{VERSION 4 0 "SUN SPARC SOLARIS" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }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 "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 O utput" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }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 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 257 "" 0 "" {TEXT -1 11 "Math 2280-2" }}{PARA 258 "" 0 " " {TEXT -1 16 "Tuesday April 17" }}{PARA 259 "" 0 "" {TEXT 256 36 "Fou rier Series and the Heat Equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 256 "These commands will be useful for the fi rst part of your final 2280 Maple assignment - which is to do the sec tion 9.5 computing project. The second part of your assignment is to \+ do the section 9.6 computing project. You may make your own solution \+ files." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "We will also use this output in class today, April 17." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart :with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name change coords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 180 "a:=n->2/L*int(f(x)*cos(Pi*n/L*x),x=0..L):\n #for a cosine serie s! Now use x because it will\n #be space variable\nb:=n->2/L*int(f( x)*sin(Pi*n/L*x),x=0..L):\n #for a sine series!" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 129 "cossum:=a(0)/2 + sum(a(n)*cos(Pi*n/L*x),n=1 ..N):\n #cosine series\nsinsum:=sum(b(n)*sin(Pi*n/L*x),n=1..N):\n \+ #sine series \n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 223 "he atsol1:=sum(b(n)*sin(Pi*n/L*x)*exp(-n^2*Pi^2*k*t/L^2),n=1..N):\n #bo undary conditions u(0,t)=u(L,t)=0\nheatsol2:=a(0)/2 + \n sum(a(n) *cos(Pi*n/L*x)*exp(-n^2*Pi^2*k*t/L^2),n=1..N):\n #insulated end boun dary conditions" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "#Exampl e 2 page 636\nf:=x->100:\nL:=50:\nN:=50: #take many terms because the \+ square wave\n #is hard to approximate\nk:=0.15: #iron" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "plot3d(heatsol1,x=0..50,t=0..1800,c olor=black,\nstyle=wireframe, title=\"Figure 9.5.4\");" }}{PARA 13 "" 1 "" {GLPLOT3D 553 334 334 {PLOTDATA 3 "6&-%%GRIDG6&;$\"\"!F($\"#]F(;F 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782A174BE61A9B99523043CBE60FCCE53EFF91BBE604FACDED0CFBABE5F49BE271EA3A BBE5DFBFFC9AEDEE1BE5CB812BB42DBE0BE5B7F14551EF5B6BE5A5193BD5527EBBE592 FC2B8724612BE58199665589500BE570EDD215F3BFCBE560F4D23169667BE551A8E318 92C88BE54304014A9F3C0BE534FFED3FCAFED-%&COLORG6&%$RGBGF'F'F'-%+AXESLAB ELSG6%%\"xG%\"tGQ!6\"-%&TITLEG6#Q-Figure~9.5.46\"-%&STYLEG6#%%LINEG" 1 6 0 1 10 0 2 1 1 2 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {PARA 0 "" 0 "" {TEXT -1 113 "Computer check of computations on page 6 37\n(a) temperature at midpoint after 1800 seconds (= 6 minutes), for \+ iron" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "evalf(subs(\{t=1800, x=25\},heatsol1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*p(*[Q%!