{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{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 "" {TEXT 268 42 "Comparison between three \+ numerical methods" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "define function " }{TEXT 267 6 "f(x,y)" }{TEXT -1 1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f:=(x,y) ->(x+y-1)^2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "define ends of th e interval [" }{TEXT 264 1 "a" }{TEXT -1 1 "," }{TEXT 265 1 "b" } {TEXT -1 28 "] and the initial condition " }{TEXT 266 8 "y(a)=y_0" } {TEXT -1 2 " :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "a:=0; b:=0.5; y_0 :=2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "define the step size:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "h:=0.1;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "N - number of calculations ; reserve space to store calcu lated values of x and y :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "N:=fl oor((b-a)/h); \nx:=Array(0..N,datatype=float):\ny:=Array(0..N,datatype =float): " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "initi al values of " }{TEXT 262 1 "x" }{TEXT -1 5 " and " }{TEXT 263 1 "y" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "x[0]:=a; y[0]:=y_0;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Euler Method" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "for k from 1 to N do\n x[k]:=a+k* h:\n y[k]:=y[k-1]+h*f(x[k-1],y[k-1]):\n \+ end do:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "store the values of \+ previous calculations in a list " }{TEXT 261 1 "p" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 35 "p:=[seq([ x[n], y[n] ], n = 0..N)]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Improved Euler Method" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 167 "for k from 1 to N do\n x[k]:=a+k*h:\n \+ y_star:=y[k-1]+h*f(x[k-1],y[k-1]):\n y[k]:=y[k-1]+h*(f(x[k-1 ],y[k-1])+f(x[k],y_star))/2:\n end do:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "store the values of previous calculations in a \+ list " }{TEXT 260 1 "q" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "q:=[seq([ x[n], y[n] ], n = 0..N)]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Run ge-Kutta Method " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 277 "for k from 1 t o N do\n x[k]:=a+k*h:\n k_1:=h*f(x[k-1],y[k-1]):\n \+ k_2:=h*f(x[k-1]+h/2,y[k-1]+k_1/2):\n k_3:=h*f(x[k-1]+h /2,y[k-1]+k_2/2):\n k_4:=h*f(x[k-1]+h,y[k-1]+k_3):\n \+ y[k]:=y[k-1]+(k_1+2*k_2+2*k_3+k_4)/6:\n end do:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "store the values of previous calcu lations in a list " }{TEXT 259 1 "r" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "r:=[seq([ x[n], y[n] ], n = 0..N)]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "plot the results stored in " }{TEXT 256 1 "p" }{TEXT -1 2 ", " }{TEXT 257 1 "q" }{TEXT -1 5 " and " }{TEXT 258 1 "r" }{TEXT -1 55 " using red line, blue line and green dots, respectively" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "plot([p,q,r],axes=boxed,color=[red, blue,green], style=[line,line,point]);" }}}}{MARK "0 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }