{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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 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 "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 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 "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Purple Emphasis" -1 261 "Times " 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Purple \+ Emphasis" -1 266 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 276 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 } {CSTYLE "" -1 277 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 278 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "Grey Emphasis" -1 279 "Times" 1 12 96 52 84 1 0 1 0 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 "Ti mes" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 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 "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 "Map le 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 "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 1 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 49 "Introduction to two point boundar y value problems" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanai mo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 25.3.2007" }}{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 279 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 one 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 40 "load extra procedures for linear systems " }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 279 8 "lins ys.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 simil ar to the one that follows, where the file path gives its location." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "read \"K:\\\\Maple/procdrs/ linsys.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 33 "Two point boundary value problems" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 52 "For a first \+ order differential equation of the form " }{XPPEDIT 18 0 "dy/dx = f(x, y);" "6#/*&%#dyG\"\"\"%#dxG!\"\"-%\"fG6$%\"xG%\"yG" }{TEXT -1 63 " the re is a unique solution curve passing through a given point" } {XPPEDIT 18 0 "``(x[0],y[0]);" "6#-%!G6$&%\"xG6#\"\"!&%\"yG6#F)" } {TEXT -1 17 " with gradient f(" }{XPPEDIT 18 0 "x[0],y[0];" "6$&%\"xG6 #\"\"!&%\"yG6#F&" }{TEXT -1 16 ") at this point." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "In this worksheet we cons ider the solution of a 2nd order diffferential equation of the form" } }{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+f(x); " "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(-%\"fG6#F+F(" } {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+g(x)*y = k(x);" "6#/,&*&%#dyG\"\" \"%#dxG!\"\"F'*&-%\"gG6#%\"xGF'%\"yGF'F'-%\"kG6#F." }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 170 "Since the general solution of such a dif ferential equation involves two arbitrary constants, we expect to find more than one solution curve passing through a given point." }}{PARA 0 "" 0 "" {TEXT -1 49 "For example, the general solution of the equati on" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2) \+ = y;" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F'" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "is " }{XPPEDIT 18 0 "y(x) = A*e xp(x)+B*exp(-x);" "6#/-%\"yG6#%\"xG,&*&%\"AG\"\"\"-%$expG6#F'F+F+*&%\" BGF+-F-6#,$F'!\"\"F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "Any solution for which the constants " }{TEXT 267 1 "A" }{TEXT -1 5 " and " }{TEXT 268 1 "B" }{TEXT -1 9 " satisfy " }{XPPEDIT 18 0 "A + B = 1" "6#/,&%\"AG\"\"\"%\"BGF&F&" }{TEXT -1 25 " passes through the \+ point" }{XPPEDIT 18 0 "`` (0,1)" "6#-%!G6$\"\"!\"\"\"" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 60 "For example, the following expression s all define solutions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "solns := [seq(i/3*exp(x)+(1-i/3)*ex p(-x),i=-6..9)]:\nfor i from 1 to 16 do print(solns[i]) end do:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#%\"xG!\"#-F%6#,$F'!\"\"\"\" $" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#%\"xG#!\"&\"\"$-F%6#,$ F'!\"\"#\"\")F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#%\"xG#! \"%\"\"$-F%6#,$F'!\"\"#\"\"(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-% $expG6#%\"xG!\"\"-F%6#,$F'F(\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, &-%$expG6#%\"xG#!\"#\"\"$-F%6#,$F'!\"\"#\"\"&F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#%\"xG#!\"\"\"\"$-F%6#,$F'F)#\"\"%F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#,$%\"xG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#%\"xG#\"\"\"\"\"$-F%6#,$F'!\"\"#\"\"#F* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#%\"xG#\"\"#\"\"$-F%6#,$ F'!\"\"#\"\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#%\"xG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#%\"xG#\"\"%\"\"$-F%6#,$F'! \"\"#F.F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#%\"xG#\"\"&\" \"$-F%6#,$F'!\"\"#!\"#F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6 #%\"xG\"\"#-F%6#,$F'!\"\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$ex pG6#%\"xG#\"\"(\"\"$-F%6#,$F'!\"\"#!\"%F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#%\"xG#\"\")\"\"$-F%6#,$F'!\"\"#!\"&F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#%\"xG\"\"$-F%6#,$F'!\"\"!\" #" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "colors := seq(COLOR(HUE,evalf(i/16)),i=0..15):\nplot( solns,x=-1..1,color=[colors],labels=[`x`,`y`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "64-%'CURVESG6$7S7$$!\"\"\"\"!$\"1]U ..m3>u!#:7$$!1nmm;p0k&*!#;$\"1>$z([]VQqF-7$$!1LL$3[F-7$$!1++](y$pZiF1$\" 16.&)ycnKXF-7$$!1LLL$yaE\"eF1$\"1YzJ7'>lC%F-7$$!1nmm\">s%HaF1$\"1Q+kB( 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!10T-a4z#\\$Fer7$Fiq$\"1Jz\\HwGk,6FPF-7$F`w$\"1g3_mV\\*)RF-7$Few$\"1EO**z!\\GD%F-7$Fjw$\"19bQhsFOX F-7$F_x$\"1H8;R9?<[F-7$Fdx$\"1xn3!yjI6&F-7$Fix$\"1$f$=w@c:aF-7$F^y$\"1 0s:/U'>q&F-7$Fcy$\"1)>;)H\">2/'F-7$Fhy$\"12%HkuJPN'F-7$F]z$\"1j(p**Gr% )p'F-7$Fbz$\"1t9kHHiRqF-7$Fgz$\"1^U..m3>uF--F\\[l6$F^[l$\"++++v$*Fd^m- %+AXESLABELSG6$%\"xG%\"yG-%%VIEWG6$;F(Fgz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" " Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "However, \+ there is a unique solution passing through two given points" } {XPPEDIT 18 0 "``(x[0],y[0]);" "6#-%!G6$&%\"xG6#\"\"!&%\"yG6#F)" } {TEXT -1 4 " and" }{XPPEDIT 18 0 "``(x[1],y[1]);" "6#-%!G6$&%\"xG6#\" \"\"&%\"yG6#F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 59 "For exa mple, the unique solution passing through the points" }{XPPEDIT 18 0 " ``(0,1)" "6#-%!G6$\"\"!\"\"\"" }{TEXT -1 4 " and" }{XPPEDIT 18 0 "`` \+ (1,0)" "6#-%!G6$\"\"\"\"\"!" }{TEXT -1 4 " is " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "y(x) = -.1565176430*exp(x)+1.15651764 3*exp(-x);" "6#/-%\"yG6#%\"xG,&*&-%&FloatG6$\"+Ik " 0 "" {MPLTEXT 1 0 253 "de := diff(y(x),x $2) = y(x);\nbc := y(0)=1,y(1)=0;\ndsolve(\{de,bc\},y(x));\nevalf(conv ert(%,exp)):\ns := unapply(rhs(%),x);\nplot([s(x),[[0,1],[1,0]]],x=-0. 2..1.2,color=[coral,blue],\n style=[line,point],symbol=circle,thickn ess=[2,1],\n labels=[`x`,`y`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#deG/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!\"\"\"/-F(6#F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&*&-%%coshG6#\"\"\"F.-%%sinhGF&F. F.-F0F-!\"\"F2-F,F&F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sGf*6#%\" xG6\"6$%)operatorG%&arrowGF(,&-%$expG6#9$$!+Ik;KH9F*$\"1a'\\(\\oa)>\"F-7$$!1nm;4'=28 \"F*$\"1X.\\ZL=b6F-7$$!1ommTEO,$)!#<$\"1%)H!pi62\"F-7$$!1qm;HtGODF@$\"1bR/tviL5F-7$$\"1c***\\P1bN$!#=$\"1B/wJ s*f&**F*7$$\"1HL$3d4cI$F@$\"1n*3PxY8d*F*7$$\"1)***\\([VhE'F@$\"1_j@>ML '>*F*7$$\"1lmm;lT6$*F@$\"1ns(*>K**=))F*7$$\"1LL$eYp$*>\"F*$\"1Nz\"e)yT $\\)F*7$$\"1+++b/L,:F*$\"1CN+&z(>M\")F*7$$\"1*****\\KJX!=F*$\"155*ePl4 y(F*7$$\"1+++X:s'4#F*$\"1&fDJ%GLZuF*7$$\"1LL3d#e?O#F*$\"1El6&*)y)\\rF* 7$$\"1mmmr#pvn#F*$\"1\\[%*)Q7F!oF*7$$\"1mmm'[[[%HF*$\"1)*>;g!HR^'F*7$$ \"1***\\PvddD$F*$\"1s)**olYQ='F*7$$\"1mmmO^'4`$F*$\"1.'4@klm*eF*7$$\"1 ***\\PD6H$QF*$\"1!)G**3Hr'e&F*7$$\"1***\\7ON/7%F*$\"1c>atMI'H&F*7$$\"1 mm;/mV?WF*$\"1?:)Q%f&z*\\F*7$$\"1mmT&RJfp%F*$\"1+76x%[zs%F*7$$\"1LL$eu *3$*\\F*$\"1R?R&)fsSWF*7$$\"1KL3dPv,`F*$\"1-B&[RJl9%F*7$$\"1***\\ioY/d &F*$\"1_'=OSP9fG?F*7$$\"1JL3x@%>\"zF*$\"1u=V]bq*y \"F*7$$\"1+++&*3T6#)F*$\"1+ey*\\q+`\"F*7$$\"1lmT?w=$\\)F*$\"1x\\y+9.(G \"F*7$$\"1++v)[Dxy)F*$\"1cJXF[2M5F*7$$\"1mmm\"4!pv!*F*$\"1&p:[cCj(yF@7 $$\"1)**\\PMirP*F*$\"1HS%*=dE.`F@7$$\"1MLL`f^n'*F*$\"1KBd_#)oHGF@7$$\" 1KL$eXWW'**F*$\"1(pD\")\\\"\\DIFP7$$\"1n;/C9*e-\"F-$!1mw&*HWR.AF@7$$\" 1+++R,&H0\"F-$!1zce!4Hx]%F@7$$\"1nm\"*zC'R3\"F-$!1T!\\75&=b/Kr\"F*-%'COLOURG6&%$RGBG$\"*+ +++\"!\")$\")AR!)\\F`[l\"\"!-%&STYLEG6#%%LINEG-%*THICKNESSG6#\"\"#-F$6 &7$7$Fc[l$\"\"\"Fc[l7$F`\\lFc[l-F[[l6&F][lFc[lFc[lF^[l-Fe[l6#%&POINTG- Fi[l6#Fa\\l-%'SYMBOLG6#%'CIRCLEG-%+AXESLABELSG6$%\"xG%\"yG-%%VIEWG6$;$ !\"#!\"\"$\"#7Fi]l%(DEFAULTG" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "The finite difference method" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 45 "C onsider the two point boundary value problem" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2) = y;" "6#/*(%\"dG\"\"#% \"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " y(0) = 1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y (1) = 0" "6#/-%\"yG6#\"\"\"\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 40 "Subdivide the interval from 0 to 1 into " }{TEXT 269 1 "n " }{TEXT -1 36 " equal sub-intervals of equal width " }{XPPEDIT 18 0 " h = 1/n;" "6#/%\"hG*&\"\"\"F&%\"nG!\"\"" }{TEXT -1 5 " by " } {XPPEDIT 18 0 "``(n+1)" "6#-%!G6#,&%\"nG\"\"\"F(F(" }{TEXT -1 9 " poin ts " }{XPPEDIT 18 0 "x[i] = x[0]+n*h,` `*i = 0,` . . . `,n;" "6&/&%\" xG6#%\"iG,&&F%6#\"\"!\"\"\"*&%\"nGF,%\"hGF,F,/*&%\"~GF,F'F,F+%(~.~.~.~ GF." }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " } {XPPEDIT 18 0 "x[0] = 0;" "6#/&%\"xG6#\"\"!F'" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "x[n] = 1;" "6#/&%\"xG6#%\"nG\"\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "We attemp t to find a discrete solution consisting of points" }{XPPEDIT 18 0 "`` (x[0],y[0]),``(x[1],y[1]),` . . . `,``(x[n],y[n]);" "6&-%!G6$&%\"xG6# \"\"!&%\"yG6#F)-F$6$&F'6#\"\"\"&F+6#F1%(~.~.~.~G-F$6$&F'6#%\"nG&F+6#F9 " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 40 "Replace the 2nd deri vative at each point" }{XPPEDIT 18 0 "``(x[i],y[i]);" "6#-%!G6$&%\"xG6 #%\"iG&%\"yG6#F)" }{TEXT -1 8 " by the " }{TEXT 266 23 "numerical appr oximation" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "(y[i+1 ]-2*y[i]+y[i-1])/(h^2);" "6#*&,(&%\"yG6#,&%\"iG\"\"\"F*F*F**&\"\"#F*&F &6#F)F*!\"\"&F&6#,&F)F*F*F/F*F**$%\"hGF,F/" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 21 "The first derivative " }{XPPEDIT 18 0 "dy/dx;" "6 #*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 90 " does not occur in the differe ntial equation, but if we are given a differential in which " } {XPPEDIT 18 0 "dy/dx;" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 39 " does occur we replace it at each point" }{XPPEDIT 18 0 "``(x[i],y[i]);" "6 #-%!G6$&%\"xG6#%\"iG&%\"yG6#F)" }{TEXT -1 33 " by the numerical approx imation: " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "(y[i+1 ]-y[i-1])/(2*h);" "6#*&,&&%\"yG6#,&%\"iG\"\"\"F*F*F*&F&6#,&F)F*F*!\"\" F.F**&\"\"#F*%\"hGF*F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 44 "The differential equation gives rise to the " }{TEXT 266 26 "system o f linear equations" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "(y[i+1]-2*y[i]+y[i- 1])/(h^2) = y[i];" "6#/*&,(&%\"yG6#,&%\"iG\"\"\"F+F+F+*&\"\"#F+&F'6#F* F+!\"\"&F'6#,&F*F+F+F0F+F+*$%\"hGF-F0&F'6#F*" }{TEXT -1 4 " , " } {XPPEDIT 18 0 "i = 1,` . . . `,n-1" "6%/%\"iG\"\"\"%(~.~.~.~G,&%\"nGF% F%!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }} {PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "y[i+1]-2*y[i]+y[i-1 ] = h^2*y[i];" "6#/,(&%\"yG6#,&%\"iG\"\"\"F*F*F**&\"\"#F*&F&6#F)F*!\" \"&F&6#,&F)F*F*F/F**&%\"hGF,&F&6#F)F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " y[i+1]+(-2-h^2)*y[i]+y[i-1] = 0;" "6#/,(&%\"yG6#,&%\"iG\"\"\"F*F*F**&, &\"\"#!\"\"*$%\"hGF-F.F*&F&6#F)F*F*&F&6#,&F)F*F*F.F*\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 51 "This system of equations written in matrix form is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[-2-h^2, 1, 0, ` . . . `, 0] , [1, -2-h^2, 1, ` . . . `, 0], [0, 1, -2-h^2, ` . . . `, 0], [`.`, `. `, `.`, `.`, `.`], [0, 0, 0, ` . . . `, -2-h^2]]);" "6#-%'matrixG6#7'7 ',&\"\"#!\"\"*$%\"hGF)F*\"\"\"\"\"!%(~.~.~.~GF.7'F-,&F)F**$F,F)F*F-F/F .7'F.F-,&F)F**$F,F)F*F/F.7'%\".GF7F7F7F77'F.F.F.F/,&F)F**$F,F)F*" } {TEXT -1 2 " " }{XPPEDIT 18 0 "matrix([[y[1]^``], [y[2]^``], [y[3]^`` ], [`.`], [y[n-1]^``]]);" "6#-%'matrixG6#7'7#)&%\"yG6#\"\"\"%!G7#)&F*6 #\"\"#F-7#)&F*6#\"\"$F-7#%\".G7#)&F*6#,&%\"nGF,F,!\"\"F-" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "matrix([[-1^``], [0^``], [0^``], [`.`], [0^``]]) ;" "6#-%'matrixG6#7'7#,$)\"\"\"%!G!\"\"7#)\"\"!F+7#)F/F+7#%\".G7#)F/F+ " }{TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 26 "where the boundary values " }{XPPEDIT 18 0 "y[0] = 1;" "6#/&%\"yG6#\"\"!\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y[1] = 0; " "6#/&%\"yG6#\"\"\"\"\"!" }{TEXT -1 84 " have been included, and appe ar (with changed sign) in the right hand column vector." }}{PARA 0 "" 0 "" {TEXT -1 85 "The following code will construct the coefficient ma trix and the vector of constants." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 193 "n := 5:\nm := n - 1:\nh := \+ 'h':\nA := Matrix(m):\nfor i from 1 to m do A[i,i] := -2 - h^2 end do: \nfor i from 1 to m-1 do\n A[i+1,i] := 1;\n A[i,i+1] := 1;\nend do :\nv := Vector(m):\nv[1] := -1:\nA,v;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6$\"*%[r]9-%'MATRIXG6#7&7&,&!\"#\"\"\"*$)%\"hG\"\"#F.!\" \"F.\"\"!F47&F.F,F.F47&F4F.F,F.7&F4F4F.F,-F$6$\"*cY.X\"-F(6#7&7#F37#F4 F?F?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "A utility \+ routine for comparing values: " }{TEXT 0 14 "comparewithfcn" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "This utility ro utine is required by examples in a later section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "comparewithfcn: usage" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 272 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 273 2 " " }{TEXT -1 36 " comp arewithfcn( pts, f ,options )" }}{PARA 0 "" 0 "" {TEXT -1 44 " com parewithfcn( pts, fx, x , options ) " }{TEXT 275 1 "\n" }{TEXT -1 0 " " }}{PARA 257 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 17 " pts - " }{TEXT 276 18 "a list of points " }{XPPEDIT 18 0 "[[x[1], y[ 1]], [x[2], y[2]] .. [x[n], y[n]]];" "6#7$7$&%\"xG6#\"\"\"&%\"yG6#F(;7 $&F&6#\"\"#&F*6#F07$&F&6#%\"nG&F*6#F6" }{TEXT 277 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT 23 15 " f or fx - \+ " }{TEXT -1 87 " a function of one variable or an expression fx def ining a function of one variable." }}{PARA 0 "" 0 "" {TEXT -1 6 " \+ " }}{PARA 0 "" 0 "" {TEXT 23 17 " x - " }{TEXT 274 65 "th e independent variable is required when the 2nd argument is an " } {TEXT -1 10 "expression" }{TEXT 278 4 " fx." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 3 " " }{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " } {TEXT 0 14 "comparewithfcn" }{TEXT -1 78 " tabulates the list of point s vertically as the first two columns of a matrix." }}{PARA 0 "" 0 "" {TEXT -1 251 "The values of the 2nd components are compared with the v alues obtained by applying the given function to the corresponding 1st components, and listing these \"exact values\" in the 3rd column of t he matrix. The relative error is given in the 4th column." }}{PARA 0 " " 0 "" {TEXT -1 68 "Alternatively, the same information can be printed out line by line." }}{PARA 0 "" 0 "" {TEXT -1 61 "The maximum of the \+ absolute or relative errors is also given." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 8 "Options:" }}{PARA 0 "" 0 "" {TEXT -1 24 "mode=linebyline / matrix" }}{PARA 0 "" 0 "" {TEXT -1 106 "With the option \"mode=linebyline\" the information is printed out li ne by line. This is the default option." }}{PARA 0 "" 0 "" {TEXT -1 112 "With the option \"mode=matrix\" the tabulated information is give n in a matrix as a return value of the procedure." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "errtype=relative/absolute " }}{PARA 0 "" 0 "" {TEXT -1 152 "This option determines whether the a bsolute or relative error is given. The default is \"errtype=relative \" in which case the relative error is tabulated." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 4 "Note" }{TEXT -1 77 ": If \+ the true value of the function is zero, the relative error is infinite . " }}{PARA 0 "" 0 "" {TEXT -1 113 "The maximum relative error is comp uted for the remaining relative errors, ignoring any infinite relative errors. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 16 "How to activate:" }{TEXT 256 1 "\n" }{TEXT -1 154 "To make the procedure active open the subsection , place the cursor anywhere after the prompt [ > and press [Enter].\n You can then close up the subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "comparewithfcn: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "comparewithfcn" {MPLTEXT 1 0 5497 "co mparewithfcn:=\nproc(pts,f)\n local xx,y1,y2,r,i,j,n,rows,fn,x,prec, prcsn,saveDigits,\n startoptions,Options,md,t,maxerr,format1,format2 ,g,e,zero,\n proctype,ertyp,xmax,prec1,prec2;\n\n proctype := fals e;\n if nargs<2 then\n error \"invalid arguments; the basic syn tax is 'comparewithfcn([[x1,y1],..,[xn,yn]],f(x),x)' or 'comparewithfc n([[x1,y1],..,[xn,yn]],f)'\"\n end if;\n if not type(pts,listlist) then\n error \"the 1st argument must be a list of points\"\n e nd if;\n if nargs>2 and not type(args[3],equation) then\n x := \+ args[3];\n if not type(x,name) then\n error \"the 3rd opt ional argument must be the name of the independent variable\"\n e nd if;\n startoptions := 4;\n if not type(f,algebraic) or no t type(indets(f,name) minus \{x\},set(realcons)) then\n error \+ \"the 2nd argument, %1, must be an expression in the single variable % 2\",f,x;\n end if;\n else\n if type(f,procedure) or (type( f,`@`) and type(\{op(f)\},set(procedure))) then\n proctype := \+ true;\n startoptions := 3;\n else\n error \"the 2 nd argument, %1, must be a function of one variable or an expression i n the one variable given as a 3rd argument\",f;\n end if;\n end if; \n\n # Get the options.\n md := 'linebyline';\n ertyp := \+ 'relative';\n if nargs>=startoptions then\n Options :=[args[sta rtoptions..nargs]];\n if not type(Options,list(equation)) then\n \+ error \"each optional argument must be an equation\"\n en d if;\n if hasoption(Options,'mode','md','Options') then\n \+ if not (md='matrix' or md='linebyline') then\n error \"\\ \"mode\\\" must be 'matrix' or 'linebyline'\"\n end if;\n \+ end if;\n if hasoption(Options,'errtype','ertyp','Options') then \n if not (ertyp='absolute' or ertyp='relative') then\n \+ error \"\\\"errtype\\\" must be 'absolute' or 'relative'\"\n \+ end if;\n end if;\n if nops(Options)>0 then\n er ror \"%1 is not a valid option for %2\",op(1,Options), procname;\n \+ end if;\n end if;\n n := nops(pts);\n prcsn := 0;\n\n # Chec k the data and find its maximum precision.\n for i to n do\n if nops(pts[i])<>2 then\n error \"the 1st argument must be a lis t of points, where each point is itself a list with two members\"\n \+ end if;\n t := pts[i,1];\n if type(t,float) and type(t,nu meric) then\n prec1 := length(convert(op(1,t),string));\n \+ elif type(t,realcons) then\n prec1 := Digits;\n else\n \+ error \"the 1st argument must be a list of points, where each p oint is itself a list of two real numbers\"\n end if;\n t := pts[i,2];\n if type(t,float) and type(t,numeric) then\n \+ prec2 := length(convert(op(1,t),string));\n elif type(t,realcons) then\n prec2 := Digits;\n else\n error \"the 1st argument must be a list of points, where each point is itself a list \+ of two real numbers\"\n end if;\n prec := max(prec1,prec2); \n if prec>prcsn then prcsn := prec end if;\n end do;\n saveD igits := Digits;\n Digits := prcsn;\n prec := trunc(prcsn/2);\n \+ \n if proctype then\n fn := f;\n else\n fn := unapply(ev alf(f),x);\n end if;\n rows := NULL;\n maxerr := 0;\n zero := \+ false;\n format1 := cat(\"%\",convert(prcsn+3,string),\".\",convert( prcsn-1,string),g);\n format2 := cat(\"%\",convert(prec+3,string) ,\".\",convert(prec-1,string),e);\n\n for i from 1 to n do\n xx := evalf(pts[i,1]);\n y1 := evalf(pts[i,2]);\n y2 := traper ror(evalf(fn(xx)));\n if y2=lasterror or not type(y2,numeric) the n\n error \"function failed to evaluate to a real floating poi nt number at %1\",xx;\n end if;\n \n if ertyp='absolute ' then\n r := evalf(abs(y1-y2));\n if r>maxerr then\n \+ maxerr := r;\n xmax := xx;\n end if;\n \+ else\n if y2<>0 then\n r := evalf(abs(y1-y2)/ab s(y2));\n if r>maxerr then\n maxerr := r;\n \+ xmax := xx;\n end if;\n else\n \+ zero := true;\n r := infinity;\n end if;\n \+ end if;\n\n if md='linebyline' then\n printf(format1,xx) ;\n printf(` `);\n printf(format1,y1);\n print f(` function val: `);\n printf(format1,y2);\n if ert yp=absolute then\n printf(` abs err: `);\n p rintf(format2,r);\n else\n printf(` rel err: `) ;\n if y2<>0 then\n printf(format2,r)\n \+ else\n printf(infinity);\n end if;\n \+ end if;\n printf(`\\n`);\n else\n rows := r ows,[xx,y1,y2,r];\n end if;\n end do;\n \n print(``);\n i f ertyp='absolute' then\n printf(` Maximum absolute \+ error: `);\n else\n printf(` Maximum relative erro r: `); \n end if;\n printf(format2,maxerr);\n\n if maxerr<> 0 then\n printf(`\\n obtained for the input value: ` );\n printf(format1,xmax);\n end if;\n \n if ertyp='relativ e' and zero then\n printf(`\\n excluding any cases w here the function value is zero.`);\n end if;\n\n Digits := saveDi gits;\n if md='matrix' then\n print(``);\n if ertyp='absol ute' then\n return array([[x,\"discrete value\",\"function val ue\",\"absolute err\"],rows]);\n else\n return array([[x, \"discrete value\",\"function value\",\"relative err\"],rows]);\n \+ end if;\n else\n return NULL;\n end if;\nend proc:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Examples appear in a later section" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "Computational example using " } {TEXT 0 15 "LinearSolve " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 58 "We perform the computations for \+ the boundary value problem" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "d^2*y/(d*x^2) = y;" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$ %\"xGF&F(!\"\"F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 1" "6#/-%\" yG6#\"\"!\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(1) = 0" "6#/-%\"y G6#\"\"\"\"\"!" }{TEXT -1 5 ". " }}{PARA 0 "" 0 "" {TEXT -1 80 "Fir st recalculate the analytical solution and set up the solution as a fu nction." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "de := diff(y(x),x$2) = y(x);\nbc := y(0)=1,y(1)=0;\nd solve(\{de,bc\},y(x));\ng := unapply(rhs(%),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!\"\"\"/-F(6#F+F* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*(-%$expG6#\"\"\"F -,&F*!\"\"-F+6#F/F-F/-F+6#,$F'F/F-F/*(F0F-,&F*F-F0F/F/-F+F&F-F/" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrow GF(,&*(-%$expG6#\"\"\"F1,&F.!\"\"-F/6#F3F1F3-F/6#,$9$F3F1F3*(F4F1,&F.F 1F4F3F3-F/6#F9F1F3F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 204 "Now set up the system of equations in matrix form . The code below gives a solution with just 10 steps, that is, with 9 \+ intermediate points along an approximate solution curve. You can chang e the value of " }{TEXT 270 1 "n" }{TEXT -1 16 " if you want to." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 216 "n := 10;\nm := n - 1:\nh := evalf(1/n);\nA := Matrix(m):\nd := ev alf(-2 - h^2):\nfor i from 1 to m do A[i,i] := d end do:\nfor i from 1 to m-1 do\n A[i+1,i] := 1;\n A[i,i+1] := 1;\nend do:\nv := Vector (m):\nv[1] := -1:\nA,v;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#5 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\"+++++5!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6$\"*3'y]9-%'MATRIXG6#7+7+$!++++5?!\"*\" \"\"\"\"!F0F0F0F0F0F07+F/F,F/F0F0F0F0F0F07+F0F/F,F/F0F0F0F0F07+F0F0F/F ,F/F0F0F0F07+F0F0F0F/F,F/F0F0F07+F0F0F0F0F/F,F/F0F07+F0F0F0F0F0F/F,F/F 07+F0F0F0F0F0F0F/F,F/7+F0F0F0F0F0F0F0F/F,-F$6$\"*%yP]9-F(6#7+7#!\"\"7# F0FAFAFAFAFAFAFA" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "This system is a tridiagonal system, but if " }{TEXT 271 1 "n" }{TEXT -1 32 " is not too large the procedure " }{TEXT 0 11 "Lin earSolve" }{TEXT -1 8 " in the " }{TEXT 0 13 "LinearAlgebra" }{TEXT -1 29 " package will work just fine." }}{PARA 0 "" 0 "" {TEXT -1 73 "N ow add in the end points and construct the solution as a list of point s." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "with(LinearAlgebra):\nUseHardwareFloats := false;\nu := LinearSolve(A,v);\nyvals := [1,op(convert(u,list)),0]:\nxvals := [ seq(h*i,i=0..n)]:\nsoln := zip((x,y)->[x,y],xvals,yvals);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%2UseHardwareFloatsG%&falseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG-%'RTABLEG6$\"*o=3X\"-%'MATRIXG6#7+7#$\"+_D-N( )!#57#$\"+J`RdvF07#$\"+l?MbkF07#$\"+;A%yT&F07#$\"+)y?XV%F07#$\"+mXa&\\ $F07#$\"+)yB:f#F07#$\"+Z#=Mr\"F07#$\"+%*)oW_)!#6" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%solnG7-7$$\"\"!F(\"\"\"7$$\"+++++5!#5$\"+_D-N()F-7$$ \"+++++?F-$\"+J`RdvF-7$$\"+++++IF-$\"+l?MbkF-7$$\"+++++SF-$\"+;A%yT&F- 7$$\"+++++]F-$\"+)y?XV%F-7$$\"+++++gF-$\"+mXa&\\$F-7$$\"+++++qF-$\"+)y B:f#F-7$$\"+++++!)F-$\"+Z#=Mr\"F-7$$\"+++++!*F-$\"+%*)oW_)!#67$$F,!\"* F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "Th e numerical solution can be compared with the analytical solution grap hically." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "plot([g(x),soln,soln],x=0..1,color=[red,green,blue], \n style=[line,line,point],thickness=[1,3,1],symbol=circle);" }} {PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6(-%'CURVESG6&7S7$ $\"\"!F)$\"\"\"F)7$$\"3emmm;arz@!#>$\"3&*\\GN%o[hr*!#=7$$\"3[LL$e9ui2% F/$\"3SlZ+U<$HZ*F27$$\"3nmmm\"z_\"4iF/$\"3srn+V`Z.#*F27$$\"3[mmmT&phN) F/$\"3ACG1U7YO*)F27$$\"3CLLe*=)H\\5F2$\"3Cc\"*oim![n)F27$$\"3gmm\"z/3u C\"F2$\"3^uL1?0wN%)F27$$\"3%)***\\7LRDX\"F2$\"3'>];O'*H<>)F27$$\"3]mm \"zR'ok;F2$\"3^bZ,hz(H%zF27$$\"3w***\\i5`h(=F2$\"35N\\oG;e)p(F27$$\"3W LLL3En$4#F2$\"3G^O$>!4y]uF27$$\"3qmm;/RE&G#F2$\"3())es\"f=WNsF27$$\"3 \")*****\\K]4]#F2$\"3?Ni8J_>'*pF27$$\"3$******\\PAvr#F2$\"3S$H,k&4CfnF 27$$\"3)******\\nHi#HF2$\"3R4$)y(p#*Q`'F27$$\"3jmm\"z*ev:JF2$\"3XI%>9J B/cNk%F 27$$\"3cmm;/T1&*\\F2$\"3Dr***z.J)QWF27$$\"3&em;zRQb@&F2$\"3[+Q`xbHGUF2 7$$\"3\\***\\(=>Y2aF2$\"3?b%*o-3qYSF27$$\"39mm;zXu9cF2$\"3e!fiK#eC_QF2 7$$\"3l******\\y))GeF2$\"3[OILU@4`OF27$$\"3'*)***\\i_QQgF2$\"37B1d04)) fMF27$$\"3@***\\7y%3TiF2$\"3%eoq>U&QuKF27$$\"35****\\P![hY'F2$\"3?Zc*z G(**pIF27$$\"3kKLL$Qx$omF2$\"3a)RwT*[n()GF27$$\"3!)*****\\P+V)oF2$\"3U NS'3k3Vp#F27$$\"3?mm\"zpe*zqF2$\"3!Q]JeRx,_#F27$$\"3%)*****\\#\\'QH(F2 $\"3h;$)pov!4L#F27$$\"3GKLe9S8&\\(F2$\"3/n;Tv[z`@F27$$\"3R***\\i?=bq(F 2$\"3M:Q,2Efp>F27$$\"3\"HLL$3s?6zF2$\"3'=ZI6kW.z\"F27$$\"3a***\\7`Wl7) F2$\"3_'[j@)*)\\.;F27$$\"3#pmmm'*RRL)F2$\"3$f>DJN[UU\"F27$$\"3Qmm;a<.Y &)F2$\"3*[a2Ipr:C\"F27$$\"3=LLe9tOc()F2$\"3[t,_4&f41\"F27$$\"3u****** \\Qk\\*)F2$\"3+:DWsS6a*)F/7$$\"3CLL$3dg6<*F2$\"3)4luw#G#31(F/7$$\"3Imm mmxGp$*F2$\"3!4F3xj./P&F/7$$\"3A++D\"oK0e*F2$\"3kP=Kq-PqNF/7$$\"3A++v= 5s#y*F2$\"3Q(R#=><,\\=F/7$F*F(-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%&S TYLEG6#%%LINEG-%*THICKNESSG6#F+-F$6&7-F'7$$\"3/+++++++5F2$\"3A+++_D-N( )F27$$\"35+++++++?F2$\"3s*****4L&RdvF27$$\"3))**************HF2$\"3S++ +l?MbkF27$$\"3A+++++++SF2$\"3\\*****f@UyT&F27$$\"3++++++++]F2$\"3#)*** **zy?XV%F27$$\"3w**************fF2$\"31+++mXa&\\$F27$$\"3a************ **pF2$\"3s*****zyB:f#F27$$\"3U+++++++!)F2$\"31+++Z#=Mr\"F27$$\"3A+++++ ++!*F2$\"3O*****R*)oW_)F/Fcz-Fez6&FgzF(FhzF(F[[l-F`[l6#\"\"$-F$6&Fd[l- Fez6&FgzF(F(Fhz-F\\[l6#%&POINTGF_[l-%'SYMBOLG6#%'CIRCLEG-%+AXESLABELSG 6$Q\"x6\"Q!Ff_l-%%VIEWG6$;F(F*%(DEFAULTG" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "The nume rical solution can also be compared with the analytical solution using the utility procedure " }{TEXT 0 14 "comparewithfcn" }{TEXT -1 5 " . \+ . " }{HYPERLNK 17 "comparewithfcn" 1 "" "comparewithfcn" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "comparewithfcn(soln,g(x),x);" }}{PARA 6 "" 1 "" {TEXT -1 81 " 0 1 function val: 1 \+ rel err: 0.0000e-01" }}{PARA 6 "" 1 "" {TEXT -1 81 " .1 \+ .873502255 function val: .873481691 rel err: 2.3543e-05" }}{PARA 6 "" 1 "" {TEXT -1 81 " .2 .755739533 function \+ val: .75570548 rel err: 4.5061e-05" }}{PARA 6 "" 1 "" {TEXT -1 81 " .3 .645534207 function val: .645492624 r el err: 6.4420e-05" }}{PARA 6 "" 1 "" {TEXT -1 81 " .4 .5 41784222 function val: .541740075 rel err: 8.1491e-05" }} {PARA 6 "" 1 "" {TEXT -1 81 " .5 .443452079 function va l: .443409442 rel err: 9.6157e-05" }}{PARA 6 "" 1 "" {TEXT -1 81 " .6 .349554457 function val: .3495166 rel \+ err: 1.0831e-04" }}{PARA 6 "" 1 "" {TEXT -1 81 " .7 .2591 52379 function val: .259121838 rel err: 1.1786e-04" }}{PARA 6 "" 1 "" {TEXT -1 81 " .8 .171341825 function val: \+ .171320455 rel err: 1.2474e-04" }}{PARA 6 "" 1 "" {TEXT -1 81 " \+ .9 .085244689 function val: .085233703 rel err: 1. 2889e-04" }}{PARA 6 "" 1 "" {TEXT -1 79 " 1 0 \+ function val: 0 rel err: infinity" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 6 "" 1 "" {TEXT -1 48 " Max imum relative error: 1.2889e-04" }}{PARA 6 "" 1 "" {TEXT -1 57 " \+ obtained for the input value: .9 " }}{PARA 6 "" 1 " " {TEXT -1 67 " excluding any cases where the function va lue is zero." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 79 "A pro cedure for solving a linear system with a tridiagonal coefficient matr ix: " }{TEXT 0 12 "TridiagSolve" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 12 "TridiagSolve " }{TEXT -1 51 " solves a system of linear equations of the form A " } {TEXT 265 1 "." }{TEXT -1 62 " x = b, where A is a tridiagonal matrix \+ given in compact form." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "TridiagSolve: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 17 "Calling Sequence:" }}{PARA 0 "" 0 "" {TEXT 263 5 "\n " }{TEXT -1 20 "TridiagSolve( A, b )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 5 " \+ " }}{PARA 0 "" 0 "" {TEXT 23 10 " A - " }{TEXT -1 73 " a tridig onal matrix given in the compact form of a matrix with 3 rows." }} {PARA 0 "" 0 "" {TEXT -1 109 " The first row cont ains the super-diagonal with a zero or dummy (unused) entry appended. " }}{PARA 0 "" 0 "" {TEXT -1 61 " The 2nd row con tains the main diagonal." }}{PARA 0 "" 0 "" {TEXT -1 106 " \+ The 3rd row contains the sub-diagonal with a zero or dummy ( unused) entry prepended." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 4 " " }{TEXT 23 8 " b - " }{TEXT -1 178 " a ve ctor with dimension the same as that of the tridigonal matrix A, that \+ is, with the same number of entries as the compact form of the tridiag onal matrix has along each row." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 257 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 12 "Tr idiagSolve" }{TEXT -1 43 " obtains a solution of the linear system A \+ " }{TEXT 264 1 "." }{TEXT -1 86 " x = b by means of a simplified metho d of LU decomposition, and backward substitution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 16 "How to activate:" }{TEXT 256 1 "\n" }{TEXT -1 155 "To ma ke the procedures active open the subsection, place the cursor anywher e after the prompt [ > and press [Enter].\nYou can then close up the \+ subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "TridiagSolve: impl ementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "T ridiagSolve" {MPLTEXT 1 0 1026 "TridiagSolve := proc(A::Matrix,b::Vect or)\n local m,n,k,beta,v,gam,j;\n\n m := LinearAlgebra['RowDimensi on'](A);\n if m<>3 then\n error \"the matrix must have 3 rows\" \n end if;\n n := LinearAlgebra['ColumnDimension'](A);\n if n<3 \+ then\n error \"the matrix must have at least 3 columns\"\n end \+ if;\n k := LinearAlgebra['Dimension'](b);\n if k<>n then\n er ror \"non matching dimensions for matrix and vector\"\n end if;\n\n \+ v := Vector(n);\n gam := Vector(n);\n beta := A[2,1]; \n if b eta=0 then\n error \"a zero pivot element has been encountered\" \n end if;\n v[1] := b[1]/beta;\n\n # Perform LU decomposition an d forward substitution.\n for j from 2 to n do\n gam[j] := A[1,j-1 ]/beta;\n beta := A[2,j] - A[3,j]*gam[j];\n if beta=0 then\n \+ error \"a zero pivot element has been encountered\"\n end if ;\n v[j] := (b[j] - A[3,j]*v[j-1])/beta;\n end do;\n \n # Perf orm the back substitution.\n for j from n-1 by -1 to 1 do\n v[j ] := v[j] - gam[j+1]*v[j+1];\n end do;\n v;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Examp les are given in the next section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 28 "Computational example using " }{TEXT 0 12 "TridiagSolve " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 77 "The example of the earlier section can also be solved by \+ using the procedure " }{TEXT 0 12 "TridiagSolve" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 84 "Again make sure that you have the analyti cal solution to the boundary value problem " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2) = y;" "6#/*(%\"dG\"\"#% \"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " y(0) = 1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y (1) = 0" "6#/-%\"yG6#\"\"\"\"\"!" }{TEXT -1 4 ", " }}{PARA 0 "" 0 " " {TEXT -1 51 "set up as a function for later comparison purposes." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "de := diff(y(x),x$2)=y(x);\nbc := y(0)=1,y(1)=0;\ndsolve(\{de,bc\} ,y(x));\ng := unapply(rhs(%),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #deG/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!\"\"\"/-F(6#F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*(-%$expG6#\"\"\"F-,&F*!\"\"-F+6#F /F-F/-F+6#,$F'F/F-F/*(F0F-,&F*F-F0F/F/-F+F&F-F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*(-%$expG6# \"\"\"F1,&F.!\"\"-F/6#F3F1F3-F/6#,$9$F3F1F3*(F4F1,&F.F1F4F3F3-F/6#F9F1 F3F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Set up an" }{XPPEDIT 18 0 " ``(n-1)" "6#-%!G6#,&%\"nG\"\"\"F(!\"\" " }{TEXT -1 21 " by 3 matrix A, as a " }{TEXT 266 15 "compact version " }{TEXT -1 70 " of the tridiagonal coefficient matrix, and the vector of constants v." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 218 "n := 10;\nm := n - 1:\nh := evalf(1/n);\nA := Matrix(3,m):\nd := evalf(-2 - h^2):\nfor i from 1 to m do A[2,i] := d end do:\nfor i from 1 to m-1 do\n A[1,i] := 1.0;\n A[3,i+1] := 1. 0;\nend do:\nv := Vector(m):\nv[1] := -1:\nA;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\" +++++5!#5" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"*k<4X\"-%'M ATRIXG6#7%7+$\"#5!\"\"F,F,F,F,F,F,F,\"\"!7+$!++++5?!\"*F1F1F1F1F1F1F1F 17+F/F,F,F,F,F,F,F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Use the procedure" }{TEXT 0 13 " TridiagSolve" } {TEXT -1 5 " . . " }{HYPERLNK 17 "TridiagSolve" 1 "" "TridiagSolve" } {TEXT -1 32 " to obtain a numerical solution." }}{PARA 0 "" 0 "" {TEXT -1 69 "Add in the end points and construct the solution as a lis t of points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "u := TridiagSolve(A,v);\nyvals := [1,op(convert(u ,list)),0]:\nxvals := [seq(h*i,i=0..n)]:\nsoln := zip((x,y)->[x,y],xva ls,yvals);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG-%'RTABLEG6$\"*))> 3X\"-%'MATRIXG6#7+7#$\"+aD-N()!#57#$\"+M`RdvF07#$\"+m?MbkF07#$\"+=A%yT &F07#$\"+*y?XV%F07#$\"+nXa&\\$F07#$\"+*yB:f#F07#$\"+[#=Mr\"F07#$\"+%*) oW_)!#6" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%solnG7-7$$\"\"!F(\"\"\"7 $$\"+++++5!#5$\"+aD-N()F-7$$\"+++++?F-$\"+M`RdvF-7$$\"+++++IF-$\"+m?Mb kF-7$$\"+++++SF-$\"+=A%yT&F-7$$\"+++++]F-$\"+*y?XV%F-7$$\"+++++gF-$\"+ nXa&\\$F-7$$\"+++++qF-$\"+*yB:f#F-7$$\"+++++!)F-$\"+[#=Mr\"F-7$$\"++++ +!*F-$\"+%*)oW_)!#67$$F,!\"*F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 80 "The numerical solution can be compared wi th the analytical solution graphically." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "plot([g(x),soln,soln], x=0..1,color=[red,green,blue],\n style=[line,line,point],thickness=[1 ,3,1],symbol=circle);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6&7S7$$\"\"!F)$\"\"\"F)7$$\"3emmm;arz@!#>$\"3 &*\\GN%o[hr*!#=7$$\"3[LL$e9ui2%F/$\"3SlZ+U<$HZ*F27$$\"3nmmm\"z_\"4iF/$ \"3srn+V`Z.#*F27$$\"3[mmmT&phN)F/$\"3ACG1U7YO*)F27$$\"3CLLe*=)H\\5F2$ \"3Cc\"*oim![n)F27$$\"3gmm\"z/3uC\"F2$\"3^uL1?0wN%)F27$$\"3%)***\\7LRD X\"F2$\"3'>];O'*H<>)F27$$\"3]mm\"zR'ok;F2$\"3^bZ,hz(H%zF27$$\"3w***\\i 5`h(=F2$\"35N\\oG;e)p(F27$$\"3WLLL3En$4#F2$\"3G^O$>!4y]uF27$$\"3qmm;/R E&G#F2$\"3())es\"f=WNsF27$$\"3\")*****\\K]4]#F2$\"3?Ni8J_>'*pF27$$\"3$ ******\\PAvr#F2$\"3S$H,k&4CfnF27$$\"3)******\\nHi#HF2$\"3R4$)y(p#*Q`'F 27$$\"3jmm\"z*ev:JF2$\"3XI%>9JB/cNk%F27$$\"3cmm;/T1&*\\F2$\"3Dr***z.J)QWF27$ $\"3&em;zRQb@&F2$\"3[+Q`xbHGUF27$$\"3\\***\\(=>Y2aF2$\"3?b%*o-3qYSF27$ $\"39mm;zXu9cF2$\"3e!fiK#eC_QF27$$\"3l******\\y))GeF2$\"3[OILU@4`OF27$ $\"3'*)***\\i_QQgF2$\"37B1d04))fMF27$$\"3@***\\7y%3TiF2$\"3%eoq>U&QuKF 27$$\"35****\\P![hY'F2$\"3?Zc*zG(**pIF27$$\"3kKLL$Qx$omF2$\"3a)RwT*[n( )GF27$$\"3!)*****\\P+V)oF2$\"3UNS'3k3Vp#F27$$\"3?mm\"zpe*zqF2$\"3!Q]Je Rx,_#F27$$\"3%)*****\\#\\'QH(F2$\"3h;$)pov!4L#F27$$\"3GKLe9S8&\\(F2$\" 3/n;Tv[z`@F27$$\"3R***\\i?=bq(F2$\"3M:Q,2Efp>F27$$\"3\"HLL$3s?6zF2$\"3 '=ZI6kW.z\"F27$$\"3a***\\7`Wl7)F2$\"3_'[j@)*)\\.;F27$$\"3#pmmm'*RRL)F2 $\"3$f>DJN[UU\"F27$$\"3Qmm;a<.Y&)F2$\"3*[a2Ipr:C\"F27$$\"3=LLe9tOc()F2 $\"3[t,_4&f41\"F27$$\"3u******\\Qk\\*)F2$\"3+:DWsS6a*)F/7$$\"3CLL$3dg6 <*F2$\"3)4luw#G#31(F/7$$\"3ImmmmxGp$*F2$\"3!4F3xj./P&F/7$$\"3A++D\"oK0 e*F2$\"3kP=Kq-PqNF/7$$\"3A++v=5s#y*F2$\"3Q(R#=><,\\=F/7$F*F(-%'COLOURG 6&%$RGBG$\"*++++\"!\")F(F(-%&STYLEG6#%%LINEG-%*THICKNESSG6#F+-F$6&7-F' 7$$\"3/+++++++5F2$\"3Q+++aD-N()F27$$\"35+++++++?F2$\"3'******RL&RdvF27 $$\"3))**************HF2$\"3[+++m?MbkF27$$\"3A+++++++SF2$\"3l*****z@Uy T&F27$$\"3++++++++]F2$\"3\"*******)y?XV%F27$$\"3w**************fF2$\"3 9+++nXa&\\$F27$$\"3a**************pF2$\"3!)******)yB:f#F27$$\"3U++++++ +!)F2$\"3()*****zC=Mr\"F27$$\"3A+++++++!*F2$\"3O*****R*)oW_)F/Fcz-Fez6 &FgzF(FhzF(F[[l-F`[l6#\"\"$-F$6&Fd[l-Fez6&FgzF(F(Fhz-F\\[l6#%&POINTGF_ [l-%'SYMBOLG6#%'CIRCLEG-%+AXESLABELSG6$Q\"x6\"Q!Ff_l-%%VIEWG6$;F(F*%(D EFAULTG" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curv e 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "The numerical sol ution can also be compared with the analytical solution using the util ity procedure " }{TEXT 0 14 "comparewithfcn" }{TEXT -1 5 " . . " } {HYPERLNK 17 "comparewithfcn" 1 "" "comparewithfcn" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "comparewithfcn(soln,g(x),x);" }}{PARA 6 "" 1 "" {TEXT -1 81 " 0 \+ 1 function val: 1 rel err: 0.00 00e-01" }}{PARA 6 "" 1 "" {TEXT -1 81 " .1 .873502255 f unction val: .873481691 rel err: 2.3543e-05" }}{PARA 6 "" 1 "" {TEXT -1 81 " .2 .755739533 function val: .75570548 \+ rel err: 4.5061e-05" }}{PARA 6 "" 1 "" {TEXT -1 81 " .3 \+ .645534207 function val: .645492624 rel err: 6.4420e-05" }}{PARA 6 "" 1 "" {TEXT -1 81 " .4 .541784222 function \+ val: .541740075 rel err: 8.1492e-05" }}{PARA 6 "" 1 "" {TEXT -1 81 " .5 .443452079 function val: .443409442 r el err: 9.6157e-05" }}{PARA 6 "" 1 "" {TEXT -1 81 " .6 .3 49554457 function val: .3495166 rel err: 1.0831e-04" }} {PARA 6 "" 1 "" {TEXT -1 81 " .7 .259152379 function va l: .259121838 rel err: 1.1786e-04" }}{PARA 6 "" 1 "" {TEXT -1 81 " .8 .171341825 function val: .171320455 rel \+ err: 1.2474e-04" }}{PARA 6 "" 1 "" {TEXT -1 81 " .9 .0852 44689 function val: .085233703 rel err: 1.2889e-04" }}{PARA 6 "" 1 "" {TEXT -1 79 " 1 0 function val: 0 rel err: infinity" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 6 "" 1 "" {TEXT -1 48 " Maximum relative error : 1.2889e-04" }}{PARA 6 "" 1 "" {TEXT -1 57 " obtained fo r the input value: .9 " }}{PARA 6 "" 1 "" {TEXT -1 67 " \+ excluding any cases where the function value is zero." }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 77 "Find a discrete numerical solution for the two-point boundary value p roblem: " }}{PARA 256 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "d^2*y /(d*x^2) = y;" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F'" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 0" "6#/-%\"yG6#\"\"!F'" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "y(1) = 1" "6#/-%\"yG6#\"\"\"F'" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{TEXT 0 6 "dsolve " }{TEXT -1 85 " can find an analytical solution, compare this solutio n with your numerical solution." }}{PARA 0 "" 0 "" {TEXT -1 43 "______ _____________________________________" }}{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 43 "____________ _______________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 " " {TEXT -1 77 "Find a discrete numerical solution for the two-point bo undary value problem: " }}{PARA 256 "" 0 "" {TEXT -1 4 " " } {XPPEDIT 18 0 "d^2*y/(d*x^2) = -y;" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(* $%\"xGF&F(!\"\",$F'F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 0" "6# /-%\"yG6#\"\"!F'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y(1) = 1" "6#/-%\"y G6#\"\"\"F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " } {TEXT 0 6 "dsolve" }{TEXT -1 85 " can find an analytical solution, com pare this solution with your numerical solution." }}{PARA 0 "" 0 "" {TEXT -1 43 "___________________________________________" }}{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 43 "___________________________________________" }}{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 }{RTABLE_HANDLES 145071484 145034656 145078608 145037784 145081868 145091764 145081988 }{RTABLE M7R0 I6RTABLE_SAVE/145071484X,%)anythingG6"6"[gl!"%!!!#1"%"%,&!"#"""*$%"hG""#!""F)"" !F.F)F'F)F.F.F)F'F)F.F.F)F'6" } {RTABLE M7R0 I6RTABLE_SAVE/145034656X*%)anythingG6"6"[gl!#%!!!"%"%!""""!F(F(6" } {RTABLE M7R0 I6RTABLE_SAVE/145078608X,%)anythingG6"6"[gl!"%!!!#\p"*"*$!++++5?!"*"""""!F+F+F+ F+F+F+F*F'F*F+F+F+F+F+F+F+F*F'F*F+F+F+F+F+F+F+F*F'F*F+F+F+F+F+F+F+F*F'F*F+F+F+F +F+F+F+F*F'F*F+F+F+F+F+F+F+F*F'F*F+F+F+F+F+F+F+F*F'F*F+F+F+F+F+F+F+F*F'6" } {RTABLE M7R0 I6RTABLE_SAVE/145037784X*%)anythingG6"6"[gl!#%!!!"*"*!""""!F(F(F(F(F(F(F(6" } {RTABLE M7R0 I6RTABLE_SAVE/145081868X*%'sfloatG6"6"[gl!#%!!!"*"*$"+_D-N()!#5$"+J`RdvF)$"+l?M bkF)$"+;A%yT&F)$"+)y?XV%F)$"+mXa&\$F)$"+)yB:f#F)$"+Z#=Mr"F)$"+%*)oW_)!#66" } {RTABLE M7R0 I6RTABLE_SAVE/145091764X,%)anythingG6"6"[gl!"%!!!#<"$"*$"#5!""$!++++5?!"*""!F'F *F'F'F*F'F'F*F'F'F*F'F'F*F'F'F*F'F'F*F'F-F*F'6" } {RTABLE M7R0 I6RTABLE_SAVE/145081988X*%)anythingG6"6"[gl!#%!!!"*"*$"+aD-N()!#5$"+M`RdvF)$"+m ?MbkF)$"+=A%yT&F)$"+*y?XV%F)$"+nXa&\$F)$"+*yB:f#F)$"+[#=Mr"F)$"+%*)oW_)!#66" }