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}{PSTYLE "" 0 274 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 277 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 281 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 284 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 286 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 287 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 289 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 302 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 303 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 305 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 306 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 259 "" 0 "" {TEXT -1 244 "Soln_u := pdsolve(PDE); Soln_u := pdsolve(PDE);Fourier Series for Heat Eq:\nu_t=K*u_\{xx\};u(0 ,t)=u(L,t)=0; u(x,0)=\{2*A*x/L,0 " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 0 14 "pde_u := diff(" }{TEXT -1 0 "" }{MPLTEXT 1 0 30 "u(x,t),t)-K*diff(u(x,t),x,x)=0" }{TEXT -1 0 "" }{TEXT 259 0 "" }{MPLTEXT 1 0 1 ";" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%&pde_uG/,&-%%diffG6$-%\"uG6$%\"xG%\"tGF.\"\"\"*&%\"KGF/-F(6$F*-%\"$G 6$F-\"\"#F/!\"\"\"\"!" }}}{EXCHG {PARA 263 "> " 0 "" {MPLTEXT 1 0 25 " Soln_u := pdsolve(pde_u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Soln_u G-%'&whereG6$/-%\"uG6$%\"xG%\"tG*&-%$_F1G6#F,\"\"\"-%$_F2G6#F-F27#<$/- %%diffG6$F3F-*(%\"KGF2&%#_cG6#F2F2F3\"\"\"/-F:6$F/-%\"$G6$F,\"\"#*&F>F AF/FA" }}}{EXCHG {PARA 265 "" 0 "" {TEXT -1 0 "" }{TEXT 256 0 "" } {TEXT 257 0 "" }{TEXT 258 0 "" }{MPLTEXT 1 0 33 "ode_x:=diff(X(x),x,x) +c^2*X(x)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode_xG/,&-%%diffG6$ -%\"XG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&)%\"cGF1\"\"\"F*F2F2\"\"!" }}} {EXCHG {PARA 267 "> " 0 "" {MPLTEXT 1 0 33 "ode_t:=diff(T(t),t)+c^2*K* T(t)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode_tG/,&-%%diffG6$-%\"T G6#%\"tGF-\"\"\"*()%\"cG\"\"#\"\"\"%\"KGF.F*F.F.\"\"!" }}}{EXCHG {PARA 269 "> " 0 "" {MPLTEXT 1 0 184 "### WARNING: `dsolve` has been e xtensively rewritten, many new result forms can occur and options are \+ slightly different, see help page for details\nsoln_x:=dsolve(\{ode_x, X(0)=0\},X(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'soln_xG/-%\"XG6# %\"xG*&%$_C1G\"\"\"-%$sinG6#*&%\"cGF,F)F,F," }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 37 "cn:=Pi*n/L;\nXn:=(x,n)->sin(Pi*n*x/L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cnG*&*&%#PiG\"\"\"%\"nGF(\"\"\"%\"LG!\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#XnGR6$%\"xG%\"nG6\"6$%)operator G%&arrowGF)-%$sinG6#*&*(%#PiG\"\"\"9%F39$F3\"\"\"%\"LG!\"\"F)F)F)" }}} {EXCHG {PARA 274 "> " 0 "" {MPLTEXT 1 0 191 "c:=cn;\n### WARNING: `dso lve` has been extensively rewritten, many new result forms can occur a nd options are slightly different, see help page for details\nsoln_t:= dsolve(\{ode_t,T(0)=1\},T(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" cG*&*&%#PiG\"\"\"%\"nGF(\"\"\"%\"LG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'soln_tG/-%\"TG6#%\"tG-%$expG6#,$*&**)%#PiG\"\"#\"\"\")%\"nGF2 F3%\"KG\"\"\"F)F7F3*$)%\"LG\"\"#F3!\"\"!\"\"" }}}{EXCHG {PARA 277 "> \+ " 0 "" {MPLTEXT 1 0 79 "f1:=x->2*A*x/L;\nf2:=x->2*A*(L-x)/L;\nf:=x->2* A*x/L+2*A*(L-2*x)*Heaviside(x-L/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#f1GR6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&*&%\"AG\"\"\"9$F0\"\"\"% \"LG!\"\"\"\"#F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2GR6#%\"xG 6\"6$%)operatorG%&arrowGF(,$*&*&%\"AG\"\"\",&%\"LGF09$!\"\"F0\"\"\"F2! \"\"\"\"#F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$ %)operatorG%&arrowGF(,&*&*&%\"AG\"\"\"9$F0\"\"\"%\"LG!\"\"\"\"#*(F/F2, &F3F0F1!\"#F0-%*HeavisideG6#,&F1F0F3#!\"\"F5F0F5F(F(F(" }}}{EXCHG {PARA 281 "> " 0 "" {MPLTEXT 1 0 94 "b1:=int(x*sin(Pi*n*x/L),x=0..L/2) ;\nb2:=int((L-x)*sin(Pi*n*x/L),x=L/2..L);\n#b:=(b1+b2)*4*A/L^2;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b1G,$*&*&)%\"LG\"\"#\"\"\",&-%$sinG 6#,$*&%#PiG\"\"\"%\"nGF3#F3F*!\"#*(F2F+F4F+-%$cosGF/F3F3F3F+*&)F2\"\"# F+)F4\"\"#F+!\"\"#!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b2G,$* &*&)%\"LG\"\"#\"\"\",(-%$sinG6#*&%#PiG\"\"\"%\"nGF2!\"#*(F1F+F3F+-%$co sG6#,$F0#F2F*F2F2-F.F8F*F2F+*&)F1\"\"#F+)F3\"\"#F+!\"\"F:" }}}{EXCHG {PARA 284 "> " 0 "" {MPLTEXT 1 0 16 "b:=`b`:\nn:=`n`:\n" }}{PARA 258 " " 0 "" {TEXT -1 105 "Next: Protect Aliases and use procedure to comput e Fourier Coefficients for\nfirst 10 integer values of n:" }}{PARA 306 "> " 0 "" {MPLTEXT 1 0 62 "b:=proc(n) option remember;\n (int(x*s in(Pi*n*x/L),x=0..L/2) \n" }{TEXT -1 0 "" }{MPLTEXT 1 0 85 " + int(( L-x)*sin(Pi*n*x/L),x=L/2..L))*4*A/L^2\nend:\nB:=array([seq([b(n)],n=1. .10)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matrixG6#7,7#,$*&% \"AG\"\"\"*$)%#PiG\"\"#F-!\"\"\"\")7#\"\"!7#,$F+#!\")\"\"*F47#,$F+#F3 \"#DF47#,$F+#F9\"#\\F47#,$F+#F3\"#\")F4" }}}{EXCHG {PARA 286 "> " 0 " " {MPLTEXT 1 0 103 "FSeries:=proc(k) option remember;\n FSeries(k-1) +b(k)*sin(Pi*k*x/L)\nend:\nFSeries(1):=b(1)*sin(Pi*x/L):" }}}{EXCHG {PARA 287 "> " 0 "" {MPLTEXT 1 0 65 "k_values:=1,3,5,7,9:\nFS:=array([ seq([k,FSeries(k)],k=k_values)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#FSG-%'matrixG6#7'7$\"\"\",$*&*&%\"AGF*-%$sinG6#*&*&%#PiGF*%\"xGF*\" \"\"%\"LG!\"\"F*F6*$)F4\"\"#F6F8\"\")7$\"\"$,&F,F<*&*&F.F6-F06#,$F2F>F *F6*$)F4\"\"#F6F8#!\")\"\"*7$\"\"&,(F,F " 0 "" {MPLTEXT 1 0 151 "L:=1 ;A:=1;\nfor k in k_values do \nk_value:=k;\nplot(\{f(x),FSeries(k)\},x =0..L,thickness=3,title=`FSeries(k)`,labels=[x,X],labelfont=[COURIER,B OLD,12])\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG\"\"\"" }} {PARA 11 "" 1 "" 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