\") " }}}{PARA 0 "" 0 "" {TEXT -1 30 " (b) temperature for concrete:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "k:=.005:\nevalf(subs(\{t=180 0,x=25\},heatsol1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+<********! \")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "evalf(subs(\{t=60*60 ,x=25\},heatsol1));\n #an hour later" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+@=Q****!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "evalf (subs(\{t=6*60*60,x=25\},heatsol1));\n #6 hours later - think of bric k buildings!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+gmF@#)!\")" }}} {PARA 0 "" 0 "" {TEXT -1 99 "We could see what the temperature looked \+ like throughout the concrete slab at this last time value:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "slice:=z->evalf(subs(\{t=6*60*60,x= z\},heatsol1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sliceGR6#%\"zG6 \"6$%)operatorG%&arrowGF(-%&evalfG6#-%%subsG6$<$/%\"xG9$/%\"tG\"&+;#%) heatsol1GF(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "plot(sli ce(x),x=0..50,color=black,\ntitle=\"temperature profile six hours late r, concrete\");" }}{PARA 13 "" 1 "" {GLPLOT2D 406 138 138 {PLOTDATA 2 "6'-%'CURVESG6#7S7$$\"\"!F)F(7$$\"3RLLL3x&)*3\"!#<$\"3c>_xAAoueF-7$$\" 3zmm\"H2P\"Q?F-$\"3c\"ysc_Kf4\"!#;7$$\"3XLL$eRwX5$F-$\"3hbueaS$=m\"F57 $$\"3pEDGbOnFF57 $$\"34LLeR-/PiF-$\"3?z]14sMfKF57$$\"3;++DcmpisF-$\"3*)[?kCpo_PF57$$\"3 uLLe*)>VB$)F-$\"3;?&e@0hLC%F57$$\"3n++DJbw!Q*F-$\"3`FWz]l[5ZF57$$\"3%o mmTIOo/\"F5$\"3p'*\\q'H))f;&F57$$\"3]LL3_>jU6F5$\"31\"fXm'QoWbF57$$\"3 E++]i^Z]7F5$\"3))z%Rc%zGWfF57$$\"3/++](=h(e8F5$\"3;dHO!G1dJ'F57$$\"3A+ +]P[6j9F5$\"3AXGg+1DWmF57$$\"3ZL$e*[z(yb\"F5$\"3)HO@%3/(o\"pF57$$\"3*p mmTXg0n\"F5$\"3p,/Y*p?%3sF57$$\"3mmmm;t,mFuF57$$\"3=+] iSj0x=F5$\"3Fn9[y+n[wF57$$\"3#omm;pW`(>F5$\"3-$yb![,#[\"yF57$$\"3L+]i! f#=$3#F5$\"3f*H&\\&R*ekzF57$$\"36+](=xpe=#F5$\"3+oz%\\pxg#F5$\"3U:qTmS5/#)F57$$\" 3D+]Pf4t.FF5$\"33S`pER\"*f\")F57$$\"3ZLLe*Gst!GF5$\"3/u&\\$=^j\"3)F57$ $\"39+++DRW9HF5$\"3LvItX<]nzF57$$\"3:++DJE>>IF5$\"3Q4B[#pEK#yF57$$\"35 +]i!RU07$F5$\"3!3>.$>[1`wF57$$\"3#)***\\(=S2LKF5$\"3q&>L@\\[\"HuF57$$ \"3nmmm\"p)=MLF5$\"37tPtXr&o>(F57$$\"3U++](=]@W$F5$\"3&*p;MaJz;pF57$$ \"35L$e*[$z*RNF5$\"3ymWqqE$\\j'F57$$\"3e++]iC$pk$F5$\"3mptq%pTpH'F57$$ \"3Rm;H2qcZPF5$\"3HI***yWs7&fF57$$\"3X+]7.\"fF&QF5$\"3X,f%pQ]Bc&F57$$ \"3amm;/OgbRF5$\"3*H-K!o;1c^F57$$\"3I+]ilAFjSF5$\"3;Z=SjUn/ZF57$$\"3(R LLL)*pp;%F5$\"3VU()\\n%okC%F57$$\"3WLL3xe,tUF5$\"3XNMuh31cPF57$$\"3Vn; HdO=yVF5$\"3bfd^Pt5]KF57$$\"3`+++D>#[Z%F5$\"3c%fvFcH+x#F57$$\"3(om;aG! e&e%F5$\"3#p8M\"HRg/AF57$$\"3vLLL$)Qk%o%F5$\"3hR0!*\\$Gwo\"F57$$\"3m+] iSjE!z%F5$\"3ydn&=#p`F6F57$$\"3u+]P40O\"*[F5$\"3xj:'y/`g&eF-7$$\"#]F)$ !3%=HtEMp'RL!#D-%+AXESLABELSG6$Q\"x6\"Q!6\"-%&TITLEG6#QNtemperature~pr ofile~six~hours~later,~concrete6\"-%'COLOURG6&%$RGBGF)F)F)-%%VIEWG6$;F (Fbz%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 99 "So, if you wanted to, you could also trace the temperature at the midpoint, as time varies..... .. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Example 3, page 640:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "f:=x->4*(x -2*(Heaviside(x-25))*(x-25)):\np lot(f(x),x=0..50,color=black);" }}{PARA 13 "" 1 "" {GLPLOT2D 228 114 114 {PLOTDATA 2 "6&-%'CURVESG6#7U7$$\"\"!F)F(7$$\"3RLLL3x&)*3\"!#<$\"3 eLLLL3VfVF-7$$\"3zmm\"H2P\"Q?F-$\"3VB$)F-$\"3]LL$ezs$HLF:7$$\"3n++DJbw!Q*F-$\"3F++ ]7iI_PF:7$$\"3%ommTIOo/\"F:$\"3Pnmm;_M(=%F:7$$\"3]LL3_>jU6F:$\"3,MLL3y _qXF:7$$\"3E++]i^Z]7F:$\"3/,++]1!>+&F:7$$\"3/++](=h(e8F:$\"39+++]Z/NaF :7$$\"3A++]P[6j9F:$\"3)3+++NfC&eF:7$$\"3ZL$e*[z(yb\"F:$\"3)QLLez6:B'F: 7$$\"3*pmmTXg0n\"F:$\"3&zmmm\"=C#o'F:7$$\"3mmmm;t,mF:$\"3Fnmmm(y8!zF:7$$ \"3L+]i!f#=$3#F:$\"3K,+]i.tK$)F:7$$\"36+](=xpe=#F:$\"3W++](3zMu)F:7$$ \"3-nm\"H28IH#F:$\"36omm\"H_?<*F:7$$\"3%om\"zpSS\"R#F:$\"3Pnm;zihl&*F: 7$$\"3?+v$41oWW#F:$\"3z++vVA(yx*F:7$$\"3cLL3_?`(\\#F:$\"3AMLL3#G,***F: 7$$\"3dL3_D1l_DF:$\"3slm\"z\\(R*y*F:7$$\"3fL$e*)>pxg#F:$\"3klm;/K#*o&* F:7$$\"3D+]Pf4t.FF:$\"3**)***\\ih2&=*F:7$$\"3ZLLe*Gst!GF:$\"38mmmT3^q( )F:7$$\"39+++DRW9HF:$\"3Y*******HCAM)F:7$$\"3:++DJE>>IF:$\"3T*****\\ZH K#zF:7$$\"35+]i!RU07$F:$\"3i****\\P/$y^(F:7$$\"3#)***\\(=S2LKF:$\"3r++ +DRqnqF:7$$\"3nmmm\"p)=MLF:$\"3KLLLL_CjmF:7$$\"3U++](=]@W$F:$\"3M)**** *\\#*RJiF:7$$\"35L$e*[$z*RNF:$\"3enm;/E3SeF:7$$\"3e++]iC$pk$F:$\"3n(** ***\\,F7aF:7$$\"3Rm;H2qcZPF:$\"3YML$3(>t4]F:7$$\"3X+]7.\"fF&QF:$\"3?)* **\\(ej*)e%F:7$$\"3amm;/OgbRF:$\"3$QLLLe&exTF:7$$\"3I+]ilAFjSF:$\"3!)) ***\\P4\"pu$F:7$$\"3(RLLL)*pp;%F:$\"37kmmm+7KLF:7$$\"3WLL3xe,tUF:$\"3E mmm\"\\Oz!HF:7$$\"3Vn;HdO=yVF:$\"3GIL$3Pls[#F:7$$\"3`+++D>#[Z%F:$\"3(y ******H725#F:7$$\"3(om;aG!e&e%F:$\"3`KLLe)ywl\"F:7$$\"3vLLL$)Qk%o%F:$ \"3,lmmmWUh7F:7$$\"3m+]iSjE!z%F:$\"3#R(***\\PY$*Q)F-7$$\"3u+]P40O\"*[F :$\"3uq***\\izbM%F-7$$\"#]F)F(-%+AXESLABELSG6$Q\"x6\"Q!6\"-%'COLOURG6& %$RGBGF)F)F)-%%VIEWG6$;F(F\\[l%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 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