{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{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 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 40 "S := 1/sqrt(4*Pi*k*t)*exp(-x^2/(4*k*t));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "k:=1;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 57 "plot3d(S,x=-8..8,t=0..8,axes=frame,orientation =[-45,45]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 97 "The following computation produces an animation of the \+ fundamental solution to the heat equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "animate(S,x=-8..8 ,t=0..8, frames=160,numpoints=100);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Now we compare 2 solutions." }} {PARA 0 "" 0 "" {TEXT -1 62 " One is the solution to the IVP with IC \+ u=phi(x)=e^(-x) H(x)." }}{PARA 0 "" 0 "" {TEXT -1 144 " The other is \+ the solution from a point source of strength \\int_\{-\\infty)^\\infty \\phi(x) dx, centered at x=ln(2), which is the centroid of phi." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "u := (.5+.5*erf(x/sqrt(4*t)-sqrt(t)))*exp(t-x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "G:=subs(x=x-ln(2),S);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "p1:=animate(G,x=-7..9,t=0..8, frame s=160,numpoints=100,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "p2:=animate(u,x=-7..9,t=0..8, frames=160,numpoints=100,color=b lue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display(\{p1,p2\}) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "14" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }