{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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 260 "T imes" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 261 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis " -1 262 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Purple \+ Emphasis" -1 263 "Times" 1 12 128 0 255 1 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" 258 264 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 265 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 260 266 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 260 267 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 268 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" 260 269 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{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 0 128 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 14 128 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 "Text Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Map le 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 "Time s" 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 "Normal" -1 256 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 "Normal" -1 257 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 28 "More boundary value problems" }} {PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }} {PARA 0 "" 0 "" {TEXT -1 19 "Version: 24.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 5 "load " }{TEXT 0 7 "desolve" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 262 7 "DEsol.m" }{TEXT -1 32 " is required by this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple session by a command similar to the o ne that follows, where the file path gives its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\\\Maple/procdrs/DEsol.m \";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 45 "load root-finding procedures etc. including: " }{TEXT 0 7 "findmin" } }{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 262 7 "roots.m " }{TEXT -1 32 " is required by this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple session by a command similar to the one that follows, where the file path gives its location." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\\\Maple/procdrs/r oots.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 93 "An example to illustrate the possible non-uniqueness of solutio ns to boundary value problems " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" {TEXT -1 147 "Acknowledgement: The exampl e in this section comes from: Introduction to Numerical Analysis, by J .Stoer and R. Bulirsch, Springer-Verlag, page 504." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 48 "Consider the non-linear boundary value problem: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "d^2*y/(d*x^2) = 3/2*y^2,y(0) = 4,y(1) = 1;" "6%/*(%\"dG \"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"*(\"\"$F(F&F,F'F&/-F'6#\"\"!\"\" %/-F'6#F(F(" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 " An anal ytical solution is given by " }{XPPEDIT 18 0 "y = 4/(1+x)^2" "6#/%\"y G*&\"\"%\"\"\"*$,&F'F'%\"xGF'\"\"#!\"\"" }{TEXT -1 30 ", as can easily be checked. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "subs(y(x)=4/ (1+x)^2,de);\neval(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#deG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%#deG" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 35 "The derivative of this solution is " }{XPPEDIT 18 0 "dy/dx = -8/((1+x)^3);" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*& \"\")F&*$,&F&F&%\"xGF&\"\"$F(F(" }{TEXT -1 22 ", which has the value \+ " }{XPPEDIT 18 0 "-8" "6#,$\"\")!\"\"" }{TEXT -1 7 ", when " } {XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Diff(4/((1 +x)^2),x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$,$* &\"\"%\"\"\",&F)F)%\"xGF)!\"#F)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, $*&\"\")\"\"\",&F&F&%\"xGF&!\"$!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 86 "We can obtain a numerical version of \+ this solution by the non-linear shooting method. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "de := diff( y(x),x$2)=3/2*y(x)^2;\nbc := y(0)=4,y(1)=1;\ngn := desolveSH(\{de,bc\} ,x=0..1,init=-9..-7,info=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG /-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#,$*&#\"\"$F0\"\"\"*$)F)F0F5F5F5 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bcG6$/-%\"yG6#\"\"!\"\"%/-F(6# \"\"\"F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%AUsing~iterative~shooting ~method.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%@Shot~1,~with~initial~de rivativeG$!\"*\"\"!%2has~end~value~->~G$!