{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 } {CSTYLE "" -1 256 "" 1 18 0 0 0 0 0 0 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 "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 {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 55 "An interesting fa mily of solutions of the heat equation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 700 "restart:\nmakesol:=proc(m0) # GENERATE TH E SOLUTION u of x-DEGREE m\n local m,k,s,i,check,eqns,sol,ii;\n m:=m 0;\n k:=0;\n s:=x^m;\n for i from 1 to infinity do\n m:=m-2;\n \+ if m < 0 then break; end if;\n k:=k+1;\n s:=s+a[k]*t^k*x^m ;\n end do:\n check:=simplify(diff(s,x,x)-diff(s,t));\n eqns:=\{seq (subs(x=j,t=k-j,check=0),j=1..k)\};\n sol:=solve(eqns);\n eval(s,sol );\nend proc:\n\ncheckhe:=proc(u) # CHECK u SOLVES HEAT E QUATION\n simplify(diff(u,x,x)-diff(u,t));\nend:\n\nget_ratio:=proc(u ) # EVALUATE THE RATIO FOR THIS u\n local ux,uxcorner,L1; \n ux:=diff(sol,x);\n uxcorner:=subs(x=1/2,t=1,ux);\n L1:=2*int(int (sol,t=0..1),x=0..1);\n uxcorner/L1;\nend proc:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 364 "u:=makesol(4);\nnt:=30: dt:=evalf(1/nt):\n plot3d(abs(u),x=-1..1,t=0..1,axes=boxed,labels=[\"x\",\"t\",\"|u|\"]); \nux:=diff(u,x);\nplot3d(abs(ux),x=-1..1,t=3/4..1,axes=boxed,labels=[ \"x\",\"t\",\"|u_x|\"]);\nplots[display](seq(plot(subs(t=it*dt,abs(u)) ,x=-1..1,thickness=3),it=0..nt),\n seq(plot(subs(t=nt*dt ,abs(u)),x=-1..1,thickness=3),it=0..nt/2),insequence=true);\n\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG,(*$)%\"xG\"\"%\"\"\"F**(\"#7F*% 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