{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 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 3 "" 0 "" {TEXT -1 23 "Wave Equation Example. " }}{PARA 0 "" 0 "" {TEXT -1 28 "Consider the wave equation " } {XPPEDIT 18 0 "a^2*u[xx] = u[tt];" "6#/*&%\"aG\"\"#&%\"uG6#%#xxG\"\"\" &F(6#%#ttG" }{TEXT -1 8 " where " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "u; " "6#%\"uG" }{TEXT -1 12 "(0,t)=0 and " }{XPPEDIT 18 0 "u;" "6#%\"uG" }{TEXT -1 8 "(L,t)=0," }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u;" "6#%\"uG" }{TEXT -1 12 "(x,0)=f(x), " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u[t];" "6#&%\"uG6#%\"tG" }{TEXT -1 11 "(x,0) \+ = 0. " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 65 "Finding the and plotting the solution from initial displacement. " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 " We'll first define the initial displacement function f(x ) plotted below. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "L := 10 ;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG\"#5" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 8 "a := 2;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"aG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "f := x -> pi ecewise(x<5,x,x>=5,10-x);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf *6#%\"xG6\"6$%)operatorG%&arrowGF(-%*piecewiseG6&29$\"\"&F01F1F0,&\"#5 \"\"\"F0!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot (f(x),x=0..10,thickness=3);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 392 392 392 {PLOTDATA 2 "6&-%'CURVESG6$7U7$$\"\"!F)F(7$$\"3emmm;arz@!#=F+7$$\" 3[LL$e9ui2%F-F/7$$\"3nmmm\"z_\"4iF-F27$$\"39ommT&phN)F-F57$$\"3KLLe*=) H\\5!#Y2aF:$\"3%)***\\73QDf%F:7$$\"3Znm;zXu9cF:$\"3_KL$3 Ub_Q%F:7$$\"34+++]y))GeF:$\"3!*******\\@6rTF:7$$\"3H++]i_QQgF:$\"3q*** *\\PZhhRF:7$$\"3b++D\"y%3TiF:$\"3Y***\\(=_\"*ePF:7$$\"3+++]P![hY'F:$\" 3+++]i>&Q`$F:7$$\"3iKLL$Qx$omF:$\"3Pnmm;EiJLF:7$$\"3Y+++v.I%)oF:$\"3`* ****\\i*p:JF:7$$\"3?mm\"zpe*zqF:$\"3zLL3-8/?HF:7$$\"3;,++D\\'QH(F:$\"3 %))****\\2Nhq#F:7$$\"3%HL$e9S8&\\(F:$\"30nmT&)f'[]#F:7$$\"3s++D1#=bq(F :$\"3G***\\Pz\"[%H#F:7$$\"3\"HLL$3s?6zF:$\"35nmm\"z#z)3#F:7$$\"3a***\\ 7`Wl7)F:$\"3Y++voaXt=F:7$$\"3enmmm*RRL)F:$\"3SKLLL+1m;F:7$$\"3%zmmTvJg a)F:$\"31KL$eCoRX\"F:7$$\"3]MLe9tOc()F:$\"3]lmT&oKOC\"F:7$$\"31,++]Qk \\*)F:$\"3%*)*****\\hN]5F:7$$\"3![LL3dg6<*F:$\"3)>lm;H%R)G)F-7$$\"3%ym mmw(Gp$*F:$\"3^@LLLB72jF-7$$\"3C++D\"oK0e*F:$\"3i(***\\(=tY>%F-7$$\"35 ,+v=5s#y*F:$\"3#*))**\\7)*ys@F-7$$\"#5F)F(-%'COLOURG6&%$RGBG$F[x!\"\"F (F(-%+AXESLABELSG6$Q\"x6\"Q!Ffx-%*THICKNESSG6#\"\"$-%%VIEWG6$;F(Fjw%(D EFAULTG" 1 2 0 1 10 3 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curv e 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Now we'll plot the odd ex tension of f(x) and define it as oddext(x)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "oddext := x -> piecewise(x<0,-f(-x),x>=0,f(x));\n " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'oddextGf*6#%\"xG6\"6$%)operator G%&arrowGF(-%*piecewiseG6&29$\"\"!,$-%\"fG6#,$F0!\"\"F71F1F0-F46#F0F(F (F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(oddext(x),x=-10 ..10,thickness=3);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 392 392 392 {PLOTDATA 2 "6&-%'CURVESG6$7W7$$!#5\"\"!$F*F*7$$!3!pmmm\"p0k&*!#<$!3#4 LLL$3VfV!#=7$$!3uKL$3s%HaF/$!3%QLL$3y_qXF/7$$!3$ GLL3x&y8_F/$!3;nm;HU@'y%F/7$$!3]******\\$*4)*\\F/Feo7$$!3g*******HF:y% F/Fho7$$!3o******\\_&\\c%F/F[p7$$!3%)******\\1aZTF/F^p7$$!3Imm;/#)[oPF /Fap7$$!3%HLLL=exJ$F/Fdp7$$!3lKLLL2$f$HF/Fgp7$$!3%)****\\PYx\"\\#F/Fjp 7$$!3gLLLL7i)4#F/F]q7$$!3o)***\\P'psm\"F/F`q7$$!3?****\\74_c7F/Fcq7$$! 3M:LL$3x%z#)F2Ffq7$$!3()HLL3s$QM%F2Fiq7$$!3]^omm;zr)*!#?F\\r7$$\"3fVLL ezw5VF2F`r7$$\"3-.++v$Q#\\\")F2Fcr7$$\"3%\\LL$e\"*[H7F/Ffr7$$\"3=++++d xd;F/Fir7$$\"3e+++D0xw?F/F\\s7$$\"35,+]i&p@[#F/F_s7$$\"3++++vgHKHF/Fbs 7$$\"3ElmmmZvOLF/Fes7$$\"3%4+++v+'oPF/Fhs7$$\"3UKL$eR<*fTF/F[t7$$\"3K- ++])Hxe%F/F^t7$$\"37MLeR*)**)y%F/Fat7$$\"3!fmm\"H!o-*\\F/Fdt7$$\"3nLL$ 3A_1?&F/$\"3Kmm;zxM*z%F/7$$\"3X,+]7k.6aF/$\"3c)***\\(ej*)e%F/7$$\"3#em mmT9C#eF/$\"3=MLL$e&exTF/7$$\"33****\\i!*3`iF/$\"3%4++v$4\"pu$F/7$$\"3 ;NLLL*zym'F/$\"3#[mmm1?@L$F/7$$\"3'eLL$3N1#4(F/$\"38kmm\"\\Oz!HF/7$$\" 3,pm;HYt7vF/$\"3+JL$3Pls[#F/7$$\"37-+++xG**yF/$\"3(y******H725#F/7$$\" 3gpmmT6KU$)F/$\"3SILLe)ywl\"F/7$$\"3qNLLLbdQ()F/$\"3IkmmmWUh7F/7$$\"3[ ++]i`1h\"*F/$\"3C&****\\PY$*Q)F27$$\"3A-+]P?Wl&*F/$\"3'y(***\\izbM%F27 $$\"#5F*F+-%'COLOURG6&%$RGBG$Fdx!\"\"F+F+-%+AXESLABELSG6$Q\"x6\"Q!F_y- %*THICKNESSG6#\"\"$-%%VIEWG6$;F(Fcx%(DEFAULTG" 1 2 0 1 10 3 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "No we'll extend this over three periods with fundamental \+ period = 20. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "h := x -> \+ piecewise(x<-10,oddext(x+20),x>10,oddext(x-20),oddext(x));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%* piecewiseG6'29$!#5-%'oddextG6#,&F0\"\"\"\"#?F62\"#5F0-F36#,&F0F6F7!\" \"-F36#F0F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "plot(h(x ),x=-20..20,thickness=3);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 392 392 392 {PLOTDATA 2 "6&-%'CURVESG6$7en7$$!#?\"\"!$F*F*7$$!3QLLL$Q6G\">!#;$\"3& =mmmmh)=()!#=7$$!3bmm;M!\\p$=F/$\"3]MLLe'40j\"!#<7$$!3ILLL))Qj^(>%F87$$!3\"*****\\_(e1a\"F/$\"3%4++]Z7Mf%F87$$!3KLL$3yO5]\"F/$\"3 %omm;>K'*)\\F87$$!3#ommT_5+Y\"F/$\"37ommT_5+YF87$$!38++]nU)*=9F/$\"3G, ++vE%)*=%F87$$!3FLL$3WDTL\"F/$\"3lKLL3WDTLF87$$!3))****\\d(Q&\\7F/$\"3 u)****\\d(Q&\\#F87$$!3ammmc4`i6F/$\"3Elmmm&4`i\"F87$$!3CLLLQW*e3\"F/$ \"3SBLLLQW*e)F27$$!3+*******p)>'***F8$!3l15+++I,QF)7$$!3Q*******\\5*H \"*F8$!3E1+++]*3q)F27$$!3m*******H\"3&H)F8$!3M++++(=\\q\"F87$$!3gKLL3k (p`(F8$!3Snmm\"fBIY#F87$$!3*emmmO;bj'F8$!35MLLLO[kLF87$$!3Ilmmm9'=(eF8 $!3pMLLL&Q\"GTF87$$!3\\KL$3P0xU&F8$!3_nm;HYHsXF87$$!3o*****\\F\\N)\\F8 F^r7$$!3WLL$3(eR!f%F8Far7$$!3AnmmmCC(>%F8Fdr7$$!3O(****\\FRXL$F8Fgr7$$ !3S)****\\#=/8DF8Fjr7$$!31jmm;a*el\"F8F]s7$$!3ufmm;Wn(o)F2F`s7$$!3IqLL L$eV(>!#>Fcs7$$\"3<(omm\"f`@')F2Fgs7$$\"3h+++vw%)H;F8Fjs7$$\"3))pmm;$y *eCF8F]t7$$\"3P++++9b:LF8F`t7$$\"3=,++]5a`TF8Fct7$$\"3o,+](3S*eXF8Fft7 $$\"3?-++D\"RV'\\F8Fit7$$\"34,+]PcY9aF8$\"3!*)***\\iV`&e%F87$$\"3++++] @fkeF8$\"3++++]ySNTF87$$\"3_ILLL&4Nn'F8$\"3[pmmm/\\ELF87$$\"3*=+++],s` (F8$\"37)******\\)ziCF87$$\"3%[mm;zM)>$)F8$\"3;NLL3_;!o\"F87$$\"3m/+++ (fa<*F8$\"3U`******HSX#)F27$$\"3!=LL$eg`!)**F8$\"3_?ommTRY>Fes7$$\"3I+ +]#G2A3\"F/$!3$*G++]#G2A)F27$$\"3;LLL$)G[k6F/$!3jJLLL)G[k\"F87$$\"3#)* ***\\7yh]7F/$!3:)****\\7yh]#F87$$\"3/nmm')fdL8F/$!3Nqmmm)fdL$F87$$\"3= nmm,FT=9F/$!3urmm;q7%=%F87$$\"3]++v8)z/Y\"F/$!3)[++v8)z/YF87$$\"3!QLLe #pa-:F/$!3)>mm;uIX(\\F87$$\"37nm\"HB-7a\"F/$!3')GL$3nxze%F87$$\"3W+++S v&)z:F/$!3t&******fC9?%F87$$\"3#RLL$GUYo;F/$!3zgmm;xN:LF87$$\"39nmm1^r ZF/$!3qb****\\#f6p)F27$$\"#?F*F+-%'COLOURG6&%$RGBG$\"#5!\"\"F+F+-%+AXE SLABELSG6$Q\"x6\"Q!F]\\l-%*THICKNESSG6#\"\"$-%%VIEWG6$;F(F`[l%(DEFAULT G" 1 2 0 1 10 3 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" } }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Now that we have the odd extens ion over three periods, we define the solution by " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "sol := (x,t) -> 1/2 * (h(x-a*t) + h(x+a*t)) ;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solGf*6$%\"xG%\"tG6\"6$%)ope ratorG%&arrowGF),&*&#\"\"\"\"\"#F0-%\"hG6#,&9$F0*&%\"aGF09%F0!\"\"F0F0 *&F/F0-F36#,&F6F0F7F0F0F0F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "and plot it over the domain of intereste x = 0..10. You can change the time. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "plot(sol(x, 10),x=0..10,thickness=3);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 392 392 392 {PLOTDATA 2 "6&-%'CURVESG6$7U7$$\"\"!F)F(7$$\"3emmm;arz@!#=F+7$$\"3[LL $e9ui2%F-F/7$$\"3nmmm\"z_\"4iF-F27$$\"39ommT&phN)F-F57$$\"3KLLe*=)H\\5 !#Y2aF:$\"3%)***\\73QDf%F:7$$\"3Znm;zXu9cF:$\"3_KL$3Ub_Q %F:7$$\"34+++]y))GeF:$\"3!*******\\@6rTF:7$$\"3H++]i_QQgF:$\"3q****\\P ZhhRF:7$$\"3b++D\"y%3TiF:$\"3Y***\\(=_\"*ePF:7$$\"3+++]P![hY'F:$\"3+++ ]i>&Q`$F:7$$\"3iKLL$Qx$omF:$\"3Pnmm;EiJLF:7$$\"3Y+++v.I%)oF:$\"3`***** \\i*p:JF:7$$\"3?mm\"zpe*zqF:$\"3zLL3-8/?HF:7$$\"3;,++D\\'QH(F:$\"3%))* ***\\2Nhq#F:7$$\"3%HL$e9S8&\\(F:$\"30nmT&)f'[]#F:7$$\"3s++D1#=bq(F:$\" 3G***\\Pz\"[%H#F:7$$\"3\"HLL$3s?6zF:$\"35nmm\"z#z)3#F:7$$\"3a***\\7`Wl 7)F:$\"3Y++voaXt=F:7$$\"3enmmm*RRL)F:$\"3SKLLL+1m;F:7$$\"3%zmmTvJga)F: $\"31KL$eCoRX\"F:7$$\"3]MLe9tOc()F:$\"3]lmT&oKOC\"F:7$$\"31,++]Qk\\*)F :$\"3%*)*****\\hN]5F:7$$\"3![LL3dg6<*F:$\"3)>lm;H%R)G)F-7$$\"3%ymmmw(G p$*F:$\"3^@LLLB72jF-7$$\"3C++D\"oK0e*F:$\"3i(***\\(=tY>%F-7$$\"35,+v=5 s#y*F:$\"3#*))**\\7)*ys@F-7$$\"#5F)F(-%'COLOURG6&%$RGBG$F[x!\"\"F(F(-% +AXESLABELSG6$Q\"x6\"Q!Ffx-%*THICKNESSG6#\"\"$-%%VIEWG6$;F(Fjw%(DEFAUL TG" 1 2 0 1 10 3 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 174 "With regards to time, the Fou rier solution shows that there is a fundamental period of 2 L / a. 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