{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 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 -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 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 -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 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Demonstration of numerical solution of " }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 21 " " }{XPPEDIT 18 0 "Di ff(u,x,x) + Diff(u,y,y)=x/y+y/x" "/,&-%%DiffG6%%\"uG%\"xGF(\"\"\"-F%6% F'%\"yGF,F),&*&F(F)F,!\"\"F)*&F,F)F(F/F)" }{TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT -1 32 " " }{XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "[1,2]" "7\$\"\"\"\" \"#" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "y" "I\"yG6\"" }{TEXT -1 7 " in " }{XPPEDIT 18 0 "[1,2]" "7\$\"\"\"\"\"#" }{TEXT -1 17 "\n \+ " }{XPPEDIT 18 0 "u(1,y)=y*ln(y)" "/-%\"uG6\$\"\"\"%\"yG*&F' F&-%#lnG6#F'F&" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "u(2,y)=2*y*ln(2*y) " "/-%\"uG6\$\"\"#%\"yG*(F&\"\"\"F'F)-%#lnG6#*&F&F)F'F)F)" }{TEXT -1 5 ", " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "u(x,1)=x*ln(x)" "/-%\"uG6\$%\" xG\"\"\"*&F&F'-%#lnG6#F&F'" }{TEXT -1 0 "" }{TEXT -1 5 ", " } {XPPEDIT 18 0 "u(x,2)=x*ln(4*x^2)" "/-%\"uG6\$%\"xG\"\"#*&F&\"\"\"-%#ln G6#*&\"\"%F)*\$F&F'F)F)" }}{PARA 0 "" 0 "" {TEXT -1 23 "The actual solu tion is " }}{PARA 0 "" 0 "" {TEXT -1 25 " " } {XPPEDIT 18 0 "u(x,y)=x*y*ln(x*y)" "/-%\"uG6\$%\"xG%\"yG*(F&\"\"\"F'F)- %#lnG6#*&F&F)F'F)F)" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "h=k" "/%\"hG%\"kG" } {TEXT -1 1 "=" }{XPPEDIT 18 0 "1/3" "*&\"\"\"F#\"\"\$!\"\"" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 12 "n:=3: m:=3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "u:=array(0..n,0..m);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG-%&arrayG6%;\"\"!\"\"\$F(7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "x:=array(0..n): y:=array(0..m):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "a:=1: b:=2: c:=1: d:=2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "h:=evalf((b-a)/n); k:=evalf((d-c)/m);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG\$\"+LLLLL!#5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"kG\$\"+LLLLL!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "for i from 0 to n do \n x[i]:=a+i*h\nod:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "for i from 0 to m do\n y[i ]:=c+i*k\nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "fa:=y->y*l n(y):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "fb:=y->2*y*ln(2*y) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "fc:=x->x*ln(x):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "fd:=x->x*ln(4*x^2):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "for i from 0 to m do \n u[ 0,i]:=evalf(fa(y[i]));\n u[n,i]:=evalf(fb(y[i]));\nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "for i from 0 to n do\n u[i,0]:=ev alf(fc(x[i]));\n u[i,m]:=evalf(fd(x[i]));\nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f:=(x,y)->x/y+y/x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6\$%\"xG%\"yG6\"6\$%)operatorG%&arrowGF),&*&9\$\"\" \"9%!\"\"F0*&F1F0F/F2F0F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "A:=matrix(4,4,[[-4,1,1,0],[1,-4,0,1],[1,0,-4,1],[0,1,1,-4]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7&7&!\"%\"\"\"F+\"\" !7&F+F*F,F+7&F+F,F*F+7&F,F+F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "d:=vector(4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG-%&arra yG6\$;\"\"\"\"\"%7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "d[1] :=h^2*f(x[1],y[2])-u[0,2]-u[1,3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> &%\"dG6#\"\"\"\$!+OH9RK!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "d[2]:=h^2*f(x[2],y[2])-u[3,2]-u[2,3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"dG6#\"\"#\$!+WJE/y!\"*" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 37 "d[3]:=h^2*f(x[1],y[1])-u[0,1]-u[1,0];" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%\"dG6#\"\"\$\$!+-(*H\\a!#5" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 37 "d[4]:=h^2*f(x[2],y[1])-u[3,1]-u[2,0];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"dG6#\"\"%\$!+OH9RK!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "z:=linsolve(A,d);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"zG-%'VECTORG6#7&\$\"+0/[v " 0 "" {MPLTEXT 1 0 55 "u[1,2]:=z[1]; \+ u[2,2]:=z[2]; u[1,1]:=z[3]; u[1,2]:=z[4];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6\$\"\"\"\"\"#\$\"+0/[v&%\"uG6\$\"\"#F'\$\"+*)f!)QG!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6\$\"\"\"F'\$\"+&psR-\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6\$\"\"\"\"\"#\$\"+0/[v " 0 "" {MPLTEXT 1 0 22 "g:=(x,y)->x*y*ln(x*y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG:6\$%\"xG%\"yG6\"6\$%)operatorG%&arrowGF)*(9\$\"\"\" 9%F/-%#lnG6#*&F.F/F0F/F/F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "g(x[1],y[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\$\"+Z:Yu " 0 "" {MPLTEXT 1 0 13 "g(x[2],y[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\$\"+N,#z\$G!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "g(x[1],y[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\$\"+ !fpG-\"!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "g(x[1],y[2]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\$\"+Z:Yu " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0" 0 }{VIEWOPTS 1 1 0 3 2 1804 }