+,:V\"e*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%@Shot~2,~with~initial~derivativeG$!\"(\"\"!%2has~en d~value~->~G$\"+**pBWO!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%@Shot~3 ,~with~initial~derivativeG$!+-P2\\\")!\"*%2has~end~value~->~G$\"+NMu$p '!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%@Shot~4,~with~initial~derivat iveG$!+veO@!)!\"*%2has~end~value~->~G$\"+FB+<&*!#5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6&%@Shot~5,~with~initial~derivativeG$!+ny^**z!\"*%2has~ end~value~->~G$\"+*[$4,5F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%@Shot~6 ,~with~initial~derivativeG$!+Z:++!)!\"*%2has~end~value~->~G$\"+;\\'*** **!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%@Shot~7,~with~initial~deriva tiveG$!+++++!)!\"*%2has~end~value~->~G$\"+(*********!#5" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "The graph of the n umerical solution appears to coincide with the graph of the analytical solution. 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "xx := evalf[15](1/sqrt(5));\ngn(xx);\nevalf(evalf[13] (g(xx)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"0e**\\&f8sW!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+c+$)4>!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+c+$)4>!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 78 "There is another solution with the initia l value of the derivative near -36. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "de := diff(y(x),x$2)=3/ 2*y(x)^2;\nbc := y(0)=4,y(1)=1;\nhn := desolveSH(\{de,bc\},x=0..1,init =-36..-35,info=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6 $-%\"yG6#%\"xG-%\"$G6$F,\"\"#,$*&#\"\"$F0\"\"\"*$)F)F0F5F5F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bcG6$/-%\"yG6#\"\"!\"\"%/-F(6#\"\"\"F/" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%AUsing~iterative~shooting~method.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%@Shot~1,~with~initial~derivativeG$! #O\"\"!%2has~end~value~->~G$\"+io+i5!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%@Shot~2,~with~initial~derivativeG$!#N\"\"!%2has~end~value~->~G$ \"+On&)fi!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%@Shot~3,~with~initial ~derivativeG$!+M*yde$!\")%2has~end~value~->~G$\"+JOn'***!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%@Shot~4,~with~initial~derivativeG$!+\"Hbee$ !\")%2has~end~value~->~G$\"+)y,++\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%@Shot~5,~with~initial~derivativeG$!+$)[&ee$!\")%2has~end~value~ ->~G$\"+++++5!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "A more accurate value for the initial value of the deriva tive is -35.85854882485548651430990. " }}{PARA 0 "" 0 "" {TEXT -1 58 " The following plot shows the graphs of the two solutions. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "plot (['gn'(x),'hn'(x)],x=0..1,color=[red,blue],thickness=1,labels=[`x`,`y( x)`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 463 352 352 {PLOTDATA 2 "6'-%'CUR VESG6$7S7$$\"\"!F)$\"\"%F)7$$\"+;arz@!#6$\"+fG;JQ!\"*7$$\"+XTFwSF/$\"+ >i!Gp$F27$$\"+\"z_\"4iF/$\"+M&yfa$F27$$\"+S&phN)F/$\"+uv%oS$F27$$\"+*= )H\\5!#5$\"+lBNwKF27$$\"+[!3uC\"FE$\"+f1&>;$F27$$\"+J$RDX\"FE$\"+)*\\p \\IF27$$\"+)R'ok;FE$\"+/MxRHF27$$\"+1J:w=FE$\"+&H9g$GF27$$\"+3En$4#FE$ \"+$Q8\\t#F27$$\"+/RE&G#FE$\"+MaF]EF27$$\"+D.&4]#FE$\"+\"z5'fDF27$$\"+ vB_F27$$\"+V&R F27$$\"+Xh-'e%FE$\"+M>7!)=F27$$\"+R\"3Gy%FE$\"+O/SI=F27 $$\"+.T1&*\\FE$\"+Y$[*yY2aFE$\"+v< *\\o\"F27$$\"+yXu9cFE$\"+PGbS;F27$$\"+\\y))GeFE$\"+7UY'f\"F27$$\"+i_QQ gFE$\"+s(H]b\"F27$$\"+!y%3TiFE$\"+(Qck^\"F27$$\"+O![hY'FE$\"+9^Gv9F27$ $\"+#Qx$omFE$\"+MWqR9F27$$\"+u.I%)oFE$\"+))o6.9F27$$\"+(pe*zqFE$\"+#Ra 6P\"F27$$\"+C\\'QH(FE$\"+*zWuL\"F27$$\"+8S8&\\(FE$\"+.\"\\oI\"F27$$\"+ 0#=bq(FE$\"+\"fwfF\"F27$$\"+2s?6zFE$\"+/(QoC\"F27$$\"+IXaE\")FE$\"+31R <7F27$$\"+l*RRL)FE$\"+\"*Q+!>\"F27$$\"+`<.Y&)FE$\"+M=%H;\"F27$$\"+8tOc ()FE$\"+:a+P6F27$$\"+\\Qk\\*)FE$\"+n*HR6\"F27$$\"+p0;r\"*FE$\"+pkL)3\" F27$$\"+lxGp$*FE$\"+&H&=m5F27$$\"+!oK0e*FE$\"+uUIV5F27$$\"+<5s#y*FE$\" +ur3A5F27$$\"\"\"F)$\"+++++5F2-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-F$6 $7SF'7$F-$\"+cGQBKF27$F4$\"+w^&Qb#F27$F9$\"+Y'p]!=F27$F>$\"+jri`5F27$F C$\"+`K%e1$FE7$FI$!+dk-fQFE7$FN$!+%4&y-6F27$FS$!+IyIV=F27$FX$!+\"G9\"z DF27$Fgn$!+wyBJLF27$F\\o$!+\"pas)RF27$Fao$!+uw5:ZF27$Ffo$!+<3GIaF27$F[ p$!+@K()*4'F27$F`p$!+p\"zpo'F27$Fep$!+VMQ`tF27$Fjp$!+fQ'f)yF27$F_q$!+k **Ri%)F27$Fdq$!+x_nG*)F27$Fiq$!+Kr%pQ*F27$F^r$!+MrIm(*F27$Fcr$!+a(y(45 F_[l7$Fhr$!+O'[S.\"F_[l7$F]s$!+?SE`5F_[l7$Fbs$!+&)=Hl5F_[l7$Fgs$!+Bv.p 5F_[l7$F\\t$!+_w*f1\"F_[l7$Fat$!+#)fr8 /&)F27$Fdv$!+/LU`zF27$Fiv$!+E)*[&R(F27$F^w$!+Hn$ox'F27$Fcw$!+t3XUhF27$ Fhw$!+/Zq_aF27$F]x$!+e`uoZF27$Fbx$!+H/6aSF27$Fgx$!+puPMLF27$F\\y$!+%[ \"RmEF27$Fay$!+/e'f*=F27$Ffy$!+4l_/7F27$F[z$!+saNjYFE7$F`z$\"+d^//CFE7 $Fez$\"+)*********FE-Fjz6&F\\[lF(F(F][l-%*THICKNESSG6#Ffz-%+AXESLABELS G6$%\"xG%%y(x)G-%%VIEWG6$;F(Fez%(DEFAULTG" 1 2 0 1 10 1 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "An analytical expressi on for the second solution is " }{XPPEDIT 18 0 "y = A*``((1-cn(c[1] *x-c[2],k))/(1+cn(c[1]*x-c[2],k))-1/sqrt(3));" "6#/%\"yG*&%\"AG\"\"\"- %!G6#,&*&,&F'F'-%#cnG6$,&*&&%\"cG6#F'F'%\"xGF'F'&F46#\"\"#!\"\"%\"kGF: F',&F'F'-F/6$,&*&&F46#F'F'F6F'F'&F46#F9F:F;F'F:F'*&F'F'-%%sqrtG6#\"\"$ F:F:F'" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "cn(x,k);" "6#-%#cnG6$% \"xG%\"kG" }{TEXT -1 50 " is the Jacobian elliptic function with param eter " }{XPPEDIT 18 0 "k=sqrt(2+sqrt(3))/2" "6#/%\"kG*&-%%sqrtG6#,&\" \"#\"\"\"-F'6#\"\"$F+F+F*!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "The constants have approximate values: " }}{PARA 256 "" 0 "" {TEXT -1 5 " A " }{TEXT 272 1 "~" }{TEXT -1 30 " 18.51675484036 711473814685, " }{XPPEDIT 18 0 "c[1]" "6#&%\"cG6#\"\"\"" }{TEXT -1 1 " " }{TEXT 270 1 "~" }{TEXT -1 30 " 4.303109903356770498341086, " } {XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 1 " " }{TEXT 271 1 "~ " }{TEXT -1 1 " " }{XPPEDIT 18 0 "2.334641957539096598324202;" "6#-%&F loatG6$\":-UK)f'4Rv&>kMB!#C" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 197 "A := 18.516754840 36711473814685;\nc1 := 4.303109903356770498341086;\nc2 := 2.3346419575 39096598324202;\nk := sqrt(2+sqrt(3))/2;\nh := x -> A*((1-JacobiCN(c1* x-c2,k))/(1+JacobiCN(c1*x-c2,k))-1/sqrt(3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG$\":&o9QZ6n.%[v;&=!#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c1G$\":'3T$)\\qnN.*4JI%!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c2G$\":-UK)f'4Rv&>kMB!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG,&*&\"\"%!\"\"\"\"'#\"\"\"\"\"#F+*&F'F(F,F*F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$%)operatorG%&arrow GF(*&%\"AG\"\"\",&*&,&F.F.-%)JacobiCNG6$,&*&%#c1GF.9$F.F.%#c2G!\"\"%\" kGF:F.,&F.F.F2F.F:F.*&F.F.-%%sqrtG6#\"\"$F:F:F.F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "The values of the tw o solutions appear to agree to 10 digits. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "xx := evalf(1/sqrt (5));\nhn(xx);\nevalf(evalf[13](h(xx)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"+af8sW!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+kl2I**! \"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+kl2I**!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "We can use either form of the solution to calculate the coordinates of the minimum point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "evalf(findmin(hn,0.54),11);\nevalf(findmin(h,0.54),11);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\",h,waU&!#6$!,#R`1p5!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\",i,waU&!#6$!,#R`1p5!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "Calculation of constants " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Construct ion of an accurate numerical solution. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "de := diff(y(x),x$2)=3 /2*y(x)^2;\nbc := y(0)=4,y(1)=1;\nkn := evalf(desolveSH(\{de,bc\},x=0. .1,init=-35.85854883..-35.85854882,info=1),25);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#,$*$)F)F0 \"\"\"#\"\"$F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bcG6$/-%\"yG6#\" \"!\"\"%/-F(6#\"\"\"F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%AUsing~iter ative~shooting~method.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%@Shot~1,~w ith~initial~derivativeG$!+$)[&ee$!\")%2has~end~value~->~G$\":(**GOO\"= `A++++\"!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%@Shot~2,~with~initial~ derivativeG$!+#)[&ee$!\")%2has~end~value~->~G$\":&[=defSty********!#D " }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%@Shot~3,~with~initial~derivativeG $!:saT^'[b[#)[&ee$!#B%2has~end~value~->~G$\":q.K*******************!#D " }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%@Shot~4,~with~initial~derivativeG $!:!*4V^'[b[#)[&ee$!#B%2has~end~value~->~G$\":++++++++++++\"!#C" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Calculati on of coordinates of minimum point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "minpt := evalf(findmin(kn, .5425476017),25);\nxmin := minpt[1];\nymin := minpt[2];" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%&minptG7$$\":&*[8iG3?h,waU&!#D$!:/ccafUg\"R`1p 5!#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%xminG$\":&*[8iG3?h,waU&!#D " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%yminG$!:/ccafUg\"R`1p5!#B" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "unassign('A','c1','c2');\nphi := (x,A,c1,c2) -> A*((1-JacobiCN(c1 *x-c2,k))/(1+JacobiCN(c1*x-c2,k))-1/sqrt(3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$phiGf*6&%\"xG%\"AG%#c1G%#c2G6\"6$%)operatorG%&arrowG F+*&9%\"\"\",&*&,&F1F1-%)JacobiCNG6$,&*&9&F19$F1F19'!\"\"%\"kGF=F1,&F1 F1F5F1F=F1*&F1F1-%%sqrtG6#\"\"$F=F=F1F+F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Use the coordinates of 3 points on the curve to calculate the 3 constants. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "eq1 := phi(0,A,c1, c2)=4:\neq2 := phi(xmin,A,c1,c2)=ymin:\neq3 := phi(1,A,c1,c2)=1:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "evalf(fsolve(\{eq1,eq2,eq3\} ,\{A=18.51675473,c1=4.30310990,c2=2.33464196\}),25);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"AG$\":&o9QZ6n.%[v;&=!#B/%#c1G$\":'3T$)\\qnN.*4 JI%!#C/%#c2G$\":-UK)f'4Rv&>kMBF-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 9 "Example 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 264 8 "Question" }{TEXT -1 1 ":" }} {PARA 0 "" 0 "" {TEXT -1 79 "(a) Find a discrete numerical solution to the two point boundary value problem " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+dy/dx+x*y = 0,y(0) = 2,y(3) = 0; " "6%/,(*(%\"dG\"\"#%\"yG\"\"\"*&F&F)*$%\"xGF'F)!\"\"F)*&%#dyGF)%#dxGF -F)*&F,F)F(F)F)\"\"!/-F(6#F2F'/-F(6#\"\"$F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 145 "which consists of a total of 21 points (includ ing the end points) spaced 0.15 units apart. Use a finite difference m ethod with no extrapolation. " }}{PARA 0 "" 0 "" {TEXT -1 21 "(Using t he procedure " }{TEXT 0 9 "desolveFD" }{TEXT -1 25 " with the single o ption \"" }{TEXT 262 13 "output=points" }{TEXT -1 51 "\", will ensure \+ that no extrapolation is performed.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "(b) Use a " }{TEXT 259 24 "finite diff erence method" }{TEXT -1 180 " as a basis for constructing a continuou s numerical solution to the boundary value problem from part (a) in th e form of a procedure which can be evaluated anywhere in the interval \+ " }{XPPEDIT 18 0 "[0,3]" "6#7$\"\"!\"\"$" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 25 "If you use the procedure " }{TEXT 0 9 "desolveFD" }{TEXT -1 119 " in the default mode to construct this continuous solu tion, the values it gives should be accurate to about 10 digits." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "(c) Plot the graphs of the continuous solution found in (b) together with the \+ discrete solution found in (a)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 70 "(d) Use the continuous solution found in \+ (b) to determine the maximum " }{TEXT 263 14 "absolute error" }{TEXT -1 39 " in the discrete solution found in (a)." }}{PARA 0 "" 0 "" {TEXT -1 15 "(The procedure " }{TEXT 0 14 "comparewithfcn" }{TEXT -1 18 " with the option \"" }{TEXT 262 16 "errtype=absolute" }{TEXT -1 29 "\" provides this information.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 10 "(e) Use a " }{TEXT 259 15 "shooting metho d" }{TEXT -1 163 " to find a continuous numerical solution to the boun dary value problem of part (a) in the form of a procedure which can be evaluated anywhere in the interval from " }{XPPEDIT 18 0 "x = -1;" "6 #/%\"xG,$\"\"\"!\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 5;" "6#/% \"xG\"\"&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 114 "(f) Construct a single plot which shows the graph s of both of the continuous solutions found in parts (b) and (e)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 8 "Solution " }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "de := diff(y(x),x$2)+diff(y(x),x)+x*y(x)=cos (x);\nic := y(0)=2,y(3)=0;\npts := desolve(\{de,ic\},y(x),type=numeric ,method=finitediff,output=points);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"-F(6$F*F-F2*&F-F2 F*F2F2-%$cosGF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\" \"!\"\"#/-F(6#\"\"$F*" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$ptsG777$$ \"\"!F($\"\"#F(7$$\"+++++:!#5$\"+Zhof$)F.7$$\"+++++IF.$!+18pv9F.7$$\"+ ++++XF.$!+#Rq%H(*F.7$$\"+++++gF.$!+nb9b;!\"*7$$\"+++++vF.$!+'e$4/AF@7$ $\"+++++!*F.$!+Y1`EEF@7$$\"++++]5F@$!+_g`FHF@7$$\"+++++7F@$!+9()y6JF@7 $$\"++++]8F@$!+<4f%=$F@7$$F-F@$!+.wm_JF@7$$\"++++];F@$!+g!RZ-$F@7$$\"+ ++++=F@$!+>!e=\"GF@7$$\"++++]>F@$!+&e,v_#F@7$$\"+++++@F@$!+KMT(=#F@7$$ \"++++]AF@$!+N:?4=F@7$$\"+++++CF@$!+U/r69F@7$$\"++++]DF@$!+ " 0 "" {MPLTEXT 1 0 139 "de : = diff(y(x),x$2)+diff(y(x),x)+x*y(x)=cos(x);\nic := y(0)=2,y(3)=0;\nfn := desolve(\{de,ic\},y(x),type=numeric,method=finitediff,info=true); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-% \"$G6$F-\"\"#\"\"\"-F(6$F*F-F2*&F-F2F*F2F2-%$cosGF," }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!\"\"#/-F(6#\"\"$F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%3constructing~orderG\"#7%Clocal~Taylor~series~ approximationsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%hnapplying~finite~ difference~method~with~extrapolation~at~levelG\"\"$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6%%Bsolving~a~tridiagonal~system~withG\"#u%*equationsG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%Bsolving~a~tridiagonal~system~wit hG\"$\\\"%*equationsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%Bsolving~a~t ridiagonal~system~withG\"$*H%*equationsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%Bsolving~a~tridiagonal~system~withG\"$*f%*equationsG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "comparewithfcn(pts,fn,errtype=absol ute);" }}{PARA 6 "" 1 "" {TEXT -1 84 " 0 2 \+ function val: 2 abs err: 0.0000e-01" }}{PARA 6 "" 1 " " {TEXT -1 84 " .15 .8359686147 function val: .81273 12637 abs err: 2.3237e-02" }}{PARA 6 "" 1 "" {TEXT -1 84 " .3 \+ -.1475691306 function val: -.1905604485 abs err: 4.29 91e-02" }}{PARA 6 "" 1 "" {TEXT -1 84 " .45 -.9729470392 \+ function val: -1.0325201658 abs err: 5.9573e-02" }}{PARA 6 "" 1 "" {TEXT -1 84 " .6 -1.655145567 function val: -1.7283 158991 abs err: 7.3170e-02" }}{PARA 6 "" 1 "" {TEXT -1 84 " .75 \+ -2.204093586 function val: -2.2879858624 abs err: 8.3 892e-02" }}{PARA 6 "" 1 "" {TEXT -1 84 " .9 -2.626530646 \+ function val: -2.7183373242 abs err: 9.1807e-02" }}{PARA 6 "" 1 "" {TEXT -1 84 " 1.050000000 -2.927536052 function val: -3.02 45052333 abs err: 9.6969e-02" }}{PARA 6 "" 1 "" {TEXT -1 84 " 1.2 00000000 -3.111788714 function val: -3.2112356032 abs err: 9 .9447e-02" }}{PARA 6 "" 1 "" {TEXT -1 84 " 1.350000000 -3.184590917 function val: -3.283927353 abs err: 9.9336e-02" }}{PARA 6 " " 1 "" {TEXT -1 84 " 1.500000000 -3.152667603 function val: -3. 2494444914 abs err: 9.6777e-02" }}{PARA 6 "" 1 "" {TEXT -1 84 " 1 .650000000 -3.024739060 function val: -3.1166966688 abs err: 9.1958e-02" }}{PARA 6 "" 1 "" {TEXT -1 84 " 1.800000000 -2.8118580 19 function val: -2.896979295 abs err: 8.5121e-02" }}{PARA 6 "" 1 "" {TEXT -1 84 " 1.950000000 -2.527501585 function val: -2 .6040639081 abs err: 7.6562e-02" }}{PARA 6 "" 1 "" {TEXT -1 84 " \+ 2.100000000 -2.187413432 function val: -2.2540346475 abs err : 6.6621e-02" }}{PARA 6 "" 1 "" {TEXT -1 84 " 2.250000000 -1.809201 535 function val: -1.864876723 abs err: 5.5675e-02" }}{PARA 6 "" 1 "" {TEXT -1 84 " 2.400000000 -1.411710442 function val: \+ -1.4558366204 abs err: 4.4126e-02" }}{PARA 6 "" 1 "" {TEXT -1 84 " 2.550000000 -1.014203217 function val: -1.0465900684 abs e rr: 3.2387e-02" }}{PARA 6 "" 1 "" {TEXT -1 84 " 2.700000000 -.6354 051815 function val: -.6562707959 abs err: 2.0866e-02" }} {PARA 6 "" 1 "" {TEXT -1 84 " 2.850000000 -.2924773449 function \+ val: -.3024288873 abs err: 9.9515e-03" }}{PARA 6 "" 1 "" {TEXT -1 84 " 3 0 function val: 0 \+ abs err: 0.0000e-01" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 6 "" 1 "" {TEXT -1 48 " Maximum absolute error: 9.9447e-0 2" }}{PARA 6 "" 1 "" {TEXT -1 58 " obtained for the input value: 1.200000000 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot('fn'(x),x=0..3);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7U7$$\"\"!F)$\"\"#F)7$$\"+DJdpK!#6$\"+UV/C< !\"*7$$\"+]i9RlF/$\"+?&ezX\"F27$$\"+XV)RQ*F/$\"+*\\(=M7F27$$\"+WA)GA\" !#5$\"+%R&R<5F27$$\"+Qeui=F@$\"+S^ZUbF@7$$\"+i3&o]#F@$\"+pod07F@7$$\"+ pX*y9$F@$!+0hZ/GF@7$$\"+WTAUPF@$!+n7]iiF@7$$\"+%*zhdVF@$!+Frr!f*F@7$$ \"+%>fS*\\F@$!+gFEx7F27$$\"+>$f%GcF@$!+Lb(*o:F27$$\"+Dy,\"G'F@$!+72HV= F27$$\"+7$F27$$\"+a&4*\\7F2$!+9+yZKF27$$\"+j=_68F2$!+1l!eF$F27$ $\"+Wy!eP\"F2$!+J'3aG$F27$$\"+UC%[V\"F2$!+1/2xKF27$$\"+J#>&)\\\"F2$!+w ;G]KF27$$\"+>:mk:F2$!+Ofm.KF27$$\"+w&QAi\"F2$!+^P>[JF27$$\"+uLU%o\"F2$ !+YM]tIF27$$\"+bjm[F2$!+5,*ei#F27$$\"+:K^+?F2$!+;sq\"\\#F27$$\"+7 ,Hl?F2$!+-H>RBF27$$\"+4w)R7#F2$!+87%R>#F27$$\"+y%f\")=#F2$!+i4%)G?F27$ $\"+/-a[AF2$!+g;zo=F27$$\"+ial6BF2$!+\"RK!)p\"F27$$\"+i@OtBF2$!+QS0H:F 27$$\"+fL'zV#F2$!+-w]^8F27$$\"+!*>=+DF2$!+Z(H8=\"F27$$\"+E&4Qc#F2$!+kJ e45F27$$\"+%>5pi#F2$!+W_SH%)F@7$$\"+bJ*[o#F2$!+^<.UpF@7$$\"+r\"[8v#F2$ !+$[/AI&F@7$$\"+Ijy5GF2$!+#z*=0RF@7$$\"+/)fT(GF2$!+sl**)\\#F@7$$\"+1j \"[$HF2$!+pu%GC\"F@7$$\"\"$F)F(-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXES LABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;F(F_[l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "de := diff( y(x),x$2)+diff(y(x),x)+x*y(x)=cos(x);\nic := y(0)=2,y(3)=0;\nfn2 := de solve(\{de,ic\},y(x),x=-1..5,type=numeric,type=shoot,info=true);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G 6$F-\"\"#\"\"\"-F(6$F*F-F2*&F-F2F*F2F2-%$cosGF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!\"\"#/-F(6#\"\"$F*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%>Using~linear~shooting~method.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%SShot,~with~initial~derivative~0,~has~end~value~- >~G$!+Q#zsu&!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%coHomogeneous~DE,~ with~initial~value~0~and~derivative~1,~has~end~value~->~G$!+:V\"oo'!#6 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%MInitial~derivative~for~required~ solution~->~G$!+sO%\\f)!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot('fn2'(x),x=-1..5);" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7W7$ $!\"\"\"\"!$\"+=ud;>!\")7$$!+v`3Y$*!#5$\"+Kax:P9pF1$\"+ _?ZB6F-7$$!+D$3XF'F1$\"+6WO&***!\"*7$$!++LSIcF1$\"+Cmv`))FM7$$!+v#)H') \\F1$\"+9%Go!yFM7$$!+i3@/PF1$\"+0vulfFM7$$!+7**F1 7$$\"*lN?c#FM$\"+6I#e[)!#67$$\"*U$e6PFM$!+Q)f,4'F17$$\"*&>q0]FM$!+\"*G %GG\"FM7$$\"*DM^I'FM$!+@r$H&=FM7$$\"*0ytb(FM$!+,.z1BFM7$$\"*RNXp)FM$!+ wE#4k#FM7$$\"+XDn/5FM$!+'y+[%HFM7$$\"+!y?#>6FM$!+'*e@DJFM7$$\"+4wY_7FM $!+MnJ\\KFM7$$\"+IOTq8FM$!+?(f`G$FM7$$\"+4\">)*\\\"FM$!+xsa\\KFM7$$\"+ EP/B;FM$!+IIKZJFM7$$\"+)o:;v\"FM$!+))=cwHFM7$$\"+%)[op=FM$!+d$y!pFFM7$ $\"+i%Qq*>FM$!+_/j*\\#FM7$$\"+RIKH@FM$!+LjW!=#FM7$$\"+^rZWAFM$!+F!y'z= FM7$$\"+[n%)oBFM$!+-oXT:FM7$$\"+5FL(\\#FM$!++`3*=\"FM7$$\"+e6.BEFM$!+4 W`I&)F17$$\"+p3lWFFM$!+$=VRY&F17$$\"+A))ozGFM$!+Bgs!Q#F17$$\"+Ik-,IFM$ \"+\"=7Q'=!#77$$\"+D-eIJFM$\"+,MPC@F17$$\"+>_(zC$FM$\"+=2^'e$F17$$\"+b *=jP$FM$\"+N^HzYF17$$\"+4/3(\\$FM$\"+E-=G_F17$$\"+C4JBOFM$\"+4NUJ`F17$ $\"+DVsYPFM$\"+XhR9]F17$$\"+>n#f(QFM$\"+T%=3J%F17$$\"+!)RO+SFM$\"+9\") ydLF17$$\"+_!>w7%FM$\"+Qso+AF17$$\"+*Q?QD%FM$\"+$pY'>(*F`p7$$\"+5jypVF M$!+\"\\anU\"F`p7$$\"+Ujp-XFM$!+57938F17$$\"+gEd@YFM$!+?OPw@F17$$\"+4' >$[ZFM$!+AR\"*fGF17$$\"+6Ejp[FM$!+$pL@C$F17$$\"\"&F*$!+$)4QPLF1-%'COLO URG6&%$RGBG$\"#5F)$F*F*Ff\\l-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;F(F \\\\l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exampl e 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 265 8 "Question" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 79 " (a) Find a discrete numerical solution to the two point boundary value problem " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/( d*x^2)+dy/dx+x^2*y = 2*sin(3*x),y(0) = 2,y(3) = 0;" "6%/,(*(%\"dG\"\"# %\"yG\"\"\"*&F&F)*$%\"xGF'F)!\"\"F)*&%#dyGF)%#dxGF-F)*&F,F'F(F)F)*&F'F )-%$sinG6#*&\"\"$F)F,F)F)/-F(6#\"\"!F'/-F(6#F7F;" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 145 "which consists of a total of 21 points ( including the end points) spaced 0.15 units apart. Use a finite differ ence method with no extrapolation. " }}{PARA 0 "" 0 "" {TEXT -1 21 "(U sing the procedure " }{TEXT 0 9 "desolveFD" }{TEXT -1 25 " with the si ngle option \"" }{TEXT 262 13 "output=points" }{TEXT -1 51 "\", will e nsure that no extrapolation is performed.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "(b) Use a " }{TEXT 259 24 "fin ite difference method" }{TEXT -1 180 " as a basis for constructing a c ontinuous numerical solution to the boundary value problem from part ( a) in the form of a procedure which can be evaluated anywhere in the i nterval " }{XPPEDIT 18 0 "[0,3]" "6#7$\"\"!\"\"$" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 25 "If you use the procedure " }{TEXT 0 9 "de solveFD" }{TEXT -1 119 " in the default mode to construct this contin uous solution, the values it gives should be accurate to about 10 digi ts." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "( c) Plot the graphs of the continuous solution found in (b) together wi th the discrete solution found in (a)." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 70 "(d) Use the continuous solution found in (b) to determine the maximum " }{TEXT 263 14 "absolute error" } {TEXT -1 39 " in the discrete solution found in (a)." }}{PARA 0 "" 0 " " {TEXT -1 15 "(The procedure " }{TEXT 0 14 "comparewithfcn" }{TEXT -1 18 " with the option \"" }{TEXT 262 16 "errtype=absolute" }{TEXT -1 29 "\" provides this information.)" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "(e) Use a " }{TEXT 259 15 "shooting m ethod" }{TEXT -1 163 " to find a continuous numerical solution to the \+ boundary value problem of part (a) in the form of a procedure which ca n be evaluated anywhere in the interval from " }{XPPEDIT 18 0 "x = -1; " "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 5;" " 6#/%\"xG\"\"&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 114 "(f) Construct a single plot which shows \+ the graphs of both of the continuous solutions found in parts (b) and \+ (e)." }}{PARA 0 "" 0 "" {TEXT -1 136 "Your plot should demonstate how \+ the solution obtained by the shooting method extends the solution give n by the finite difference method." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 267 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "de := \+ diff(y(x),x$2)+diff(y(x),x)+x^2*y(x)=2*sin(3*x);\nic := y(0)=1,y(3)=0; \npts := desolve(\{de,ic\},y(x),type=numeric,method=finitediff,output= points);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6 #%\"xG-%\"$G6$F-\"\"#\"\"\"-F(6$F*F-F2*&)F-F1F2F*F2F2,$-%$sinG6#,$F-\" \"$F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!\"\"\"/ -F(6#\"\"$F*" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$ptsG777$$\"\"!F($\" \"\"F(7$$\"+++++:!#5$\"+%*\\KW))F.7$$\"+++++IF.$\"++0#y-)F.7$$\"+++++X F.$\"+\")*G!QwF.7$$\"+++++gF.$\"+.lpywF.7$$\"+++++vF.$\"+\\&)[j!)F.7$$ \"+++++!*F.$\"+G-OD')F.7$$\"++++]5!\"*$\"+pW]T\"*F.7$$\"+++++7FM$\"+RP ;r$*F.7$$\"++++]8FM$\"+)*Q4,\"*F.7$$F-FM$\"+\"[j9>)F.7$$\"++++];FM$\"+ .))z8mF.7$$\"+++++=FM$\"+j!fDZ%F.7$$\"++++]>FM$\"+SAJ.?F.7$$\"+++++@FM $!+brTlX!#67$$\"++++]AFM$!+'*>)R_#F.7$$\"+++++CFM$!+CB6ZQF.7$$\"++++]D FM$!+_ge*=%F.7$$\"+++++FFM$!+4*[T]$F.7$$\"++++]GFM$!+Ueot>F.7$$\"\"$F( F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "de := diff(y(x),x$2)+diff(y(x),x)+x^2*y(x)=2*sin(3*x );\nic := y(0)=1,y(3)=0;\nfn := desolve(\{de,ic\},y(x),type=numeric,me thod=finitediff,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/ ,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"-F(6$F*F-F2*&)F-F1F2F*F2 F2,$-%$sinG6#,$F-\"\"$F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/- %\"yG6#\"\"!\"\"\"/-F(6#\"\"$F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%3c onstructing~orderG\"#7%Clocal~Taylor~series~approximationsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%hnapplying~finite~difference~method~with~ex trapolation~at~levelG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%Bsolvin g~a~tridiagonal~system~withG\"#u%*equationsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%Bsolving~a~tridiagonal~system~withG\"$\\\"%*equationsG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%Bsolving~a~tridiagonal~system~wit hG\"$*H%*equationsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%Bsolving~a~tri diagonal~system~withG\"$*f%*equationsG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "comparewithfcn(pts,fn,errtype=absolute);" }}{PARA 6 " " 1 "" {TEXT -1 84 " 0 1 function val: 1 \+ abs err: 0.0000e-01" }}{PARA 6 "" 1 "" {TEXT -1 84 " \+ .15 .8844324994 function val: .8810145495 abs err: 3.4179e-03" }}{PARA 6 "" 1 "" {TEXT -1 84 " .3 .8027820 5 function val: .7962044101 abs err: 6.5776e-03" }}{PARA 6 "" 1 "" {TEXT -1 84 " .45 .7638028981 function val: \+ .7539719994 abs err: 9.8309e-03" }}{PARA 6 "" 1 "" {TEXT -1 84 " \+ .6 .7678696503 function val: .7544812667 abs err : 1.3388e-02" }}{PARA 6 "" 1 "" {TEXT -1 84 " .75 .806348 8549 function val: .7890735883 abs err: 1.7275e-02" }}{PARA 6 "" 1 "" {TEXT -1 84 " .9 .8625360228 function val: \+ .8412282151 abs err: 2.1308e-02" }}{PARA 6 "" 1 "" {TEXT -1 84 " 1.050000000 .9141504469 function val: .8890518012 abs e rr: 2.5099e-02" }}{PARA 6 "" 1 "" {TEXT -1 84 " 1.200000000 .9371 163739 function val: .9090186817 abs err: 2.8098e-02" }} {PARA 6 "" 1 "" {TEXT -1 84 " 1.350000000 .9101093898 function \+ val: .8804367575 abs err: 2.9673e-02" }}{PARA 6 "" 1 "" {TEXT -1 84 " 1.500000000 .8191463481 function val: .7899187825 \+ abs err: 2.9228e-02" }}{PARA 6 "" 1 "" {TEXT -1 84 " 1.650000000 \+ .6613798803 function val: .6350299979 abs err: 2.6350e-02" }}{PARA 6 "" 1 "" {TEXT -1 84 " 1.800000000 .4472559063 functio n val: .4262915926 abs err: 2.0964e-02" }}{PARA 6 "" 1 "" {TEXT -1 84 " 1.950000000 .200331224 function val: .1868704 754 abs err: 1.3461e-02" }}{PARA 6 "" 1 "" {TEXT -1 84 " 2.100000 000 -.0456541716 function val: -.0504063963 abs err: 4.7522 e-03" }}{PARA 6 "" 1 "" {TEXT -1 84 " 2.250000000 -.2523981996 f unction val: -.2486149941 abs err: 3.7832e-03" }}{PARA 6 "" 1 " " {TEXT -1 84 " 2.400000000 -.3847112324 function val: -.37419 96688 abs err: 1.0512e-02" }}{PARA 6 "" 1 "" {TEXT -1 84 " 2.5500 00000 -.4189586052 function val: -.4050523369 abs err: 1.39 06e-02" }}{PARA 6 "" 1 "" {TEXT -1 84 " 2.700000000 -.3504148909 \+ function val: -.3373989387 abs err: 1.3016e-02" }}{PARA 6 "" 1 "" {TEXT -1 84 " 2.850000000 -.1973685842 function val: -.1894 481095 abs err: 7.9205e-03" }}{PARA 6 "" 1 "" {TEXT -1 84 " 3 \+ 0 function val: 0 abs err: 0.0 000e-01" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 6 "" 1 "" {TEXT -1 48 " Maximum absolute error: 2.9673e-02" }} {PARA 6 "" 1 "" {TEXT -1 58 " obtained for the input valu e: 1.350000000 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 21 "plot('fn'(x),x=0..3);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7W7$$\"\"!F)$\"\"\"F) 7$$\"+]i9Rl!#6$\"+@FcZ%*!#57$$\"+WA)GA\"F2$\"+AP!z+*F27$$\"+Qeui=F2$\" +%=#Hp&)F27$$\"+i3&o]#F2$\"+?sW'>)F27$$\"+pX*y9$F2$\"+U!y2!zF27$$\"+WT AUPF2$\"+1;o(p(F27$$\"+%*zhdVF2$\"+1-\"3c(F27$$\"+%>fS*\\F2$\"+&>]m\\( F27$$\"+>$f%GcF2$\"+Qp(p](F27$$\"+Dy,\"G'F2$\"+Wjo(e(F27$$\"+7&)\\\"Ffp$\"+(GK8\"zF27$$\"+>:mk:Ffp$\"+t$RtI(F27$$\"+w&QAi\"Ffp$\"+9 Z]#o'F27$$\"+uLU%o\"Ffp$\"+_%)Q7fF27$$\"+bjm[Ffp$\"+Xa=M?F 27$$\"+:K^+?Ffp$\"+8?3\\5F27$$\"+7,Hl?Ffp$\"+xmrwB!#77$$\"+4w)R7#Ffp$! +%fP\"o&)F/7$$\"+y%f\")=#Ffp$!+4>cR=+DFfp$!+\\f zfSF27$$\"+ed*>`#Ffp$!+<]umSF27$$\"+E&4Qc#Ffp$!+:bFGSF27$$\"+g)f`f#Ffp $!+D=)e%RF27$$\"+%>5pi#Ffp$!+./\\?QF27$$\"+bJ*[o#Ffp$!+-kF$[$F27$$\"+r \"[8v#Ffp$!+*Q6A%HF27$$\"+Ijy5GFfp$!+R^@SBF27$$\"+/)fT(GFfp$!+)R?[g\"F 27$$\"+1j\"[$HFfp$!+5>VE%)F/7$$\"\"$F)F(-%'COLOURG6&%$RGBG$\"#5!\"\"F( F(-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;F(Fj[l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "de := \+ diff(y(x),x$2)+diff(y(x),x)+x^2*y(x)=2*sin(3*x);\nic := y(0)=1,y(3)=0; \nfn2 := desolve(\{de,ic\},y(x),x=-1..5,type=numeric,type=shoot,info=t rue);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#% \"xG-%\"$G6$F-\"\"#\"\"\"-F(6$F*F-F2*&)F-F1F2F*F2F2,$-%$sinG6#,$F-\"\" $F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!\"\"\"/-F (6#\"\"$F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>Using~linear~shooting~ method.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%SShot,~with~initial~deriv ative~0,~has~end~value~->~G$!+G4N==!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 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Find a discrete numerical solution to the two point boundary value problem " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "d^2*y/(d*x^2)+sin(x);" "6#,&*(%\" dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(-%$sinG6#F+F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+y=0,y(0) = 1, y(2) = 0" "6%/,&*&%#dyG\"\"\"%# dxG!\"\"F'%\"yGF'\"\"!/-F*6#F+F'/-F*6#\"\"#F+" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 144 "which consists of a total of 21 points ( including the end points) spaced 0.1 units apart. Use a finite differe nce method with no extrapolation. " }}{PARA 0 "" 0 "" {TEXT -1 21 "(Us ing the procedure " }{TEXT 0 9 "desolveFD" }{TEXT -1 25 " with the sin gle option \"" }{TEXT 262 13 "output=points" }{TEXT -1 51 "\", will en sure that no extrapolation is performed.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "(b) Use a " }{TEXT 259 24 "finite difference method" }{TEXT -1 185 " as a basis for constructing a cont inuous numerical solution to the boundary value problem from part (a) \+ in the form of a procedure which can be evaluated anywhere in the inte rval from " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 2" "6#/%\"xG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 25 "If you use the procedure " }{TEXT 0 9 "desolveFD" } {TEXT -1 116 " in the default mode to construct this continuous solut ion, the values it gives should accurate to about 10 digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "(c) Plot the g raphs of the continuous solution found in (b) together with the discre te solution found in (a)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 86 "(d) Compare the values of the two the solutions fo und in (a) and (b) at the mid-point " }{XPPEDIT 18 0 "x = 1" "6#/%\"xG \"\"\"" }{TEXT -1 17 " of the interval " }{XPPEDIT 18 0 "x = 0;" "6#/% \"xG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 2" "6#/%\"xG\"\"#" } {TEXT -1 106 ". Use the continuous solution to find the relative error in the value obtained from the discrete solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "(e) Use a " }{TEXT 259 15 "shooting method" }{TEXT -1 163 " to find a continuous numerical so lution to the boundary value problem of part (a) in the form of a proc edure which can be evaluated anywhere in the interval from " } {XPPEDIT 18 0 "x = -1;" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 4 " to " } {XPPEDIT 18 0 "x = 3;" "6#/%\"xG\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "(f) Construct a singl e plot which shows the graphs of both of the two continuous solutions \+ found in parts (b) and (e)." }}{PARA 0 "" 0 "" {TEXT -1 136 "Your plot should demonstate how the solution obtained by the shooting method ex tends the solution given by the finite difference method." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 115 "(g) Construct a s ingle plot which shows the graphs of both of the continuous solutions \+ found in parts (b) and (e). " }}{PARA 0 "" 0 "" {TEXT -1 41 "_________ ________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 269 0 "" }{TEXT -1 41 "________________ _________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }