%-_EnvInFsolveG%%trueG>8(-&%&evalfG 6#F46#9$>F=-F@FB>8%-%%evalG6#&%8 &-_FG%$GetG6#Q/soln_procedureF1@$4-%%typeG6$F=.%(numericG@+/F=Q.solnpr ocedureF1O-FI6#FP-%'memberG6$F=7)Q&startF1Q%leftF1Q&rightF1Q*errorproc F1Q(rawdataF1Q&orderF1Q&errorF1O-FPFB/F=Q(sysvarsF1O-FR6#F[p09!%(unkno wnGO-.F`pFBC$>8'-%(pointtoG6#&-FR6#Q0soln_proceduresF16#\"\"!O-.FgpFBZ %C$>8$-FP6#F=7$/%\"xG&FgqFM-%$seqG6$/&.7$-%\"uG6#F\\r-%%diffG6$FerF\\r 6#8)&Fgq6#,&F\\sFNFNFN/F\\s;FN\"\"#F1YF1F1F1F1" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 75 "# Plot the solution and compare with the asymp totic expansion:\nwith(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "odeplot(pp, [[x, u(x)], [x, unif]]);" }}{PARA 13 "" 1 "" {GLPLOT2D 541 541 541 {PLOTDATA 2 "6%-%'CURVESG6$X,%)anythingG6\"6\"[g l'!%\"!!#_q\"S\"#0000000000000000BFFFFFFFFFFFFFFD3F94E5E0A2A03116C0005 65B796EBAC73FA4E5E0A533E612C000AE9F6698CF7A3FAF58D0F917B39AC00108D540D B073C3FB4E5E0A60FCD11C00165013A461D463FBA1F58D001B3D5C001C31B8AC533FB3 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