{VERSION 6 0 "IBM INTEL LINUX" "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 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 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 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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT -1 74 "Evans p 57. A solution of the heat equation used in the proof of Thm 6. " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "restart: n:=1;\nv:=(-mu/(T+eps-t)^( n/2))*exp((x-y)^2/(4*(T+eps-t)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"nG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG,$*(%#muG\"\"\",( %\"TGF(%$epsGF(%\"tG!\"\"#F-\"\"#-%$expG6#*&,&%\"xGF(%\"yGF-F/,(*&\"\" %F(F*F(F(*&F9F(F+F(F(*&F9F(F,F(F-F-F(F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "heat_lhs:=diff(v,t)-diff(v,x,x);\nsimplify(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%)heat_lhsG,&*&#\"\"\"\"\"#F(*&,&#\"# @\"#?F(%\"tG!\"\"#!\"$F)-%$expG6#*&%\"xGF),&#F-\"\"&F(*&\"\"%F(F/F(F0F 0F(F(F0**F)F(F+#F0F)F8F0F3F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\" !" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "T:=1;\neps:=1/20;\nmu :=1;\ny:=0;\nsubs(t=0,v);\nsubs(t=T,v);\nr:=0.2;\nplot([subs(t=0,v),su bs(t=T,v)],x=-r..r);\nplot3d(v,x=-r..r,t=0..T,axes=boxed);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$epsG#\"\"\"\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#muG\"\" \"" }}{PARA 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